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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 ` fld )  e.  _V
110 cnfldtps 21876 . . . . . . . . . 10  |-fld  e.  TopSp
111 cnfldbas 19051 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
112 eqid 2471 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
113111, 112tpsuni 20030 . . . . . . . . . 10  |-  (fld  e.  TopSp  ->  CC  =  U. ( TopOpen
` fld
) )
114110, 113ax-mp 5 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
115114restid 15410 . . . . . . . 8  |-  ( (
TopOpen ` fld )  e.  _V  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
116109, 115ax-mp 5 . . . . . . 7  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
117116eqcomi 2480 . . . . . 6  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
118112cnfldtop 21882 . . . . . . . 8  |-  ( TopOpen ` fld )  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 ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
126125blopn 21593 . . . . . . . . . 10  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld ) )
127123, 124, 63, 126mp3an 1390 . . . . . . . . 9  |-  ( 0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld )
128122, 127eqeltri 2545 . . . . . . . 8  |-  D  e.  ( TopOpen ` fld )
129 isopn3i 20175 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  D  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  D
)  =  D )
130118, 128, 129mp2an 686 . . . . . . 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 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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