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Theorem binomcxplemdvbinom 36772
Description: Lemma for binomcxp 36776. By the power and chain rules, calculate the derivative of ( ( 1 + b ) ^c -u C ), with respect to b in the disk of convergence D. We later multiply the derivative in the later binomcxplemdvsum 36774 by this derivative to show that ( ( 1 + b ) ^c C ) (with a non-negated C) and the later sum, since both at b = 0 equal one, are the same. (Contributed by Steve Rodriguez, 22-Apr-2020.)
Hypotheses
Ref Expression
binomcxp.a |- ( ph -> A e. RR+ )
binomcxp.b |- ( ph -> B e. RR )
binomcxp.lt |- ( ph -> ( abs ` B ) < ( abs ` A ) )
binomcxp.c |- ( ph -> C e. CC )
binomcxplem.f |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
binomcxplem.s |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
binomcxplem.r |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
binomcxplem.e |- E = ( b e. CC |-> ( k e. NN |-> ( ( k x. ( F ` k ) ) x. ( b ^ ( k - 1 ) ) ) ) )
binomcxplem.d |- D = ( `' abs " ( 0 [,) R ) )
Assertion
Ref Expression
binomcxplemdvbinom |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
Distinct variable groups:   j, k, ph   k, b, C   C, j   F, b, k   S, r   r, b
Allowed substitution hints:   ph( r, b)   A( j, k, r, b)   B( j, k, r, b)   C( r)   D( j, k, r, b)   R( j, k, r, b)   S( j, k, b)   E( j, k, r, b)   F( j, r)
Proof of Theorem binomcxplemdvbinom
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 binomcxplem.d . . . . 5 |- D = ( `' abs " ( 0 [,) R ) )
2 nfcv 2612 . . . . . 6 |- F/_ b `' abs
3 nfcv 2612 . . . . . . 7 |- F/_ b 0
4 nfcv 2612 . . . . . . 7 |- F/_ b [,)
5 binomcxplem.r . . . . . . . 8 |- R = sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
6 nfcv 2612 . . . . . . . . . . . 12 |- F/_ b +
7 binomcxplem.s . . . . . . . . . . . . . 14 |- S = ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
8 nfmpt1 4485 . . . . . . . . . . . . . 14 |- F/_ b ( b e. CC |-> ( k e. NN0 |-> ( ( F ` k ) x. ( b ^ k ) ) ) )
97, 8nfcxfr 2610 . . . . . . . . . . . . 13 |- F/_ b S
10 nfcv 2612 . . . . . . . . . . . . 13 |- F/_ b r
119, 10nffv 5886 . . . . . . . . . . . 12 |- F/_ b ( S ` r )
123, 6, 11nfseq 12261 . . . . . . . . . . 11 |- F/_ b seq 0 ( + , ( S ` r ) )
1312nfel1 2626 . . . . . . . . . 10 |- F/ b seq 0 ( + , ( S ` r ) ) e. dom ~~>
14 nfcv 2612 . . . . . . . . . 10 |- F/_ b RR
1513, 14nfrab 2958 . . . . . . . . 9 |- F/_ b { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> }
16 nfcv 2612 . . . . . . . . 9 |- F/_ b RR*
17 nfcv 2612 . . . . . . . . 9 |- F/_ b <
1815, 16, 17nfsup 7983 . . . . . . . 8 |- F/_ b sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < )
195, 18nfcxfr 2610 . . . . . . 7 |- F/_ b R
203, 4, 19nfov 6334 . . . . . 6 |- F/_ b ( 0 [,) R )
212, 20nfima 5182 . . . . 5 |- F/_ b ( `' abs " ( 0 [,) R ) )
221, 21nfcxfr 2610 . . . 4 |- F/_ b D
23 nfcv 2612 . . . 4 |- F/_ y D
24 nfcv 2612 . . . 4 |- F/_ y ( ( 1 + b ) ^c -u C )
25 nfcv 2612 . . . 4 |- F/_ b ( ( 1 + y ) ^c -u C )
26 oveq2 6316 . . . . 5 |- ( b = y -> ( 1 + b ) = ( 1 + y ) )
2726oveq1d 6323 . . . 4 |- ( b = y -> ( ( 1 + b ) ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
2822, 23, 24, 25, 27cbvmptf 4486 . . 3 |- ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) = ( y e. D |-> ( ( 1 + y ) ^c -u C ) )
2928oveq2i 6319 . 2 |- ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) )
30 cnelprrecn 9650 . . . . 5 |- CC e. { RR , CC }
3130a1i 11 . . . 4 |- ( ( ph /\ -. C e. NN0 ) -> CC e. { RR , CC } )
32 1cnd 9677 . . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> 1 e. CC )
33 cnvimass 5194 . . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) C_ dom abs
341, 33eqsstri 3448 . . . . . . . . 9 |- D C_ dom abs
35 absf 13477 . . . . . . . . . 10 |- abs : CC --> RR
3635fdmi 5746 . . . . . . . . 9 |- dom abs = CC
3734, 36sseqtri 3450 . . . . . . . 8 |- D C_ CC
3837a1i 11 . . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> D C_ CC )
3938sselda 3418 . . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> y e. CC )
4032, 39addcld 9680 . . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. CC )
41 simpr 468 . . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR )
42 1cnd 9677 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. CC )
4339adantr 472 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. CC )
4442, 43pncan2d 10007 . . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) = y )
45 1red 9676 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 1 e. RR )
4641, 45resubcld 10068 . . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( ( 1 + y ) - 1 ) e. RR )
4744, 46eqeltrrd 2550 . . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> y e. RR )
48 1pneg1e0 10740 . . . . . . . . 9 |- ( 1 + -u 1 ) = 0
49 1red 9676 . . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 1 e. RR )
5049renegcld 10067 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 e. RR )
51 simpr 468 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> y e. RR )
52 ffn 5739 . . . . . . . . . . . . . . . . . . . 20 |- ( abs : CC --> RR -> abs Fn CC )
53 elpreima 6017 . . . . . . . . . . . . . . . . . . . 20 |- ( abs Fn CC -> ( y e. ( `' abs " ( 0 [,) R ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) R ) ) ) )
5435, 52, 53mp2b 10 . . . . . . . . . . . . . . . . . . 19 |- ( y e. ( `' abs " ( 0 [,) R ) ) <-> ( y e. CC /\ ( abs ` y ) e. ( 0 [,) R ) ) )
5554simprbi 471 . . . . . . . . . . . . . . . . . 18 |- ( y e. ( `' abs " ( 0 [,) R ) ) -> ( abs ` y ) e. ( 0 [,) R ) )
5655, 1eleq2s 2567 . . . . . . . . . . . . . . . . 17 |- ( y e. D -> ( abs ` y ) e. ( 0 [,) R ) )
57 0re 9661 . . . . . . . . . . . . . . . . . 18 |- 0 e. RR
58 ssrab2 3500 . . . . . . . . . . . . . . . . . . . . 21 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR
59 ressxr 9702 . . . . . . . . . . . . . . . . . . . . 21 |- RR C_ RR*
6058, 59sstri 3427 . . . . . . . . . . . . . . . . . . . 20 |- { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR*
61 supxrcl 11625 . . . . . . . . . . . . . . . . . . . 20 |- ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } C_ RR* -> sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR* )
6260, 61ax-mp 5 . . . . . . . . . . . . . . . . . . 19 |- sup ( { r e. RR | seq 0 ( + , ( S ` r ) ) e. dom ~~> } , RR* , < ) e. RR*
635, 62eqeltri 2545 . . . . . . . . . . . . . . . . . 18 |- R e. RR*
64 elico2 11723 . . . . . . . . . . . . . . . . . 18 |- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) ) )
6557, 63, 64mp2an 686 . . . . . . . . . . . . . . . . 17 |- ( ( abs ` y ) e. ( 0 [,) R ) <-> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
6656, 65sylib 201 . . . . . . . . . . . . . . . 16 |- ( y e. D -> ( ( abs ` y ) e. RR /\ 0 <_ ( abs ` y ) /\ ( abs ` y ) < R ) )
6766simp3d 1044 . . . . . . . . . . . . . . 15 |- ( y e. D -> ( abs ` y ) < R )
6867adantl 473 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < R )
69 binomcxp.a . . . . . . . . . . . . . . . 16 |- ( ph -> A e. RR+ )
70 binomcxp.b . . . . . . . . . . . . . . . 16 |- ( ph -> B e. RR )
71 binomcxp.lt . . . . . . . . . . . . . . . 16 |- ( ph -> ( abs ` B ) < ( abs ` A ) )
72 binomcxp.c . . . . . . . . . . . . . . . 16 |- ( ph -> C e. CC )
73 binomcxplem.f . . . . . . . . . . . . . . . 16 |- F = ( j e. NN0 |-> ( CC𝑐 j ) )
7469, 70, 71, 72, 73, 7, 5binomcxplemradcnv 36771 . . . . . . . . . . . . . . 15 |- ( ( ph /\ -. C e. NN0 ) -> R = 1 )
7574adantr 472 . . . . . . . . . . . . . 14 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> R = 1 )
7668, 75breqtrd 4420 . . . . . . . . . . . . 13 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( abs ` y ) < 1 )
7776adantr 472 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( abs ` y ) < 1 )
7851, 49absltd 13568 . . . . . . . . . . . 12 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( ( abs ` y ) < 1 <-> ( -u 1 < y /\ y < 1 ) ) )
7977, 78mpbid 215 . . . . . . . . . . 11 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( -u 1 < y /\ y < 1 ) )
8079simpld 466 . . . . . . . . . 10 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> -u 1 < y )
8150, 51, 49, 80ltadd2dd 9811 . . . . . . . . 9 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> ( 1 + -u 1 ) < ( 1 + y ) )
8248, 81syl5eqbrr 4430 . . . . . . . 8 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ y e. RR ) -> 0 < ( 1 + y ) )
8347, 82syldan 478 . . . . . . 7 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> 0 < ( 1 + y ) )
8441, 83elrpd 11361 . . . . . 6 |- ( ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) /\ ( 1 + y ) e. RR ) -> ( 1 + y ) e. RR+ )
8584ex 441 . . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) )
86 eqid 2471 . . . . . 6 |- ( CC \ ( -oo (,] 0 ) ) = ( CC \ ( -oo (,] 0 ) )
8786ellogdm 23663 . . . . 5 |- ( ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) <-> ( ( 1 + y ) e. CC /\ ( ( 1 + y ) e. RR -> ( 1 + y ) e. RR+ ) ) )
8840, 85, 87sylanbrc 677 . . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( 1 + y ) e. ( CC \ ( -oo (,] 0 ) ) )
89 eldifi 3544 . . . . . 6 |- ( x e. ( CC \ ( -oo (,] 0 ) ) -> x e. CC )
9089adantl 473 . . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> x e. CC )
9172adantr 472 . . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> C e. CC )
9291negcld 9992 . . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> -u C e. CC )
9392adantr 472 . . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> -u C e. CC )
9490, 93cxpcld 23732 . . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( x ^c -u C ) e. CC )
95 ovex 6336 . . . . 5 |- ( -u C x. ( x ^c ( -u C - 1 ) ) ) e. _V
9695a1i 11 . . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. ( CC \ ( -oo (,] 0 ) ) ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) e. _V )
97 1cnd 9677 . . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 1 e. CC )
98 simpr 468 . . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> x e. CC )
9997, 98addcld 9680 . . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> ( 1 + x ) e. CC )
100 c0ex 9655 . . . . . . . . 9 |- 0 e. _V
101100a1i 11 . . . . . . . 8 |- ( ( ( ph /\ -. C e. NN0 ) /\ x e. CC ) -> 0 e. _V )
102 1cnd 9677 . . . . . . . . 9 |- ( ( ph /\ -. C e. NN0 ) -> 1 e. CC )
10331, 102dvmptc 22991 . . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> 1 ) ) = ( x e. CC |-> 0 ) )
10431dvmptid 22990 . . . . . . . 8 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
10531, 97, 101, 103, 98, 97, 104dvmptadd 22993 . . . . . . 7 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> ( 0 + 1 ) ) )
106 0p1e1 10743 . . . . . . . 8 |- ( 0 + 1 ) = 1
107106mpteq2i 4479 . . . . . . 7 |- ( x e. CC |-> ( 0 + 1 ) ) = ( x e. CC |-> 1 )
108105, 107syl6eq 2521 . . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. CC |-> ( 1 + x ) ) ) = ( x e. CC |-> 1 ) )
109 fvex 5889 . . . . . . . 8 |- ( TopOpen ` ℂfld ) e. _V
110 cnfldtps 21876 . . . . . . . . . 10 |- ℂfld e. TopSp
111 cnfldbas 19051 . . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
112 eqid 2471 . . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
113111, 112tpsuni 20030 . . . . . . . . . 10 |- (ℂfld e. TopSp -> CC = U. ( TopOpen ` ℂfld ) )
114110, 113ax-mp 5 . . . . . . . . 9 |- CC = U. ( TopOpen ` ℂfld )
115114restid 15410 . . . . . . . 8 |- ( ( TopOpen ` ℂfld ) e. _V -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
116109, 115ax-mp 5 . . . . . . 7 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
117116eqcomi 2480 . . . . . 6 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
118112cnfldtop 21882 . . . . . . . 8 |- ( TopOpen ` ℂfld ) e. Top
119 eqid 2471 . . . . . . . . . . . 12 |- ( abs o. - ) = ( abs o. - )
120119cnbl0 21872 . . . . . . . . . . 11 |- ( R e. RR* -> ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R ) )
12163, 120ax-mp 5 . . . . . . . . . 10 |- ( `' abs " ( 0 [,) R ) ) = ( 0 ( ball ` ( abs o. - ) ) R )
1221, 121eqtri 2493 . . . . . . . . 9 |- D = ( 0 ( ball ` ( abs o. - ) ) R )
123 cnxmet 21871 . . . . . . . . . 10 |- ( abs o. - ) e. ( *Met ` CC )
124 0cn 9653 . . . . . . . . . 10 |- 0 e. CC
125112cnfldtopn 21880 . . . . . . . . . . 11 |- ( TopOpen ` ℂfld ) = ( MetOpen ` ( abs o. - ) )
126125blopn 21593 . . . . . . . . . 10 |- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ R e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld ) )
127123, 124, 63, 126mp3an 1390 . . . . . . . . 9 |- ( 0 ( ball ` ( abs o. - ) ) R ) e. ( TopOpen ` ℂfld )
128122, 127eqeltri 2545 . . . . . . . 8 |- D e. ( TopOpen ` ℂfld )
129 isopn3i 20175 . . . . . . . 8 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ D e. ( TopOpen ` ℂfld ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
130118, 128, 129mp2an 686 . . . . . . 7 |- ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D
131130a1i 11 . . . . . 6 |- ( ( ph /\ -. C e. NN0 ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` D ) = D )
13231, 99, 97, 108, 38, 117, 112, 131dvmptres2 22995 . . . . 5 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( x e. D |-> 1 ) )
133 oveq2 6316 . . . . . . 7 |- ( x = y -> ( 1 + x ) = ( 1 + y ) )
134133cbvmptv 4488 . . . . . 6 |- ( x e. D |-> ( 1 + x ) ) = ( y e. D |-> ( 1 + y ) )
135134oveq2i 6319 . . . . 5 |- ( CC _D ( x e. D |-> ( 1 + x ) ) ) = ( CC _D ( y e. D |-> ( 1 + y ) ) )
136 eqidd 2472 . . . . . 6 |- ( x = y -> 1 = 1 )
137136cbvmptv 4488 . . . . 5 |- ( x e. D |-> 1 ) = ( y e. D |-> 1 )
138132, 135, 1373eqtr3g 2528 . . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( 1 + y ) ) ) = ( y e. D |-> 1 ) )
13986dvcncxp1 23762 . . . . 5 |- ( -u C e. CC -> ( CC _D ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( x ^c -u C ) ) ) = ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( -u C x. ( x ^c ( -u C - 1 ) ) ) ) )
14092, 139syl 17 . . . 4 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( x ^c -u C ) ) ) = ( x e. ( CC \ ( -oo (,] 0 ) ) |-> ( -u C x. ( x ^c ( -u C - 1 ) ) ) ) )
141 oveq1 6315 . . . 4 |- ( x = ( 1 + y ) -> ( x ^c -u C ) = ( ( 1 + y ) ^c -u C ) )
142 oveq1 6315 . . . . 5 |- ( x = ( 1 + y ) -> ( x ^c ( -u C - 1 ) ) = ( ( 1 + y ) ^c ( -u C - 1 ) ) )
143142oveq2d 6324 . . . 4 |- ( x = ( 1 + y ) -> ( -u C x. ( x ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) )
14431, 31, 88, 32, 94, 96, 138, 140, 141, 143dvmptco 23005 . . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) ) = ( y e. D |-> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) ) )
14591adantr 472 . . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> C e. CC )
146145negcld 9992 . . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> -u C e. CC )
147146, 32subcld 10005 . . . . . . 7 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C - 1 ) e. CC )
14840, 147cxpcld 23732 . . . . . 6 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( 1 + y ) ^c ( -u C - 1 ) ) e. CC )
149146, 148mulcld 9681 . . . . 5 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) e. CC )
150149mulid1d 9678 . . . 4 |- ( ( ( ph /\ -. C e. NN0 ) /\ y e. D ) -> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) = ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) )
151150mpteq2dva 4482 . . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( y e. D |-> ( ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) x. 1 ) ) = ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) )
152 nfcv 2612 . . . . 5 |- F/_ b ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) )
153 nfcv 2612 . . . . 5 |- F/_ y ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) )
154 oveq2 6316 . . . . . . 7 |- ( y = b -> ( 1 + y ) = ( 1 + b ) )
155154oveq1d 6323 . . . . . 6 |- ( y = b -> ( ( 1 + y ) ^c ( -u C - 1 ) ) = ( ( 1 + b ) ^c ( -u C - 1 ) ) )
156155oveq2d 6324 . . . . 5 |- ( y = b -> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) = ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) )
15723, 22, 152, 153, 156cbvmptf 4486 . . . 4 |- ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) )
158157a1i 11 . . 3 |- ( ( ph /\ -. C e. NN0 ) -> ( y e. D |-> ( -u C x. ( ( 1 + y ) ^c ( -u C - 1 ) ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
159144, 151, 1583eqtrd 2509 . 2 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( y e. D |-> ( ( 1 + y ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
16029, 159syl5eq 2517 1 |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {crab 2760   _Vcvv 3031   \ cdif 3387   C_ wss 3390   {cpr 3961   U.cuni 4190   class class class wbr 4395   |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   "cima 4842   o. ccom 4843   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   -oocmnf 9691   RR*cxr 9692   < clt 9693   <_ cle 9694   - cmin 9880   -ucneg 9881   NNcn 10631   NN0cn0 10893   RR+crp 11325   (,]cioc 11661   [,)cico 11662   seqcseq 12251   ^cexp 12310   abscabs 13374   ~~> cli 13625   ↾t crest 15397   TopOpenctopn 15398   *Metcxmt 19032   ballcbl 19034  ℂfldccnfld 19047   Topctop 19994   TopSpctps 19996   intcnt 20109   _D cdv 22897   ^c ccxp 23584  C𝑐cbcc 36755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-prod 14037  df-fallfac 14137  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-bcc 36756
This theorem is referenced by:  binomcxplemnotnn0  36775
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