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Theorem binomcxplemdvbinom 36702
Description: Lemma for binomcxp 36706. 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 36704 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 2592 . . . . . 6  |-  F/_ b `' abs
3 nfcv 2592 . . . . . . 7  |-  F/_ b
0
4 nfcv 2592 . . . . . . 7  |-  F/_ b [,)
5 binomcxplem.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
6 nfcv 2592 . . . . . . . . . . . 12  |-  F/_ b  +
7 binomcxplem.s . . . . . . . . . . . . . 14  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
8 nfmpt1 4492 . . . . . . . . . . . . . 14  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
97, 8nfcxfr 2590 . . . . . . . . . . . . 13  |-  F/_ b S
10 nfcv 2592 . . . . . . . . . . . . 13  |-  F/_ b
r
119, 10nffv 5872 . . . . . . . . . . . 12  |-  F/_ b
( S `  r
)
123, 6, 11nfseq 12223 . . . . . . . . . . 11  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1312nfel1 2606 . . . . . . . . . 10  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
14 nfcv 2592 . . . . . . . . . 10  |-  F/_ b RR
1513, 14nfrab 2972 . . . . . . . . 9  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
16 nfcv 2592 . . . . . . . . 9  |-  F/_ b RR*
17 nfcv 2592 . . . . . . . . 9  |-  F/_ b  <
1815, 16, 17nfsup 7965 . . . . . . . 8  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
195, 18nfcxfr 2590 . . . . . . 7  |-  F/_ b R
203, 4, 19nfov 6316 . . . . . 6  |-  F/_ b
( 0 [,) R
)
212, 20nfima 5176 . . . . 5  |-  F/_ b
( `' abs " (
0 [,) R ) )
221, 21nfcxfr 2590 . . . 4  |-  F/_ b D
23 nfcv 2592 . . . 4  |-  F/_ y D
24 nfcv 2592 . . . 4  |-  F/_ y
( ( 1  +  b )  ^c  -u C )
25 nfcv 2592 . . . 4  |-  F/_ b
( ( 1  +  y )  ^c  -u C )
26 oveq2 6298 . . . . 5  |-  ( b  =  y  ->  (
1  +  b )  =  ( 1  +  y ) )
2726oveq1d 6305 . . . 4  |-  ( b  =  y  ->  (
( 1  +  b )  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
2822, 23, 24, 25, 27cbvmptf 4493 . . 3  |-  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C
) )  =  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) )
2928oveq2i 6301 . 2  |-  ( CC 
_D  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C ) ) )  =  ( CC 
_D  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) ) )
30 cnelprrecn 9632 . . . . 5  |-  CC  e.  { RR ,  CC }
3130a1i 11 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  CC  e.  { RR ,  CC } )
32 1cnd 9659 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  1  e.  CC )
33 cnvimass 5188 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
341, 33eqsstri 3462 . . . . . . . . 9  |-  D  C_  dom  abs
35 absf 13400 . . . . . . . . . 10  |-  abs : CC
--> RR
3635fdmi 5734 . . . . . . . . 9  |-  dom  abs  =  CC
3734, 36sseqtri 3464 . . . . . . . 8  |-  D  C_  CC
3837a1i 11 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  D  C_  CC )
3938sselda 3432 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  y  e.  CC )
4032, 39addcld 9662 . . . . 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 9659 . . . . . . . . . 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 9988 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  =  y )
45 1red 9658 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  1  e.  RR )
4641, 45resubcld 10047 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  e.  RR )
4744, 46eqeltrrd 2530 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  y  e.  RR )
48 1pneg1e0 10718 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
49 1red 9658 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  1  e.  RR )
5049renegcld 10046 . . . . . . . . . 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 5728 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
53 elpreima 6002 . . . . . . . . . . . . . . . . . . . 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 2547 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  D  ->  ( abs `  y )  e.  ( 0 [,) R
) )
57 0re 9643 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
58 ssrab2 3514 . . . . . . . . . . . . . . . . . . . . 21  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR
59 ressxr 9684 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  RR*
6058, 59sstri 3441 . . . . . . . . . . . . . . . . . . . 20  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR*
61 supxrcl 11600 . . . . . . . . . . . . . . . . . . . 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 2525 . . . . . . . . . . . . . . . . . 18  |-  R  e. 
RR*
64 elico2 11698 . . . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . 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 36701 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
7574adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  R  =  1 )
7668, 75breqtrd 4427 . . . . . . . . . . . . 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 13491 . . . . . . . . . . . 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 9794 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( 1  +  -u 1 )  < 
( 1  +  y ) )
8248, 81syl5eqbrr 4437 . . . . . . . 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 11338 . . . . . 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 2451 . . . . . 6  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
8786ellogdm 23584 . . . . 5  |-  ( ( 1  +  y )  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( ( 1  +  y )  e.  CC  /\  ( ( 1  +  y )  e.  RR  ->  (
1  +  y )  e.  RR+ ) ) )
8840, 85, 87sylanbrc 670 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( 1  +  y )  e.  ( CC 
\  ( -oo (,] 0 ) ) )
89 eldifi 3555 . . . . . 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 9973 . . . . . 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 23653 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  -> 
( x  ^c  -u C )  e.  CC )
95 ovex 6318 . . . . 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 9659 . . . . . . 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 9662 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  ( 1  +  x
)  e.  CC )
100 c0ex 9637 . . . . . . . . 9  |-  0  e.  _V
101100a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  0  e.  _V )
102 1cnd 9659 . . . . . . . . 9  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  1  e.  CC )
10331, 102dvmptc 22912 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  1 ) )  =  ( x  e.  CC  |->  0 ) )
10431dvmptid 22911 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
10531, 97, 101, 103, 98, 97, 104dvmptadd 22914 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  ( 0  +  1 ) ) )
106 0p1e1 10721 . . . . . . . 8  |-  ( 0  +  1 )  =  1
107106mpteq2i 4486 . . . . . . 7  |-  ( x  e.  CC  |->  ( 0  +  1 ) )  =  ( x  e.  CC  |->  1 )
108105, 107syl6eq 2501 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  1 ) )
109 fvex 5875 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  _V
110 cnfldtps 21798 . . . . . . . . . 10  |-fld  e.  TopSp
111 cnfldbas 18974 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
112 eqid 2451 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
113111, 112tpsuni 19953 . . . . . . . . . 10  |-  (fld  e.  TopSp  ->  CC  =  U. ( TopOpen
` fld
) )
114110, 113ax-mp 5 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
115114restid 15332 . . . . . . . 8  |-  ( (
TopOpen ` fld )  e.  _V  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
116109, 115ax-mp 5 . . . . . . 7  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
117116eqcomi 2460 . . . . . 6  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
118112cnfldtop 21804 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Top
119 eqid 2451 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
120119cnbl0 21794 . . . . . . . . . . 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 2473 . . . . . . . . 9  |-  D  =  ( 0 ( ball `  ( abs  o.  -  ) ) R )
123 cnxmet 21793 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
124 0cn 9635 . . . . . . . . . 10  |-  0  e.  CC
125112cnfldtopn 21802 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
126125blopn 21515 . . . . . . . . . 10  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld ) )
127123, 124, 63, 126mp3an 1364 . . . . . . . . 9  |-  ( 0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld )
128122, 127eqeltri 2525 . . . . . . . 8  |-  D  e.  ( TopOpen ` fld )
129 isopn3i 20098 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  D  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  D
)  =  D )
130118, 128, 129mp2an 678 . . . . . . 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 22916 . . . . 5  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( x  e.  D  |->  1 ) )
133 oveq2 6298 . . . . . . 7  |-  ( x  =  y  ->  (
1  +  x )  =  ( 1  +  y ) )
134133cbvmptv 4495 . . . . . 6  |-  ( x  e.  D  |->  ( 1  +  x ) )  =  ( y  e.  D  |->  ( 1  +  y ) )
135134oveq2i 6301 . . . . 5  |-  ( CC 
_D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )
136 eqidd 2452 . . . . . 6  |-  ( x  =  y  ->  1  =  1 )
137136cbvmptv 4495 . . . . 5  |-  ( x  e.  D  |->  1 )  =  ( y  e.  D  |->  1 )
138132, 135, 1373eqtr3g 2508 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )  =  ( y  e.  D  |->  1 ) )
13986dvcncxp1 23683 . . . . 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 6297 . . . 4  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
142 oveq1 6297 . . . . 5  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  y )  ^c  (
-u C  -  1 ) ) )
143142oveq2d 6306 . . . 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 22926 . . 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 9973 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  -> 
-u C  e.  CC )
147146, 32subcld 9986 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  - 
1 )  e.  CC )
14840, 147cxpcld 23653 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( ( 1  +  y )  ^c 
( -u C  -  1 ) )  e.  CC )
149146, 148mulcld 9663 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  x.  ( ( 1  +  y )  ^c 
( -u C  -  1 ) ) )  e.  CC )
150149mulid1d 9660 . . . 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 4489 . . 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 2592 . . . . 5  |-  F/_ b
( -u C  x.  (
( 1  +  y )  ^c  (
-u C  -  1 ) ) )
153 nfcv 2592 . . . . 5  |-  F/_ y
( -u C  x.  (
( 1  +  b )  ^c  (
-u C  -  1 ) ) )
154 oveq2 6298 . . . . . . 7  |-  ( y  =  b  ->  (
1  +  y )  =  ( 1  +  b ) )
155154oveq1d 6305 . . . . . 6  |-  ( y  =  b  ->  (
( 1  +  y )  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  b )  ^c  (
-u C  -  1 ) ) )
156155oveq2d 6306 . . . . 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 4493 . . . 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 2489 . 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 2497 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 985    = wceq 1444    e. wcel 1887   {crab 2741   _Vcvv 3045    \ cdif 3401    C_ wss 3404   {cpr 3970   U.cuni 4198   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   "cima 4837    o. ccom 4838    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   -oocmnf 9673   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860   -ucneg 9861   NNcn 10609   NN0cn0 10869   RR+crp 11302   (,]cioc 11636   [,)cico 11637    seqcseq 12213   ^cexp 12272   abscabs 13297    ~~> cli 13548   ↾t crest 15319   TopOpenctopn 15320   *Metcxmt 18955   ballcbl 18957  ℂfldccnfld 18970   Topctop 19917   TopSpctps 19919   intcnt 20032    _D cdv 22818    ^c ccxp 23505  C𝑐cbcc 36685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-mod 12097  df-seq 12214  df-exp 12273  df-fac 12460  df-bc 12488  df-hash 12516  df-shft 13130  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-limsup 13526  df-clim 13552  df-rlim 13553  df-sum 13753  df-prod 13960  df-fallfac 14060  df-ef 14121  df-sin 14123  df-cos 14124  df-tan 14125  df-pi 14126  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-lp 20152  df-perf 20153  df-cn 20243  df-cnp 20244  df-haus 20331  df-cmp 20402  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cncf 21910  df-limc 22821  df-dv 22822  df-log 23506  df-cxp 23507  df-bcc 36686
This theorem is referenced by:  binomcxplemnotnn0  36705
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