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Theorem binomcxplemdvbinom 36604
Description: Lemma for binomcxp 36608. 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 36606 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 2585 . . . . . 6  |-  F/_ b `' abs
3 nfcv 2585 . . . . . . 7  |-  F/_ b
0
4 nfcv 2585 . . . . . . 7  |-  F/_ b [,)
5 binomcxplem.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
6 nfcv 2585 . . . . . . . . . . . 12  |-  F/_ b  +
7 binomcxplem.s . . . . . . . . . . . . . 14  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
8 nfmpt1 4511 . . . . . . . . . . . . . 14  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
97, 8nfcxfr 2583 . . . . . . . . . . . . 13  |-  F/_ b S
10 nfcv 2585 . . . . . . . . . . . . 13  |-  F/_ b
r
119, 10nffv 5886 . . . . . . . . . . . 12  |-  F/_ b
( S `  r
)
123, 6, 11nfseq 12224 . . . . . . . . . . 11  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1312nfel1 2601 . . . . . . . . . 10  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
14 nfcv 2585 . . . . . . . . . 10  |-  F/_ b RR
1513, 14nfrab 3011 . . . . . . . . 9  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
16 nfcv 2585 . . . . . . . . 9  |-  F/_ b RR*
17 nfcv 2585 . . . . . . . . 9  |-  F/_ b  <
1815, 16, 17nfsup 7969 . . . . . . . 8  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
195, 18nfcxfr 2583 . . . . . . 7  |-  F/_ b R
203, 4, 19nfov 6329 . . . . . 6  |-  F/_ b
( 0 [,) R
)
212, 20nfima 5193 . . . . 5  |-  F/_ b
( `' abs " (
0 [,) R ) )
221, 21nfcxfr 2583 . . . 4  |-  F/_ b D
23 nfcv 2585 . . . 4  |-  F/_ y D
24 nfcv 2585 . . . 4  |-  F/_ y
( ( 1  +  b )  ^c  -u C )
25 nfcv 2585 . . . 4  |-  F/_ b
( ( 1  +  y )  ^c  -u C )
26 oveq2 6311 . . . . 5  |-  ( b  =  y  ->  (
1  +  b )  =  ( 1  +  y ) )
2726oveq1d 6318 . . . 4  |-  ( b  =  y  ->  (
( 1  +  b )  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
2822, 23, 24, 25, 27cbvmptf 4512 . . 3  |-  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C
) )  =  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) )
2928oveq2i 6314 . 2  |-  ( CC 
_D  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C ) ) )  =  ( CC 
_D  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) ) )
30 cnelprrecn 9634 . . . . 5  |-  CC  e.  { RR ,  CC }
3130a1i 11 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  CC  e.  { RR ,  CC } )
32 1cnd 9661 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  1  e.  CC )
33 cnvimass 5205 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
341, 33eqsstri 3495 . . . . . . . . 9  |-  D  C_  dom  abs
35 absf 13394 . . . . . . . . . 10  |-  abs : CC
--> RR
3635fdmi 5749 . . . . . . . . 9  |-  dom  abs  =  CC
3734, 36sseqtri 3497 . . . . . . . 8  |-  D  C_  CC
3837a1i 11 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  D  C_  CC )
3938sselda 3465 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  y  e.  CC )
4032, 39addcld 9664 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( 1  +  y )  e.  CC )
41 simpr 463 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
1  +  y )  e.  RR )
42 1cnd 9661 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  1  e.  CC )
4339adantr 467 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  y  e.  CC )
4442, 43pncan2d 9990 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  =  y )
45 1red 9660 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  1  e.  RR )
4641, 45resubcld 10049 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  e.  RR )
4744, 46eqeltrrd 2512 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  y  e.  RR )
48 1pneg1e0 10720 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
49 1red 9660 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  1  e.  RR )
5049renegcld 10048 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  -u 1  e.  RR )
51 simpr 463 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  y  e.  RR )
52 ffn 5744 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
53 elpreima 6015 . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( `' abs " ( 0 [,) R
) )  ->  ( abs `  y )  e.  ( 0 [,) R
) )
5655, 1eleq2s 2531 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  D  ->  ( abs `  y )  e.  ( 0 [,) R
) )
57 0re 9645 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
58 ssrab2 3547 . . . . . . . . . . . . . . . . . . . . 21  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR
59 ressxr 9686 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  RR*
6058, 59sstri 3474 . . . . . . . . . . . . . . . . . . . 20  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR*
61 supxrcl 11602 . . . . . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . . . . 18  |-  R  e. 
RR*
64 elico2 11700 . . . . . . . . . . . . . . . . . 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 677 . . . . . . . . . . . . . . . . 17  |-  ( ( abs `  y )  e.  ( 0 [,) R )  <->  ( ( abs `  y )  e.  RR  /\  0  <_ 
( abs `  y
)  /\  ( abs `  y )  <  R
) )
6656, 65sylib 200 . . . . . . . . . . . . . . . 16  |-  ( y  e.  D  ->  (
( abs `  y
)  e.  RR  /\  0  <_  ( abs `  y
)  /\  ( abs `  y )  <  R
) )
6766simp3d 1020 . . . . . . . . . . . . . . 15  |-  ( y  e.  D  ->  ( abs `  y )  < 
R )
6867adantl 468 . . . . . . . . . . . . . 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 36603 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
7574adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  R  =  1 )
7668, 75breqtrd 4446 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( abs `  y
)  <  1 )
7776adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( abs `  y )  <  1
)
7851, 49absltd 13485 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( ( abs `  y )  <  1  <->  ( -u 1  <  y  /\  y  <  1 ) ) )
7977, 78mpbid 214 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( -u
1  <  y  /\  y  <  1 ) )
8079simpld 461 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  -u 1  <  y )
8150, 51, 49, 80ltadd2dd 9796 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( 1  +  -u 1 )  < 
( 1  +  y ) )
8248, 81syl5eqbrr 4456 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  0  < 
( 1  +  y ) )
8347, 82syldan 473 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  0  <  ( 1  +  y ) )
8441, 83elrpd 11340 . . . . . 6  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
1  +  y )  e.  RR+ )
8584ex 436 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( ( 1  +  y )  e.  RR  ->  ( 1  +  y )  e.  RR+ )
)
86 eqid 2423 . . . . . 6  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
8786ellogdm 23576 . . . . 5  |-  ( ( 1  +  y )  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( ( 1  +  y )  e.  CC  /\  ( ( 1  +  y )  e.  RR  ->  (
1  +  y )  e.  RR+ ) ) )
8840, 85, 87sylanbrc 669 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( 1  +  y )  e.  ( CC 
\  ( -oo (,] 0 ) ) )
89 eldifi 3588 . . . . . 6  |-  ( x  e.  ( CC  \ 
( -oo (,] 0 ) )  ->  x  e.  CC )
9089adantl 468 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  ->  x  e.  CC )
9172adantr 467 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  C  e.  CC )
9291negcld 9975 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  -u C  e.  CC )
9392adantr 467 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  ->  -u C  e.  CC )
9490, 93cxpcld 23645 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  -> 
( x  ^c  -u C )  e.  CC )
95 ovex 6331 . . . . 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 9661 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  1  e.  CC )
98 simpr 463 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  x  e.  CC )
9997, 98addcld 9664 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  ( 1  +  x
)  e.  CC )
100 c0ex 9639 . . . . . . . . 9  |-  0  e.  _V
101100a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  0  e.  _V )
102 1cnd 9661 . . . . . . . . 9  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  1  e.  CC )
10331, 102dvmptc 22904 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  1 ) )  =  ( x  e.  CC  |->  0 ) )
10431dvmptid 22903 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
10531, 97, 101, 103, 98, 97, 104dvmptadd 22906 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  ( 0  +  1 ) ) )
106 0p1e1 10723 . . . . . . . 8  |-  ( 0  +  1 )  =  1
107106mpteq2i 4505 . . . . . . 7  |-  ( x  e.  CC  |->  ( 0  +  1 ) )  =  ( x  e.  CC  |->  1 )
108105, 107syl6eq 2480 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  1 ) )
109 fvex 5889 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  _V
110 cnfldtps 21790 . . . . . . . . . 10  |-fld  e.  TopSp
111 cnfldbas 18967 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
112 eqid 2423 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
113111, 112tpsuni 19945 . . . . . . . . . 10  |-  (fld  e.  TopSp  ->  CC  =  U. ( TopOpen
` fld
) )
114110, 113ax-mp 5 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
115114restid 15325 . . . . . . . 8  |-  ( (
TopOpen ` fld )  e.  _V  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
116109, 115ax-mp 5 . . . . . . 7  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
117116eqcomi 2436 . . . . . 6  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
118112cnfldtop 21796 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Top
119 eqid 2423 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
120119cnbl0 21786 . . . . . . . . . . 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 2452 . . . . . . . . 9  |-  D  =  ( 0 ( ball `  ( abs  o.  -  ) ) R )
123 cnxmet 21785 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
124 0cn 9637 . . . . . . . . . 10  |-  0  e.  CC
125112cnfldtopn 21794 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
126125blopn 21507 . . . . . . . . . 10  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld ) )
127123, 124, 63, 126mp3an 1361 . . . . . . . . 9  |-  ( 0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld )
128122, 127eqeltri 2507 . . . . . . . 8  |-  D  e.  ( TopOpen ` fld )
129 isopn3i 20090 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  D  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  D
)  =  D )
130118, 128, 129mp2an 677 . . . . . . 7  |-  ( ( int `  ( TopOpen ` fld )
) `  D )  =  D
131130a1i 11 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  (
( int `  ( TopOpen
` fld
) ) `  D
)  =  D )
13231, 99, 97, 108, 38, 117, 112, 131dvmptres2 22908 . . . . 5  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( x  e.  D  |->  1 ) )
133 oveq2 6311 . . . . . . 7  |-  ( x  =  y  ->  (
1  +  x )  =  ( 1  +  y ) )
134133cbvmptv 4514 . . . . . 6  |-  ( x  e.  D  |->  ( 1  +  x ) )  =  ( y  e.  D  |->  ( 1  +  y ) )
135134oveq2i 6314 . . . . 5  |-  ( CC 
_D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )
136 eqidd 2424 . . . . . 6  |-  ( x  =  y  ->  1  =  1 )
137136cbvmptv 4514 . . . . 5  |-  ( x  e.  D  |->  1 )  =  ( y  e.  D  |->  1 )
138132, 135, 1373eqtr3g 2487 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )  =  ( y  e.  D  |->  1 ) )
13986dvcncxp1 23675 . . . . 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 6310 . . . 4  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
142 oveq1 6310 . . . . 5  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  y )  ^c  (
-u C  -  1 ) ) )
143142oveq2d 6319 . . . 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 22918 . . 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 467 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  C  e.  CC )
146145negcld 9975 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  -> 
-u C  e.  CC )
147146, 32subcld 9988 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  - 
1 )  e.  CC )
14840, 147cxpcld 23645 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( ( 1  +  y )  ^c 
( -u C  -  1 ) )  e.  CC )
149146, 148mulcld 9665 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  x.  ( ( 1  +  y )  ^c 
( -u C  -  1 ) ) )  e.  CC )
150149mulid1d 9662 . . . 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 4508 . . 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 2585 . . . . 5  |-  F/_ b
( -u C  x.  (
( 1  +  y )  ^c  (
-u C  -  1 ) ) )
153 nfcv 2585 . . . . 5  |-  F/_ y
( -u C  x.  (
( 1  +  b )  ^c  (
-u C  -  1 ) ) )
154 oveq2 6311 . . . . . . 7  |-  ( y  =  b  ->  (
1  +  y )  =  ( 1  +  b ) )
155154oveq1d 6318 . . . . . 6  |-  ( y  =  b  ->  (
( 1  +  y )  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  b )  ^c  (
-u C  -  1 ) ) )
156155oveq2d 6319 . . . . 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 4512 . . . 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 2468 . 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 2476 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 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   {crab 2780   _Vcvv 3082    \ cdif 3434    C_ wss 3437   {cpr 3999   U.cuni 4217   class class class wbr 4421    |-> cmpt 4480   `'ccnv 4850   dom cdm 4851   "cima 4854    o. ccom 4855    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303   supcsup 7958   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546   -oocmnf 9675   RR*cxr 9676    < clt 9677    <_ cle 9678    - cmin 9862   -ucneg 9863   NNcn 10611   NN0cn0 10871   RR+crp 11304   (,]cioc 11638   [,)cico 11639    seqcseq 12214   ^cexp 12273   abscabs 13291    ~~> cli 13541   ↾t crest 15312   TopOpenctopn 15313   *Metcxmt 18948   ballcbl 18950  ℂfldccnfld 18963   Topctop 19909   TopSpctps 19911   intcnt 20024    _D cdv 22810    ^c ccxp 23497  C𝑐cbcc 36587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ioo 11641  df-ioc 11642  df-ico 11643  df-icc 11644  df-fz 11787  df-fzo 11918  df-fl 12029  df-mod 12098  df-seq 12215  df-exp 12274  df-fac 12461  df-bc 12489  df-hash 12517  df-shft 13124  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-prod 13953  df-fallfac 14053  df-ef 14114  df-sin 14116  df-cos 14117  df-tan 14118  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19913  df-bases 19914  df-topon 19915  df-topsp 19916  df-cld 20026  df-ntr 20027  df-cls 20028  df-nei 20106  df-lp 20144  df-perf 20145  df-cn 20235  df-cnp 20236  df-haus 20323  df-cmp 20394  df-tx 20569  df-hmeo 20762  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-xms 21327  df-ms 21328  df-tms 21329  df-cncf 21902  df-limc 22813  df-dv 22814  df-log 23498  df-cxp 23499  df-bcc 36588
This theorem is referenced by:  binomcxplemnotnn0  36607
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