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Theorem binomcxplemdvbinom 36106
Description: Lemma for binomcxp 36110. 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 36108 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 2564 . . . . . 6  |-  F/_ b `' abs
3 nfcv 2564 . . . . . . 7  |-  F/_ b
0
4 nfcv 2564 . . . . . . 7  |-  F/_ b [,)
5 binomcxplem.r . . . . . . . 8  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
6 nfcv 2564 . . . . . . . . . . . 12  |-  F/_ b  +
7 binomcxplem.s . . . . . . . . . . . . . 14  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
8 nfmpt1 4484 . . . . . . . . . . . . . 14  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
97, 8nfcxfr 2562 . . . . . . . . . . . . 13  |-  F/_ b S
10 nfcv 2564 . . . . . . . . . . . . 13  |-  F/_ b
r
119, 10nffv 5856 . . . . . . . . . . . 12  |-  F/_ b
( S `  r
)
123, 6, 11nfseq 12161 . . . . . . . . . . 11  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1312nfel1 2580 . . . . . . . . . 10  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
14 nfcv 2564 . . . . . . . . . 10  |-  F/_ b RR
1513, 14nfrab 2989 . . . . . . . . 9  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
16 nfcv 2564 . . . . . . . . 9  |-  F/_ b RR*
17 nfcv 2564 . . . . . . . . 9  |-  F/_ b  <
1815, 16, 17nfsup 7944 . . . . . . . 8  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
195, 18nfcxfr 2562 . . . . . . 7  |-  F/_ b R
203, 4, 19nfov 6304 . . . . . 6  |-  F/_ b
( 0 [,) R
)
212, 20nfima 5165 . . . . 5  |-  F/_ b
( `' abs " (
0 [,) R ) )
221, 21nfcxfr 2562 . . . 4  |-  F/_ b D
23 nfcv 2564 . . . 4  |-  F/_ y D
24 nfcv 2564 . . . 4  |-  F/_ y
( ( 1  +  b )  ^c  -u C )
25 nfcv 2564 . . . 4  |-  F/_ b
( ( 1  +  y )  ^c  -u C )
26 oveq2 6286 . . . . 5  |-  ( b  =  y  ->  (
1  +  b )  =  ( 1  +  y ) )
2726oveq1d 6293 . . . 4  |-  ( b  =  y  ->  (
( 1  +  b )  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
2822, 23, 24, 25, 27cbvmptf 4485 . . 3  |-  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C
) )  =  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) )
2928oveq2i 6289 . 2  |-  ( CC 
_D  ( b  e.  D  |->  ( ( 1  +  b )  ^c  -u C ) ) )  =  ( CC 
_D  ( y  e.  D  |->  ( ( 1  +  y )  ^c  -u C ) ) )
30 cnelprrecn 9615 . . . . 5  |-  CC  e.  { RR ,  CC }
3130a1i 11 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  CC  e.  { RR ,  CC } )
32 1cnd 9642 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  1  e.  CC )
33 cnvimass 5177 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
341, 33eqsstri 3472 . . . . . . . . 9  |-  D  C_  dom  abs
35 absf 13319 . . . . . . . . . 10  |-  abs : CC
--> RR
3635fdmi 5719 . . . . . . . . 9  |-  dom  abs  =  CC
3734, 36sseqtri 3474 . . . . . . . 8  |-  D  C_  CC
3837a1i 11 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  D  C_  CC )
3938sselda 3442 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  y  e.  CC )
4032, 39addcld 9645 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( 1  +  y )  e.  CC )
41 simpr 459 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
1  +  y )  e.  RR )
42 1cnd 9642 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  1  e.  CC )
4339adantr 463 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  y  e.  CC )
4442, 43pncan2d 9969 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  =  y )
45 1red 9641 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  1  e.  RR )
4641, 45resubcld 10028 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
( 1  +  y )  -  1 )  e.  RR )
4744, 46eqeltrrd 2491 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  y  e.  RR )
48 1pneg1e0 10685 . . . . . . . . 9  |-  ( 1  +  -u 1 )  =  0
49 1red 9641 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  1  e.  RR )
5049renegcld 10027 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  -u 1  e.  RR )
51 simpr 459 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  y  e.  RR )
52 ffn 5714 . . . . . . . . . . . . . . . . . . . 20  |-  ( abs
: CC --> RR  ->  abs 
Fn  CC )
53 elpreima 5985 . . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( `' abs " ( 0 [,) R
) )  ->  ( abs `  y )  e.  ( 0 [,) R
) )
5655, 1eleq2s 2510 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  D  ->  ( abs `  y )  e.  ( 0 [,) R
) )
57 0re 9626 . . . . . . . . . . . . . . . . . 18  |-  0  e.  RR
58 ssrab2 3524 . . . . . . . . . . . . . . . . . . . . 21  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR
59 ressxr 9667 . . . . . . . . . . . . . . . . . . . . 21  |-  RR  C_  RR*
6058, 59sstri 3451 . . . . . . . . . . . . . . . . . . . 20  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } 
C_  RR*
61 supxrcl 11559 . . . . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . . 18  |-  R  e. 
RR*
64 elico2 11642 . . . . . . . . . . . . . . . . . 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 670 . . . . . . . . . . . . . . . . 17  |-  ( ( abs `  y )  e.  ( 0 [,) R )  <->  ( ( abs `  y )  e.  RR  /\  0  <_ 
( abs `  y
)  /\  ( abs `  y )  <  R
) )
6656, 65sylib 196 . . . . . . . . . . . . . . . 16  |-  ( y  e.  D  ->  (
( abs `  y
)  e.  RR  /\  0  <_  ( abs `  y
)  /\  ( abs `  y )  <  R
) )
6766simp3d 1011 . . . . . . . . . . . . . . 15  |-  ( y  e.  D  ->  ( abs `  y )  < 
R )
6867adantl 464 . . . . . . . . . . . . . 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 36105 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  R  =  1 )
7574adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  R  =  1 )
7668, 75breqtrd 4419 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( abs `  y
)  <  1 )
7776adantr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( abs `  y )  <  1
)
7851, 49absltd 13410 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( ( abs `  y )  <  1  <->  ( -u 1  <  y  /\  y  <  1 ) ) )
7977, 78mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( -u
1  <  y  /\  y  <  1 ) )
8079simpld 457 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  -u 1  <  y )
8150, 51, 49, 80ltadd2dd 9775 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  ( 1  +  -u 1 )  < 
( 1  +  y ) )
8248, 81syl5eqbrr 4429 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  y  e.  RR )  ->  0  < 
( 1  +  y ) )
8347, 82syldan 468 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  0  <  ( 1  +  y ) )
8441, 83elrpd 11301 . . . . . 6  |-  ( ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D
)  /\  ( 1  +  y )  e.  RR )  ->  (
1  +  y )  e.  RR+ )
8584ex 432 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( ( 1  +  y )  e.  RR  ->  ( 1  +  y )  e.  RR+ )
)
86 eqid 2402 . . . . . 6  |-  ( CC 
\  ( -oo (,] 0 ) )  =  ( CC  \  ( -oo (,] 0 ) )
8786ellogdm 23314 . . . . 5  |-  ( ( 1  +  y )  e.  ( CC  \ 
( -oo (,] 0 ) )  <->  ( ( 1  +  y )  e.  CC  /\  ( ( 1  +  y )  e.  RR  ->  (
1  +  y )  e.  RR+ ) ) )
8840, 85, 87sylanbrc 662 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( 1  +  y )  e.  ( CC 
\  ( -oo (,] 0 ) ) )
89 eldifi 3565 . . . . . 6  |-  ( x  e.  ( CC  \ 
( -oo (,] 0 ) )  ->  x  e.  CC )
9089adantl 464 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  ->  x  e.  CC )
9172adantr 463 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  C  e.  CC )
9291negcld 9954 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  -u C  e.  CC )
9392adantr 463 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  ->  -u C  e.  CC )
9490, 93cxpcld 23383 . . . 4  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  ( CC  \  ( -oo (,] 0
) ) )  -> 
( x  ^c  -u C )  e.  CC )
95 ovex 6306 . . . . 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 9642 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  1  e.  CC )
98 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  x  e.  CC )
9997, 98addcld 9645 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  ( 1  +  x
)  e.  CC )
100 c0ex 9620 . . . . . . . . 9  |-  0  e.  _V
101100a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  x  e.  CC )  ->  0  e.  _V )
102 1cnd 9642 . . . . . . . . 9  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  1  e.  CC )
10331, 102dvmptc 22653 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  1 ) )  =  ( x  e.  CC  |->  0 ) )
10431dvmptid 22652 . . . . . . . 8  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  x ) )  =  ( x  e.  CC  |->  1 ) )
10531, 97, 101, 103, 98, 97, 104dvmptadd 22655 . . . . . . 7  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  ( 0  +  1 ) ) )
106 0p1e1 10688 . . . . . . . 8  |-  ( 0  +  1 )  =  1
107106mpteq2i 4478 . . . . . . 7  |-  ( x  e.  CC  |->  ( 0  +  1 ) )  =  ( x  e.  CC  |->  1 )
108105, 107syl6eq 2459 . . . . . 6  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  CC  |->  ( 1  +  x ) ) )  =  ( x  e.  CC  |->  1 ) )
109 fvex 5859 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  _V
110 cnfldtps 21577 . . . . . . . . . 10  |-fld  e.  TopSp
111 cnfldbas 18744 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
112 eqid 2402 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
113111, 112tpsuni 19731 . . . . . . . . . 10  |-  (fld  e.  TopSp  ->  CC  =  U. ( TopOpen
` fld
) )
114110, 113ax-mp 5 . . . . . . . . 9  |-  CC  =  U. ( TopOpen ` fld )
115114restid 15048 . . . . . . . 8  |-  ( (
TopOpen ` fld )  e.  _V  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
116109, 115ax-mp 5 . . . . . . 7  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
117116eqcomi 2415 . . . . . 6  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
118112cnfldtop 21583 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  Top
119 eqid 2402 . . . . . . . . . . . 12  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
120119cnbl0 21573 . . . . . . . . . . 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 2431 . . . . . . . . 9  |-  D  =  ( 0 ( ball `  ( abs  o.  -  ) ) R )
123 cnxmet 21572 . . . . . . . . . 10  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
124 0cn 9618 . . . . . . . . . 10  |-  0  e.  CC
125112cnfldtopn 21581 . . . . . . . . . . 11  |-  ( TopOpen ` fld )  =  ( MetOpen `  ( abs  o.  -  ) )
126125blopn 21295 . . . . . . . . . 10  |-  ( ( ( abs  o.  -  )  e.  ( *Met `  CC )  /\  0  e.  CC  /\  R  e.  RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld ) )
127123, 124, 63, 126mp3an 1326 . . . . . . . . 9  |-  ( 0 ( ball `  ( abs  o.  -  ) ) R )  e.  (
TopOpen ` fld )
128122, 127eqeltri 2486 . . . . . . . 8  |-  D  e.  ( TopOpen ` fld )
129 isopn3i 19876 . . . . . . . 8  |-  ( ( ( TopOpen ` fld )  e.  Top  /\  D  e.  ( TopOpen ` fld )
)  ->  ( ( int `  ( TopOpen ` fld ) ) `  D
)  =  D )
130118, 128, 129mp2an 670 . . . . . . 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 22657 . . . . 5  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( x  e.  D  |->  1 ) )
133 oveq2 6286 . . . . . . 7  |-  ( x  =  y  ->  (
1  +  x )  =  ( 1  +  y ) )
134133cbvmptv 4487 . . . . . 6  |-  ( x  e.  D  |->  ( 1  +  x ) )  =  ( y  e.  D  |->  ( 1  +  y ) )
135134oveq2i 6289 . . . . 5  |-  ( CC 
_D  ( x  e.  D  |->  ( 1  +  x ) ) )  =  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )
136 eqidd 2403 . . . . . 6  |-  ( x  =  y  ->  1  =  1 )
137136cbvmptv 4487 . . . . 5  |-  ( x  e.  D  |->  1 )  =  ( y  e.  D  |->  1 )
138132, 135, 1373eqtr3g 2466 . . . 4  |-  ( (
ph  /\  -.  C  e.  NN0 )  ->  ( CC  _D  ( y  e.  D  |->  ( 1  +  y ) ) )  =  ( y  e.  D  |->  1 ) )
13986dvcncxp1 23413 . . . . 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 6285 . . . 4  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  -u C )  =  ( ( 1  +  y )  ^c  -u C ) )
142 oveq1 6285 . . . . 5  |-  ( x  =  ( 1  +  y )  ->  (
x  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  y )  ^c  (
-u C  -  1 ) ) )
143142oveq2d 6294 . . . 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 22667 . . 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 463 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  C  e.  CC )
146145negcld 9954 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  -> 
-u C  e.  CC )
147146, 32subcld 9967 . . . . . . 7  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  - 
1 )  e.  CC )
14840, 147cxpcld 23383 . . . . . 6  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( ( 1  +  y )  ^c 
( -u C  -  1 ) )  e.  CC )
149146, 148mulcld 9646 . . . . 5  |-  ( ( ( ph  /\  -.  C  e.  NN0 )  /\  y  e.  D )  ->  ( -u C  x.  ( ( 1  +  y )  ^c 
( -u C  -  1 ) ) )  e.  CC )
150149mulid1d 9643 . . . 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 4481 . . 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 2564 . . . . 5  |-  F/_ b
( -u C  x.  (
( 1  +  y )  ^c  (
-u C  -  1 ) ) )
153 nfcv 2564 . . . . 5  |-  F/_ y
( -u C  x.  (
( 1  +  b )  ^c  (
-u C  -  1 ) ) )
154 oveq2 6286 . . . . . . 7  |-  ( y  =  b  ->  (
1  +  y )  =  ( 1  +  b ) )
155154oveq1d 6293 . . . . . 6  |-  ( y  =  b  ->  (
( 1  +  y )  ^c  (
-u C  -  1 ) )  =  ( ( 1  +  b )  ^c  (
-u C  -  1 ) ) )
156155oveq2d 6294 . . . . 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 4485 . . . 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 2447 . 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 2455 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 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {crab 2758   _Vcvv 3059    \ cdif 3411    C_ wss 3414   {cpr 3974   U.cuni 4191   class class class wbr 4395    |-> cmpt 4453   `'ccnv 4822   dom cdm 4823   "cima 4826    o. ccom 4827    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   supcsup 7934   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527   -oocmnf 9656   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841   -ucneg 9842   NNcn 10576   NN0cn0 10836   RR+crp 11265   (,]cioc 11583   [,)cico 11584    seqcseq 12151   ^cexp 12210   abscabs 13216    ~~> cli 13456   ↾t crest 15035   TopOpenctopn 15036   *Metcxmt 18723   ballcbl 18725  ℂfldccnfld 18740   Topctop 19686   TopSpctps 19689   intcnt 19810    _D cdv 22559    ^c ccxp 23235  C𝑐cbcc 36089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-prod 13865  df-fallfac 13952  df-ef 14012  df-sin 14014  df-cos 14015  df-tan 14016  df-pi 14017  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-cxp 23237  df-bcc 36090
This theorem is referenced by:  binomcxplemnotnn0  36109
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