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digama0/itexp.txt

Created April 12, 2019 10:07
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Metamath proof tutorial
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itexp.txt
*** The goal today is to prove the following theorem:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** Jon thinks it would be easier to do this without the "ph ->", but I think he may regret that decision later.
*** Anyway it is epsilon harder to have a context so let's just go with it.
*** Let's try ftc2 with the right substitution:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
!54:: |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
!55:: |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
!56:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
57:50,51,52,54,55,56:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** We prove the derivative is equal to something, which we use for 54 and 55:
!d1:: |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = &C1 )
!d2:: |- ( ph -> &C1 e. ( ( A (,) B ) -cn-> CC ) )
54:d1,d2:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
*** and apply dvmptres2 and substitute:
!d3:: |- ( ph -> RR e. { RR , CC } )
!d4:: |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
!d5:: |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. &C5 )
!d6:: |- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!d7:: |- ( ph -> ( A [,] B ) C_ RR )
!d8:: |- &C3 = ( &C4 |`t RR )
!d9:: |- &C4 = ( TopOpen ` CCfld )
!d10:: |- ( ph -> ( ( int ` &C3 ) ` ( A [,] B ) ) = ( A (,) B ) )
d1:d3,d4,d5,d6,d7,d8,d9,d10:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
*** let's clean up some side conditions:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
54::reelprrecn |- RR e. { RR , CC }
55:54:a1i |- ( ph -> RR e. { RR , CC } )
56::simpr |- ( ( ph /\ t e. RR ) -> t e. RR )
!57:: |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
58:56,57:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. RR )
59:58:recnd |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
!60:: |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. &C5 )
!61:: |- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
62::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
63:50,51,62:syl2anc |- ( ph -> ( A [,] B ) C_ RR )
!64:: |- &C3 = ( &C4 |`t RR )
!65:: |- &C4 = ( TopOpen ` CCfld )
!66:: |- ( ph -> ( ( int ` &C3 ) ` ( A [,] B ) ) = ( A (,) B ) )
67:55,59,60,61,63,64,65,66:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!68:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
69:67,68:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
!70:: |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
!71:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
72:50,51,52,69,70,71:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** Oops, I forgot the name of this one:
! |- ( ( ph /\ t e. RR ) -> N e. NN0 )
!57:: |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
*** thanks mmj2!
!d1:: |- ( ( ph /\ t e. RR ) -> N e. NN0 )
d2::peano2nn0 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
57:d1,d2:syl |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
*** Okay, cleaned up:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
54::reelprrecn |- RR e. { RR , CC }
55:54:a1i |- ( ph -> RR e. { RR , CC } )
56::simpr |- ( ( ph /\ t e. RR ) -> t e. RR )
57:53:nnnn0d |- ( ph -> N e. NN0 )
58:57:adantr |- ( ( ph /\ t e. RR ) -> N e. NN0 )
59::peano2nn0 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
60:58,59:syl |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
61:56,60:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. RR )
62:61:recnd |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
63:60:nn0red |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. RR )
64:56,58:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ N ) e. RR )
65:63,64:remulcld |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. RR )
!66:: |- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
67::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
68:50,51,67:syl2anc |- ( ph -> ( A [,] B ) C_ RR )
69::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
70:69:tgioo2 |- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
!71:: |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
72:55,62,65,66,68,70,69,71:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!73:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
74:72,73:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
!75:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
76:72,75:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
!77:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
78:50,51,52,74,76,77:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** Now line 66 is provable without too much trouble.
*** let's ask mmj2 for dvexp:
!dvexp
d1::dvexp |- ( &C1 e. NN -> ( CC _D ( &S1 e. CC |-> ( &S1 ^ &C1 ) ) ) = ( &S1 e. CC |-> ( &C1 x. ( &S1 ^ ( &C1 - 1 ) ) ) ) )
*** so we need to further relax the domain to CC, that's dvmptres3:
!d2:: |- &C2 = ( TopOpen ` CCfld )
!d3:: |- ( ph -> CC e. &C2 )
!d4:: |- ( ph -> ( RR i^i CC ) = RR )
!d5:: |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
!d6:: |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. &C3 )
!d7:: |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
66:d2,55,d3,d4,d5,d6,d7:dvmptres3 |- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
*** Here d3 says CC is open, toponmax is useful for that, and d4 is just set theory:
!d2::eqid |- &C2 = ( TopOpen ` CCfld )
!d8:: |- &C2 e. ( TopOn ` CC )
d9::toponmax |- ( &C2 e. ( TopOn ` CC ) -> CC e. &C2 )
d3:d8,d9:mp1i |- ( ph -> CC e. &C2 )
*** thanks mmj2:
d2::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
d8:d2:cnfldtopon |- ( TopOpen ` CCfld ) e. ( TopOn ` CC )
d9::toponmax |- ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) -> CC e. ( TopOpen ` CCfld ) )
d3:d8,d9:mp1i |- ( ph -> CC e. ( TopOpen ` CCfld ) )
*** for the other, apply sylib:
!d10:: |- ( ph -> &W1 )
!d11:: |- ( &W1 <-> ( RR i^i CC ) = RR )
d4:d10,d11:sylib |- ( ph -> ( RR i^i CC ) = RR )
*** and double click on d11, because I know there is a theorem about this one but I forget the name:
!d10:: |- ( ph -> RR C_ CC )
d11::df-ss |- ( RR C_ CC <-> ( RR i^i CC ) = RR )
d4:d10,d11:sylib |- ( ph -> ( RR i^i CC ) = RR )
*** Turns out it's the definition. Moving on...
d12::ax-resscn |- RR C_ CC
d10:d12:a1i |- ( ph -> RR C_ CC )
*** Now we have these closure lemmas; unfortunately the previous closure lemmas don't apply because we are on CC now:
!d5:: |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
!d6:: |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. &C3 )
*** (I typed expcld and then adantr to get the following:)
d13::simpr |- ( ( ph /\ t e. CC ) -> t e. CC )
d15:57,59:syl |- ( ph -> ( N + 1 ) e. NN0 )
d14:d15:adantr |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. NN0 )
d5:d13,d14:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
*** (and expcld and adantr again:)
d18:57:adantr |- ( ( ph /\ t e. CC ) -> N e. NN0 )
d17:d13,d18:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ N ) e. CC )
d6:d16,d17:mulcld |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
*** Finally, we have the real goal:
!d7:: |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
*** which unfortunately doesn't quite match dvexp because it uses -1 instead of +1. So we will use transitivity:
d21:53:peano2nnd |- ( ph -> ( N + 1 ) e. NN )
d19:d21,d1:syl |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
!d20:: |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
d7:d19,d20:eqtrd |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
*** use mpteq2dva, oveq2 twice then pncan, and we're done.
d26:53:nncnd |- ( ph -> N e. CC )
d27::1cnd |- ( ph -> 1 e. CC )
d25:d26,d27:pncand |- ( ph -> ( ( N + 1 ) - 1 ) = N )
d24:d25:adantr |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) - 1 ) = N )
d23:d24:oveq2d |- ( ( ph /\ t e. CC ) -> ( t ^ ( ( N + 1 ) - 1 ) ) = ( t ^ N ) )
d22:d23:oveq2d |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) = ( ( N + 1 ) x. ( t ^ N ) ) )
d20:d22:mpteq2dva |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
*** Let's clean up and take stock, which means use Ctrl-Shift-U to unify/renumber and Ctrl-R to reformat:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
54::reelprrecn |- RR e. { RR , CC }
55:54:a1i |- ( ph -> RR e. { RR , CC } )
56::simpr |- ( ( ph /\ t e. RR ) -> t e. RR )
57:53:nnnn0d |- ( ph -> N e. NN0 )
58:57:adantr |- ( ( ph /\ t e. RR ) -> N e. NN0 )
59::peano2nn0 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
60:58,59:syl |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
61:56,60:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. RR )
62:61:recnd |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
63:60:nn0red |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. RR )
64:56,58:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ N ) e. RR )
65:63,64:remulcld |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. RR )
66::dvexp |- ( ( N + 1 ) e. NN -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
67::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
68:67:cnfldtopon |- ( TopOpen ` CCfld ) e. ( TopOn ` CC )
69::toponmax |- ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) -> CC e. ( TopOpen ` CCfld ) )
70:68,69:mp1i |- ( ph -> CC e. ( TopOpen ` CCfld ) )
71::ax-resscn |- RR C_ CC
72:71:a1i |- ( ph -> RR C_ CC )
73::df-ss |- ( RR C_ CC <-> ( RR i^i CC ) = RR )
74:72,73:sylib |- ( ph -> ( RR i^i CC ) = RR )
75::simpr |- ( ( ph /\ t e. CC ) -> t e. CC )
76:57,59:syl |- ( ph -> ( N + 1 ) e. NN0 )
77:76:adantr |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. NN0 )
78:75,77:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
79:77:nn0cnd |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. CC )
80:57:adantr |- ( ( ph /\ t e. CC ) -> N e. NN0 )
81:75,80:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ N ) e. CC )
82:79,81:mulcld |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
83:53:peano2nnd |- ( ph -> ( N + 1 ) e. NN )
84:83,66:syl |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
85:53:nncnd |- ( ph -> N e. CC )
86::1cnd |- ( ph -> 1 e. CC )
87:85,86:pncand |- ( ph -> ( ( N + 1 ) - 1 ) = N )
88:87:adantr |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) - 1 ) = N )
89:88:oveq2d |- ( ( ph /\ t e. CC ) -> ( t ^ ( ( N + 1 ) - 1 ) ) = ( t ^ N ) )
90:89:oveq2d |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) = ( ( N + 1 ) x. ( t ^ N ) ) )
91:90:mpteq2dva |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
92:84,91:eqtrd |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
93:67,55,70,74,78,82,92:dvmptres3
|- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
94::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95:50,51,94:syl2anc |- ( ph -> ( A [,] B ) C_ RR )
96::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
97:96:tgioo2 |- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
!98:: |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
99:55,62,65,93,95,97,96,98:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!100:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
101:99,100:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
!102:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
103:99,102:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
!104:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
105:50,51,52,101,103,104:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** The next goal is:
!98:: |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
*** I know there is some theorem about this, I'll use syl2anc because I know it takes the assumptions A e. RR and B e. RR
!d1:: |- ( ph -> &W1 )
!d2:: |- ( ph -> &W2 )
!d3:: |- ( ( &W1 /\ &W2 ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
98:d1,d2,d3:syl2anc |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
*** and double click so mmj2 finds it for me:
d3::iccntr |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
98:50,51,d3:syl2anc |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
*** Okay, so we are getting to the hard goals:
!100:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
!102:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
!104:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
*** well the first one isn't that hard. That's a thing about restrictions of continuous functions:
rescncf
d4::rescncf |- ( &C3 C_ &C1 -> ( &C4 e. ( &C1 -cn-> &C2 ) -> ( &C4 |` &C3 ) e. ( &C3 -cn-> &C2 ) ) )
*** Oh, it uses explicit restrictions so we need to use the theorem on restrictions of a mapping.
d4::rescncf |- ( ( A (,) B ) C_ &C1 ->
( ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( &C1 -cn-> CC ) -> ( ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
)
!d9:: |- ( ph -> ( A (,) B ) C_ &C5 )
d10::resmpt |- ( ( A (,) B ) C_ &C5 -> ( ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
d5:d9,d10:syl |- ( ph -> ( ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!d7:: |- ( ph -> ( A (,) B ) C_ &C1 )
!d8:: |- ( ph -> ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( &C1 -cn-> CC ) )
d6:d7,d8,d4:sylc |- ( ph -> ( ( t e. &C5 |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
100:d5,d6:eqeltrrd |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
*** A bit of duplication here, I want &C5 and &C1 to be CC:
!d9:: |- ( ph -> ( A (,) B ) C_ CC )
d10::resmpt |- ( ( A (,) B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
d5:d9,d10:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
!d8:: |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) )
d6:d9,d8,d4:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
*** We've already proved most of this subsetting so it's easy:
d12::ioossicc |- ( A (,) B ) C_ ( A [,] B )
d11:d12,95:syl5ss |- ( ph -> ( A (,) B ) C_ RR )
d9:d11,71:syl6ss |- ( ph -> ( A (,) B ) C_ CC )
*** To prove d8, we use the mapping functions for proving continuity:
!d13:: |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
!d14::a1i |- ( ph -> x. e. ( ( &C1 tX &C1 ) Cn &C1 ) )
!d15:: |- ( ph -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
!d16:: |- ( ph -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
d8:d13,d14,d15,d16:cncfmpt2f |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) )
d17:67:mulcn |- x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
d14:d17:a1i |- ( ph -> x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) )
!d15:: |- ( ph -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
!d16:: |- ( ph -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
d8:67,d14,d15,d16:cncfmpt2f |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) )
d18:85,86:addcld |- ( ph -> ( N + 1 ) e. CC )
!d19:: |- ( ph -> CC C_ CC )
d21::cncfmptc |- ( ( ( N + 1 ) e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
d15:d18,d19,d19,d21:syl3anc |- ( ph -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
d22::ssid |- CC C_ CC
d19:d22:a1i |- ( ph -> CC C_ CC )
d23::expcncf |- ( N e. NN0 -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
d16:57,d23:syl |- ( ph -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
*** Okay, that finishes continuity on ( A (,) B ). Luckily we can overlap the work for continuity on ( A [,] B ):
!d29:: |- ( ph -> ( A [,] B ) C_ &C4 )
d30::resmpt |- ( ( A [,] B ) C_ &C4 -> ( ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
d24:d29,d30:syl |- ( ph -> ( ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
!d26:: |- ( ph -> ( A [,] B ) C_ &C2 )
!d27:: |- ( ph -> ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) e. ( &C2 -cn-> CC ) )
d28::rescncf |- ( ( A [,] B ) C_ &C2 ->
( ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) e. ( &C2 -cn-> CC ) -> ( ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
d25:d26,d27,d28:sylc |- ( ph -> ( ( t e. &C4 |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
104:d24,d25:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
d29:95,72:sstrd |- ( ph -> ( A [,] B ) C_ CC )
d30::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
d24:d29,d30:syl |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
!d27:: |- ( ph -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
d28::rescncf |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
d25:d29,d27,d28:sylc |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
104:d24,d25:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
d32::expcncf |- ( ( N + 1 ) e. NN0 -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
d27:76,d32:syl |- ( ph -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
*** Yay, almost done. We have left the best for last:
!102:: |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** and also we haven't started on connecting this whole proof to the qed step.
105:50,51,52,101,103,104:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** I looked around the library and found iblss, which helps:
d33:d12:a1i |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
d37::ioombl |- ( A (,) B ) e. dom vol
d34:d37:a1i |- ( ph -> ( A (,) B ) e. dom vol )
!d35:: |- ( ( ph /\ t e. ( A [,] B ) ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. &C2 )
!d36:: |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
102:d33,d34,d35,d36:iblss |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** We can do d35 by reference to a previous proof of it in a different context:
d38:d29:sselda |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. CC )
d35:d38,82:syldan |- ( ( ph /\ t e. ( A [,] B ) ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
*** (I had to move d38 up from below to avoid duplicate steps, although it's not a big deal as it would
*** get deduplicated anyway in the final proof.)
*** For d36 we use cniccibl:
!d39:: |- ( ph -> &C1 e. RR )
!d40:: |- ( ph -> &C2 e. RR )
!d41:: |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( &C1 [,] &C2 ) -cn-> CC ) )
d42::cniccibl |- ( ( &C1 e. RR /\ &C2 e. RR /\ ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( &C1 [,] &C2 ) -cn-> CC ) ) ->
( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
d36:d39,d40,d41,d42:syl3anc |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
!d41:: |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
d42::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) ) ->
( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
d36:50,51,d41,d42:syl3anc |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** I guess we still haven't proven this variation yet. Once again, let's take it to CC so we can use the previous result.
d46::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
d43:d29,d46:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
d49::rescncf |- ( ( A [,] B ) C_ CC ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
)
d44:d29,d8,d49:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
d41:d43,d44:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
*** Wow, we're done! Let's clean up and renumber again:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
54::reelprrecn |- RR e. { RR , CC }
55:54:a1i |- ( ph -> RR e. { RR , CC } )
56::simpr |- ( ( ph /\ t e. RR ) -> t e. RR )
57:53:nnnn0d |- ( ph -> N e. NN0 )
58:57:adantr |- ( ( ph /\ t e. RR ) -> N e. NN0 )
59::peano2nn0 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
60:58,59:syl |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
61:56,60:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. RR )
62:61:recnd |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
63:60:nn0red |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. RR )
64:56,58:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ N ) e. RR )
65:63,64:remulcld |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. RR )
66::dvexp |- ( ( N + 1 ) e. NN -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
67::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
68:67:cnfldtopon |- ( TopOpen ` CCfld ) e. ( TopOn ` CC )
69::toponmax |- ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) -> CC e. ( TopOpen ` CCfld ) )
70:68,69:mp1i |- ( ph -> CC e. ( TopOpen ` CCfld ) )
71::ax-resscn |- RR C_ CC
72:71:a1i |- ( ph -> RR C_ CC )
73::df-ss |- ( RR C_ CC <-> ( RR i^i CC ) = RR )
74:72,73:sylib |- ( ph -> ( RR i^i CC ) = RR )
75::simpr |- ( ( ph /\ t e. CC ) -> t e. CC )
76:57,59:syl |- ( ph -> ( N + 1 ) e. NN0 )
77:76:adantr |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. NN0 )
78:75,77:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
79:77:nn0cnd |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. CC )
80:57:adantr |- ( ( ph /\ t e. CC ) -> N e. NN0 )
81:75,80:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ N ) e. CC )
82:79,81:mulcld |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
83:53:peano2nnd |- ( ph -> ( N + 1 ) e. NN )
84:83,66:syl |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
85:53:nncnd |- ( ph -> N e. CC )
86::1cnd |- ( ph -> 1 e. CC )
87:85,86:pncand |- ( ph -> ( ( N + 1 ) - 1 ) = N )
88:87:adantr |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) - 1 ) = N )
89:88:oveq2d |- ( ( ph /\ t e. CC ) -> ( t ^ ( ( N + 1 ) - 1 ) ) = ( t ^ N ) )
90:89:oveq2d |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) = ( ( N + 1 ) x. ( t ^ N ) ) )
91:90:mpteq2dva |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
92:84,91:eqtrd |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
93:67,55,70,74,78,82,92:dvmptres3
|- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
94::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95:50,51,94:syl2anc |- ( ph -> ( A [,] B ) C_ RR )
96::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
97:96:tgioo2 |- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
98::iccntr |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
99:50,51,98:syl2anc |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
100:55,62,65,93,95,97,96,99:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
101::rescncf |- ( ( A (,) B ) C_ CC ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) )
102::ioossicc |- ( A (,) B ) C_ ( A [,] B )
103:102,95:syl5ss |- ( ph -> ( A (,) B ) C_ RR )
104:103,71:syl6ss |- ( ph -> ( A (,) B ) C_ CC )
105::resmpt |- ( ( A (,) B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
106:104,105:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
107:67:mulcn |- x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
108:107:a1i |- ( ph -> x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) )
109:85,86:addcld |- ( ph -> ( N + 1 ) e. CC )
110::ssid |- CC C_ CC
111:110:a1i |- ( ph -> CC C_ CC )
112::cncfmptc |- ( ( ( N + 1 ) e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
113:109,111,111,112:syl3anc
|- ( ph -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
114::expcncf |- ( N e. NN0 -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
115:57,114:syl |- ( ph -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
116:67,108,113,115:cncfmpt2f
|- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) )
117:104,116,101:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
118:106,117:eqeltrrd |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
119:100,118:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
120:102:a1i |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
121::ioombl |- ( A (,) B ) e. dom vol
122:121:a1i |- ( ph -> ( A (,) B ) e. dom vol )
123:95,72:sstrd |- ( ph -> ( A [,] B ) C_ CC )
124:123:sselda |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. CC )
125:124,82:syldan |- ( ( ph /\ t e. ( A [,] B ) ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
126::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
127:123,126:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
128::rescncf |- ( ( A [,] B ) C_ CC ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
)
129:123,116,128:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
130:127,129:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
131::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) ) ->
( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
132:50,51,130,131:syl3anc
|- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
133:120,122,125,132:iblss
|- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
134:100,133:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
135::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
136:123,135:syl |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
137::expcncf |- ( ( N + 1 ) e. NN0 -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
138:76,137:syl |- ( ph -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
139::rescncf |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
140:123,138,139:sylc |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
141:136,140:eqeltrrd
|- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
142:50,51,52,119,134,141:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** One step left, which is to connect this all to the goal.
!d1:: |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A [,] B ) ( x ^ N ) _d x )
!d2:: |- ( ph ->
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) =
( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
qed:142,d1,d2:3eqtr3d |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** Oh, that first one is inconvenient, it uses the wrong interval. That's a measure zero difference,
*** surely there's a theorem about that. (Later:) Aha, it's itgioo.
!d3:: |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
d6:123:sselda |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
d7:57:adantr |- ( ( ph /\ x e. ( A [,] B ) ) -> N e. NN0 )
d5:d6,d7:expcld |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x ^ N ) e. CC )
d4:50,51,d5:itgioo |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x = S. ( A [,] B ) ( x ^ N ) _d x )
*** Thank goodness for mmj2 coming up with that closure proof so I don't have to.
*** (I had to give expcld and it figured out the rest)
*** Now back to equality lemmas:
!d3::itgeq2dva |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
*** Oops, invalid ref. What's it called again? (Later:) It's just itgeq2, no one bothered to make the deduction version.
!d10:: |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) )
d8:d10:ralrimiva |- ( ph -> A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) )
d9::itgeq2 |- ( A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
d3:d8,d9:syl |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
*** Okay, let's be clever here, and prove that the left bit is the derivative mapping and the right bit is the substitution:
!d13:: |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = &C2 )
d11:d13:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( &C2 ` x ) )
!d12:: |- ( x e. ( A (,) B ) -> ( &C2 ` x ) = ( x ^ N ) )
d10:d11,d12:sylan9eq |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) )
*** I'm sure d13 is above somewhere: (Later: it's 100)
!d13::#100 |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = &C2 )
d11:100:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) )
!d12:: |- ( x e. ( A (,) B ) -> ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) = ( x ^ N ) )
d10:d11,d12:sylan9eq |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) )
!d13:: |- ( t = x -> ( ( N + 1 ) x. ( t ^ N ) ) = ( x ^ N ) )
d14::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) )
d15::ovex |- ( x ^ N ) e. _V
d12:d13,d14,d15:fvmpt |- ( x e. ( A (,) B ) -> ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) = ( x ^ N ) )
*** Oh no! I wasn't paying enough attention to the goal, and I've got some nonsense in the proof.
*** The integrands aren't actually equal, they differ by a constant. Okay, let's delete all the
*** garbage steps and double check our place, starting at the ftc proof:
142:50,51,52,119,134,141:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
143:100:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) )
144::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) )
145::ovex |- ( x ^ N ) e. _V
146::itgeq2 |- ( A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
147:123:sselda |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
148:57:adantr |- ( ( ph /\ x e. ( A [,] B ) ) -> N e. NN0 )
149:147,148:expcld |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x ^ N ) e. CC )
150:50,51,149:itgioo
|- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x = S. ( A [,] B ) ( x ^ N ) _d x )
qed:: |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** What we should actually be doing here is move the ( N + 1 ) to the other side
!d2:: |- ( ph -> ( &C2 / ( N + 1 ) ) = S. ( A (,) B ) ( x ^ N ) _d x )
!d4:: |- ( ph -> &C2 = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
d3:d4:oveq1d |- ( ph -> ( &C2 / ( N + 1 ) ) = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
d1:d2,d3:eqtr3d |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
qed:150,d1:eqtr3d |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
*** We will prove d4 by transitivity to the ftc.
!d5:: |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x e. CC )
d6:83:nnne0d |- ( ph -> ( N + 1 ) =/= 0 )
d2:d5,109,d6:divcan3d
|- ( ph -> ( ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) / ( N + 1 ) ) = S. ( A (,) B ) ( x ^ N ) _d x )
!d7:: |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x ^ N ) e. &C1 )
!d8:: |- ( ph -> ( x e. ( A (,) B ) |-> ( x ^ N ) ) e. L^1 )
d5:d7,d8:itgcl |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x e. CC )
*** Just in case you haven't seen enough of integrability proofs.
*** I don't like using x as a bound variable though, all our earlier results used t, so let's back up and change it:
d9::oveq1 |- ( x = t -> ( x ^ N ) = ( t ^ N ) )
d7:d9:cbvitgv |- S. ( A (,) B ) ( x ^ N ) _d x = S. ( A (,) B ) ( t ^ N ) _d t
!d8:: |- ( ph -> S. ( A (,) B ) ( t ^ N ) _d t e. CC )
d5:d7,d8:syl5eqel |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x e. CC )
*** Look at all those previous results that match. So nice.
d13:124,81:syldan |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t ^ N ) e. CC )
d15:102:sseli |- ( t e. ( A (,) B ) -> t e. ( A [,] B ) )
d10:d15,d13:sylan2 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( t ^ N ) e. CC )
!d14:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. L^1 )
d11:120,122,d13,d14:iblss |- ( ph -> ( t e. ( A (,) B ) |-> ( t ^ N ) ) e. L^1 )
d8:d10,d11:itgcl |- ( ph -> S. ( A (,) B ) ( t ^ N ) _d t e. CC )
!d18:: |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. ( ( A [,] B ) -cn-> CC ) )
d19::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. L^1 )
d14:50,51,d18,d19:syl3anc |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. L^1 )
*** You know the drill.
d23::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ N ) ) )
d20:123,d23:syl |- ( ph -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ N ) ) )
d26::rescncf |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
d21:123,115,d26:sylc |- ( ph -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
d18:d20,d21:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. ( ( A [,] B ) -cn-> CC ) )
*** In hindsight, I might not have bothered to prove that
*** |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** if I had known we would do it for this function too. Let's move that proof (step 133) right here:
d11:120,122,d13,d14:iblss |- ( ph -> ( t e. ( A (,) B ) |-> ( t ^ N ) ) e. L^1 )
133:120,122,125,132:iblss
|- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** and now mmj2 reorders the steps, moving ftc down to restore the dag order:
d11:120,122,d13,d14:iblss |- ( ph -> ( t e. ( A (,) B ) |-> ( t ^ N ) ) e. L^1 )
133:120,122,125,132:iblss
|- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
134:100,133:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
142:50,51,52,119,134,141:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
*** Instead of iblss, let's use iblmulc2:
!d27:: |- ( ( ph /\ t e. ( A (,) B ) ) -> ( t ^ N ) e. &C1 )
133:109,d27,d11:iblmulc2
|- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** Nice! No side conditions except the easy one...
133:109,d10,d11:iblmulc2
|- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
*** and we've already done it.
*** One last step, and then we're done. Luckily it's an easy one.
142:50,51,52,119,134,141:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
!d4:: |- ( ph -> ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
!d30:: |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x ^ N ) e. CC )
!d31:: |- ( ph -> ( x e. ( A (,) B ) |-> ( x ^ N ) ) e. L^1 )
d27:109,d30,d31:itgmulc2
|- ( ph -> ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
!d28:: |- ( ph -> S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x = &C2 )
!d29:: |- ( ph -> &C2 = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
d4:d27,d28,d29:3eqtrd |- ( ph -> ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
*** Eh, we're back to x again. I won't rename it this time since it's going to interact with t in ftc2.
*** Still, we proved d30 already, on a different domain:
d32:102:sseli |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
d30:d32,149:sylan2 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x ^ N ) e. CC )
*** Definitely not proving integrability again
d33:d9:cbvmptv |- ( x e. ( A (,) B ) |-> ( x ^ N ) ) = ( t e. ( A (,) B ) |-> ( t ^ N ) )
d31:d33,d11:syl5eqel |- ( ph -> ( x e. ( A (,) B ) |-> ( x ^ N ) ) e. L^1 )
*** Okay, for these we want to use ftc2
!d28:: |- ( ph -> S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x = &C2 )
!d29:: |- ( ph -> &C2 = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
*** so we need another transitivity
!d28:,142:eqtr3d |- ( ph -> S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x = &C2 )
!d29:: |- ( ph -> &C2 = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
!d34:: |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
d28:d34,142:eqtr3d |- ( ph ->
S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x = ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
!d29:: |- ( ph -> ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
*** As we saw, there is no itgeq2dva, so we make do
!d37:: |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
d35:d37:ralrimiva |- ( ph -> A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
d36::itgeq2 |- ( A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
d34:d35,d36:syl |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
*** and I'm bound and determined to use my trick
!d40::#100 |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = &C2 )
d38:d40:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( &C2 ` x ) )
!d39:: |- ( x e. ( A (,) B ) -> ( &C2 ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
d37:d38,d39:sylan9eq |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
!d40:: |- ( &S1 = x -> &C1 = ( ( N + 1 ) x. ( x ^ N ) ) )
!d41::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( &S1 e. ( A (,) B ) |-> &C1 )
d42::ovex |- ( ( N + 1 ) x. ( x ^ N ) ) e. _V
d39:d40,d41,d42:fvmpt |- ( x e. ( A (,) B ) -> ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
*** Hey look it works this time
d43::oveq1 |- ( t = x -> ( t ^ N ) = ( x ^ N ) )
d40:d43:oveq2d |- ( t = x -> ( ( N + 1 ) x. ( t ^ N ) ) = ( ( N + 1 ) x. ( x ^ N ) ) )
d41::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) )
d42::ovex |- ( ( N + 1 ) x. ( x ^ N ) ) e. _V
d39:d40,d41,d42:fvmpt |- ( x e. ( A (,) B ) -> ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
*** One last proof:
!d29:: |- ( ph -> ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
!d46:: |- ( ph -> B e. ( A [,] B ) )
d48::oveq1 |- ( t = B -> ( t ^ ( N + 1 ) ) = ( B ^ ( N + 1 ) ) )
d49::eqid |- ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) )
d50::ovex |- ( B ^ ( N + 1 ) ) e. _V
d47:d48,d49,d50:fvmpt |- ( B e. ( A [,] B ) -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) = ( B ^ ( N + 1 ) ) )
d44:d46,d47:syl |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) = ( B ^ ( N + 1 ) ) )
!d45:: |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) = ( A ^ ( N + 1 ) ) )
d29:d44,d45:oveq12d |- ( ph -> ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
*** d46 has a name, let's use syl3anc and double click to look it up (scroll to the bottom past the boring lemmas):
*** It's ubicc2 and the rest gets filled in.
d51:50:rexrd |- ( ph -> A e. RR* )
d52:51:rexrd |- ( ph -> B e. RR* )
d54::ubicc2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
d46:d51,d52,52,d54:syl3anc |- ( ph -> B e. ( A [,] B ) )
*** Do it again for the other goal:
d63::lbicc2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
d55:d51,d52,52,d63:syl3anc |- ( ph -> A e. ( A [,] B ) )
d57::oveq1 |- ( t = A -> ( t ^ ( N + 1 ) ) = ( A ^ ( N + 1 ) ) )
d58::eqid |- ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) )
d59::ovex |- ( A ^ ( N + 1 ) ) e. _V
d56:d57,d58,d59:fvmpt |- ( A e. ( A [,] B ) -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) = ( A ^ ( N + 1 ) ) )
d45:d55,d56:syl |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) = ( A ^ ( N + 1 ) ) )
*** And we're done! Cleanup and renumber again, and here's the final result:
h50::itexp.1 |- ( ph -> A e. RR )
h51::itexp.2 |- ( ph -> B e. RR )
h52::itexp.3 |- ( ph -> A <_ B )
h53::itexp.4 |- ( ph -> N e. NN )
54::reelprrecn |- RR e. { RR , CC }
55:54:a1i |- ( ph -> RR e. { RR , CC } )
56::simpr |- ( ( ph /\ t e. RR ) -> t e. RR )
57:53:nnnn0d |- ( ph -> N e. NN0 )
58:57:adantr |- ( ( ph /\ t e. RR ) -> N e. NN0 )
59::peano2nn0 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
60:58,59:syl |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. NN0 )
61:56,60:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. RR )
62:61:recnd |- ( ( ph /\ t e. RR ) -> ( t ^ ( N + 1 ) ) e. CC )
63:60:nn0red |- ( ( ph /\ t e. RR ) -> ( N + 1 ) e. RR )
64:56,58:reexpcld |- ( ( ph /\ t e. RR ) -> ( t ^ N ) e. RR )
65:63,64:remulcld |- ( ( ph /\ t e. RR ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. RR )
66::dvexp |- ( ( N + 1 ) e. NN -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
67::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
68:67:cnfldtopon |- ( TopOpen ` CCfld ) e. ( TopOn ` CC )
69::toponmax |- ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) -> CC e. ( TopOpen ` CCfld ) )
70:68,69:mp1i |- ( ph -> CC e. ( TopOpen ` CCfld ) )
71::ax-resscn |- RR C_ CC
72:71:a1i |- ( ph -> RR C_ CC )
73::df-ss |- ( RR C_ CC <-> ( RR i^i CC ) = RR )
74:72,73:sylib |- ( ph -> ( RR i^i CC ) = RR )
75::simpr |- ( ( ph /\ t e. CC ) -> t e. CC )
76:57,59:syl |- ( ph -> ( N + 1 ) e. NN0 )
77:76:adantr |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. NN0 )
78:75,77:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ ( N + 1 ) ) e. CC )
79:77:nn0cnd |- ( ( ph /\ t e. CC ) -> ( N + 1 ) e. CC )
80:57:adantr |- ( ( ph /\ t e. CC ) -> N e. NN0 )
81:75,80:expcld |- ( ( ph /\ t e. CC ) -> ( t ^ N ) e. CC )
82:79,81:mulcld |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
83:53:peano2nnd |- ( ph -> ( N + 1 ) e. NN )
84:83,66:syl |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) )
85:53:nncnd |- ( ph -> N e. CC )
86::1cnd |- ( ph -> 1 e. CC )
87:85,86:pncand |- ( ph -> ( ( N + 1 ) - 1 ) = N )
88:87:adantr |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) - 1 ) = N )
89:88:oveq2d |- ( ( ph /\ t e. CC ) -> ( t ^ ( ( N + 1 ) - 1 ) ) = ( t ^ N ) )
90:89:oveq2d |- ( ( ph /\ t e. CC ) -> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) = ( ( N + 1 ) x. ( t ^ N ) ) )
91:90:mpteq2dva |- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ ( ( N + 1 ) - 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
92:84,91:eqtrd |- ( ph -> ( CC _D ( t e. CC |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
93:67,55,70,74,78,82,92:dvmptres3
|- ( ph -> ( RR _D ( t e. RR |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. RR |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
94::iccssre |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
95:50,51,94:syl2anc |- ( ph -> ( A [,] B ) C_ RR )
96::eqid |- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
97:96:tgioo2 |- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR )
98::iccntr |- ( ( A e. RR /\ B e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
99:50,51,98:syl2anc |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( A [,] B ) ) = ( A (,) B ) )
100:55,62,65,93,95,97,96,99:dvmptres2
|- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
101::rescncf |- ( ( A (,) B ) C_ CC ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) ) )
102::ioossicc |- ( A (,) B ) C_ ( A [,] B )
103:102,95:syl5ss |- ( ph -> ( A (,) B ) C_ RR )
104:103,71:syl6ss |- ( ph -> ( A (,) B ) C_ CC )
105::resmpt |- ( ( A (,) B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
106:104,105:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
107:67:mulcn |- x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) )
108:107:a1i |- ( ph -> x. e. ( ( ( TopOpen ` CCfld ) tX ( TopOpen ` CCfld ) ) Cn ( TopOpen ` CCfld ) ) )
109:85,86:addcld |- ( ph -> ( N + 1 ) e. CC )
110::ssid |- CC C_ CC
111:110:a1i |- ( ph -> CC C_ CC )
112::cncfmptc |- ( ( ( N + 1 ) e. CC /\ CC C_ CC /\ CC C_ CC ) -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
113:109,111,111,112:syl3anc
|- ( ph -> ( t e. CC |-> ( N + 1 ) ) e. ( CC -cn-> CC ) )
114::expcncf |- ( N e. NN0 -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
115:57,114:syl |- ( ph -> ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) )
116:67,108,113,115:cncfmpt2f
|- ( ph -> ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) )
117:104,116,101:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A (,) B ) ) e. ( ( A (,) B ) -cn-> CC ) )
118:106,117:eqeltrrd |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
119:100,118:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. ( ( A (,) B ) -cn-> CC ) )
120:102:a1i |- ( ph -> ( A (,) B ) C_ ( A [,] B ) )
121::ioombl |- ( A (,) B ) e. dom vol
122:121:a1i |- ( ph -> ( A (,) B ) e. dom vol )
123:95,72:sstrd |- ( ph -> ( A [,] B ) C_ CC )
124:123:sselda |- ( ( ph /\ t e. ( A [,] B ) ) -> t e. CC )
125:124,82:syldan |- ( ( ph /\ t e. ( A [,] B ) ) -> ( ( N + 1 ) x. ( t ^ N ) ) e. CC )
126::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
127:123,126:syl |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) )
128::rescncf |- ( ( A [,] B ) C_ CC ->
( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
129:123,116,128:sylc |- ( ph -> ( ( t e. CC |-> ( ( N + 1 ) x. ( t ^ N ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
130:127,129:eqeltrrd
|- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
131::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. ( ( A [,] B ) -cn-> CC ) ) ->
( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
132:50,51,130,131:syl3anc
|- ( ph -> ( t e. ( A [,] B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
133::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
134:123,133:syl |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) )
135::expcncf |- ( ( N + 1 ) e. NN0 -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
136:76,135:syl |- ( ph -> ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) )
137::rescncf |- ( ( A [,] B ) C_ CC ->
( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
138:123,136,137:sylc |- ( ph -> ( ( t e. CC |-> ( t ^ ( N + 1 ) ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
139:134,138:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) e. ( ( A [,] B ) -cn-> CC ) )
140:100:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) )
141::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) )
142::ovex |- ( x ^ N ) e. _V
143::itgeq2 |- ( A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( x ^ N ) ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( x ^ N ) _d x )
144:123:sselda |- ( ( ph /\ x e. ( A [,] B ) ) -> x e. CC )
145:57:adantr |- ( ( ph /\ x e. ( A [,] B ) ) -> N e. NN0 )
146:144,145:expcld |- ( ( ph /\ x e. ( A [,] B ) ) -> ( x ^ N ) e. CC )
147:50,51,146:itgioo
|- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x = S. ( A [,] B ) ( x ^ N ) _d x )
148::oveq1 |- ( x = t -> ( x ^ N ) = ( t ^ N ) )
149:148:cbvitgv |- S. ( A (,) B ) ( x ^ N ) _d x = S. ( A (,) B ) ( t ^ N ) _d t
150:124,81:syldan |- ( ( ph /\ t e. ( A [,] B ) ) -> ( t ^ N ) e. CC )
151:102:sseli |- ( t e. ( A (,) B ) -> t e. ( A [,] B ) )
152:151,150:sylan2 |- ( ( ph /\ t e. ( A (,) B ) ) -> ( t ^ N ) e. CC )
153::resmpt |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ N ) ) )
154:123,153:syl |- ( ph -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) = ( t e. ( A [,] B ) |-> ( t ^ N ) ) )
155::rescncf |- ( ( A [,] B ) C_ CC -> ( ( t e. CC |-> ( t ^ N ) ) e. ( CC -cn-> CC ) -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) ) )
156:123,115,155:sylc |- ( ph -> ( ( t e. CC |-> ( t ^ N ) ) |` ( A [,] B ) ) e. ( ( A [,] B ) -cn-> CC ) )
157:154,156:eqeltrrd |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. ( ( A [,] B ) -cn-> CC ) )
158::cniccibl |- ( ( A e. RR /\ B e. RR /\ ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. ( ( A [,] B ) -cn-> CC ) ) -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. L^1 )
159:50,51,157,158:syl3anc |- ( ph -> ( t e. ( A [,] B ) |-> ( t ^ N ) ) e. L^1 )
160:120,122,150,159:iblss
|- ( ph -> ( t e. ( A (,) B ) |-> ( t ^ N ) ) e. L^1 )
161:109,152,160:iblmulc2 |- ( ph -> ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) e. L^1 )
162:100,161:eqeltrd |- ( ph -> ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) e. L^1 )
163:50,51,52,119,162,139:ftc2
|- ( ph ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x =
( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
164:152,160:itgcl |- ( ph -> S. ( A (,) B ) ( t ^ N ) _d t e. CC )
165:149,164:syl5eqel |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x e. CC )
166:83:nnne0d |- ( ph -> ( N + 1 ) =/= 0 )
167:165,109,166:divcan3d
|- ( ph -> ( ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) / ( N + 1 ) ) = S. ( A (,) B ) ( x ^ N ) _d x )
168:102:sseli |- ( x e. ( A (,) B ) -> x e. ( A [,] B ) )
169:168,146:sylan2 |- ( ( ph /\ x e. ( A (,) B ) ) -> ( x ^ N ) e. CC )
170:148:cbvmptv |- ( x e. ( A (,) B ) |-> ( x ^ N ) ) = ( t e. ( A (,) B ) |-> ( t ^ N ) )
171:170,160:syl5eqel |- ( ph -> ( x e. ( A (,) B ) |-> ( x ^ N ) ) e. L^1 )
172:109,169,171:itgmulc2
|- ( ph -> ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
173:100:fveq1d |- ( ph -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) )
174::oveq1 |- ( t = x -> ( t ^ N ) = ( x ^ N ) )
175:174:oveq2d |- ( t = x -> ( ( N + 1 ) x. ( t ^ N ) ) = ( ( N + 1 ) x. ( x ^ N ) ) )
176::eqid |- ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) = ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) )
177::ovex |- ( ( N + 1 ) x. ( x ^ N ) ) e. _V
178:175,176,177:fvmpt |- ( x e. ( A (,) B ) -> ( ( t e. ( A (,) B ) |-> ( ( N + 1 ) x. ( t ^ N ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
179:173,178:sylan9eq |- ( ( ph /\ x e. ( A (,) B ) ) -> ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
180:179:ralrimiva |- ( ph -> A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) )
181::itgeq2 |- ( A. x e. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) = ( ( N + 1 ) x. ( x ^ N ) ) ->
S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
182:180,181:syl |- ( ph -> S. ( A (,) B ) ( ( RR _D ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ) ` x ) _d x = S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x )
183:182,163:eqtr3d |- ( ph ->
S. ( A (,) B ) ( ( N + 1 ) x. ( x ^ N ) ) _d x = ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) )
184:50:rexrd |- ( ph -> A e. RR* )
185:51:rexrd |- ( ph -> B e. RR* )
186::ubicc2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
187:184,185,52,186:syl3anc
|- ( ph -> B e. ( A [,] B ) )
188::oveq1 |- ( t = B -> ( t ^ ( N + 1 ) ) = ( B ^ ( N + 1 ) ) )
189::eqid |- ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) )
190::ovex |- ( B ^ ( N + 1 ) ) e. _V
191:188,189,190:fvmpt |- ( B e. ( A [,] B ) -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) = ( B ^ ( N + 1 ) ) )
192:187,191:syl |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) = ( B ^ ( N + 1 ) ) )
193::lbicc2 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
194:184,185,52,193:syl3anc
|- ( ph -> A e. ( A [,] B ) )
195::oveq1 |- ( t = A -> ( t ^ ( N + 1 ) ) = ( A ^ ( N + 1 ) ) )
196::eqid |- ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) = ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) )
197::ovex |- ( A ^ ( N + 1 ) ) e. _V
198:195,196,197:fvmpt |- ( A e. ( A [,] B ) -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) = ( A ^ ( N + 1 ) ) )
199:194,198:syl |- ( ph -> ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) = ( A ^ ( N + 1 ) ) )
200:192,199:oveq12d |- ( ph -> ( ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` B ) - ( ( t e. ( A [,] B ) |-> ( t ^ ( N + 1 ) ) ) ` A ) ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
201:172,183,200:3eqtrd
|- ( ph -> ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) = ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) )
202:201:oveq1d |- ( ph -> ( ( ( N + 1 ) x. S. ( A (,) B ) ( x ^ N ) _d x ) / ( N + 1 ) ) = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
203:167,202:eqtr3d |- ( ph -> S. ( A (,) B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
qed:147,203:eqtr3d |- ( ph -> S. ( A [,] B ) ( x ^ N ) _d x = ( ( ( B ^ ( N + 1 ) ) - ( A ^ ( N + 1 ) ) ) / ( N + 1 ) ) )
$d t x
$d A t
$d A x
$d B t
$d B x
$d N t
$d N x
$d ph t
$d ph x
$= ( itexp.1 itexp.2 vt co cexp wcel cc cr wss a1i cmul cmpt wceq syl cfv cioo
itexp.4 itexp.3 cv citg cicc c1 caddc cmin cdiv wa iccssre ax-resscn sselda
syl2anc sstrd cn0 nnnn0d adantr expcld itgioo oveq1 cbvitgv ioossicc syldan
sseli simpr sylan2 cvol cdm ioombl ccncf cibl cres expcncf rescncf eqeltrrd
resmpt sylc cniccibl syl3anc iblss itgcl nncnd 1cnd addcld peano2nnd nnne0d
syl5eqel divcan3d cbvmptv itgmulc2 cdv wral crn ctg ccnfld ctopn reelprrecn
cpr peano2nn0 reexpcld nn0red remulcld eqid ctopon cnfldtopon toponmax mp1i
recnd cin df-ss sylib nn0cnd mulcld dvexp pncand oveq2d mpteq2dva dvmptres3
cn eqtrd tgioo2 cnt iccntr dvmptres2 fveq1d fvmpt sylan9eq ralrimiva itgeq2
ovex syl5ss syl6ss ctx ccn eqeltrd eqtr3d cxr rexrd ssid cncfmptc cncfmpt2f
mulcn iblmulc2 ftc2 cle wbr ubicc2 lbicc2 oveq12d 3eqtrd oveq1d ) ABCDUAIZB
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UJWAHCUXBUVAUURUXCUVOCUUSJVBVUICUUSJYLYHSUUKUULUUMYRYR $.
*** I've never been so happy to see a perfectly rectangular encrypted mess. :)
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