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Theorem binomcxplemdvsum 36704
Description: Lemma for binomcxp 36706. The derivative of the generalized sum in binomcxplemnn0 36698. Part of remark "This convergence allows us to apply term-by-term differentiation..." in the Wikibooks proof. (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 ) )
binomcxplem.p  |-  P  =  ( b  e.  D  |-> 
sum_ k  e.  NN0  ( ( S `  b ) `  k
) )
Assertion
Ref Expression
binomcxplemdvsum  |-  ( ph  ->  ( CC  _D  P
)  =  ( b  e.  D  |->  sum_ k  e.  NN  ( ( E `
 b ) `  k ) ) )
Distinct variable groups:    k, b, F    ph, b, k    r,
b, k, F    j,
k, ph    C, j
Allowed substitution hints:    ph( r)    A( j, k, r, b)    B( j, k, r, b)    C( k, r, b)    D( j, k, r, b)    P( j, k, r, b)    R( j, k, r, b)    S( j, k, r, b)    E( j, k, r, b)    F( j)

Proof of Theorem binomcxplemdvsum
Dummy variables  m  n  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 binomcxplem.s . . . 4  |-  S  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
2 binomcxplem.p . . . . 5  |-  P  =  ( b  e.  D  |-> 
sum_ k  e.  NN0  ( ( S `  b ) `  k
) )
3 binomcxplem.d . . . . . . 7  |-  D  =  ( `' abs " (
0 [,) R ) )
4 nfcv 2592 . . . . . . . 8  |-  F/_ b `' abs
5 nfcv 2592 . . . . . . . . 9  |-  F/_ b
0
6 nfcv 2592 . . . . . . . . 9  |-  F/_ b [,)
7 binomcxplem.r . . . . . . . . . 10  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
8 nfcv 2592 . . . . . . . . . . . . . 14  |-  F/_ b  +
9 nfmpt1 4492 . . . . . . . . . . . . . . . 16  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
101, 9nfcxfr 2590 . . . . . . . . . . . . . . 15  |-  F/_ b S
11 nfcv 2592 . . . . . . . . . . . . . . 15  |-  F/_ b
r
1210, 11nffv 5872 . . . . . . . . . . . . . 14  |-  F/_ b
( S `  r
)
135, 8, 12nfseq 12223 . . . . . . . . . . . . 13  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1413nfel1 2606 . . . . . . . . . . . 12  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
15 nfcv 2592 . . . . . . . . . . . 12  |-  F/_ b RR
1614, 15nfrab 2972 . . . . . . . . . . 11  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
17 nfcv 2592 . . . . . . . . . . 11  |-  F/_ b RR*
18 nfcv 2592 . . . . . . . . . . 11  |-  F/_ b  <
1916, 17, 18nfsup 7965 . . . . . . . . . 10  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
207, 19nfcxfr 2590 . . . . . . . . 9  |-  F/_ b R
215, 6, 20nfov 6316 . . . . . . . 8  |-  F/_ b
( 0 [,) R
)
224, 21nfima 5176 . . . . . . 7  |-  F/_ b
( `' abs " (
0 [,) R ) )
233, 22nfcxfr 2590 . . . . . 6  |-  F/_ b D
24 nfcv 2592 . . . . . 6  |-  F/_ y D
25 nfcv 2592 . . . . . 6  |-  F/_ y sum_ k  e.  NN0  (
( S `  b
) `  k )
26 nfcv 2592 . . . . . . 7  |-  F/_ b NN0
27 nfcv 2592 . . . . . . . . 9  |-  F/_ b
y
2810, 27nffv 5872 . . . . . . . 8  |-  F/_ b
( S `  y
)
29 nfcv 2592 . . . . . . . 8  |-  F/_ b
m
3028, 29nffv 5872 . . . . . . 7  |-  F/_ b
( ( S `  y ) `  m
)
3126, 30nfsum 13757 . . . . . 6  |-  F/_ b sum_ m  e.  NN0  (
( S `  y
) `  m )
32 simpl 459 . . . . . . . . . 10  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
b  =  y )
3332fveq2d 5869 . . . . . . . . 9  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( S `  b
)  =  ( S `
 y ) )
3433fveq1d 5867 . . . . . . . 8  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( ( S `  b ) `  k
)  =  ( ( S `  y ) `
 k ) )
3534sumeq2dv 13769 . . . . . . 7  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ k  e.  NN0  ( ( S `  y ) `
 k ) )
36 nfcv 2592 . . . . . . . 8  |-  F/_ m
( ( S `  y ) `  k
)
37 nfcv 2592 . . . . . . . . . . . 12  |-  F/_ k CC
38 nfmpt1 4492 . . . . . . . . . . . 12  |-  F/_ k
( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) )
3937, 38nfmpt 4491 . . . . . . . . . . 11  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
401, 39nfcxfr 2590 . . . . . . . . . 10  |-  F/_ k S
41 nfcv 2592 . . . . . . . . . 10  |-  F/_ k
y
4240, 41nffv 5872 . . . . . . . . 9  |-  F/_ k
( S `  y
)
43 nfcv 2592 . . . . . . . . 9  |-  F/_ k
m
4442, 43nffv 5872 . . . . . . . 8  |-  F/_ k
( ( S `  y ) `  m
)
45 fveq2 5865 . . . . . . . 8  |-  ( k  =  m  ->  (
( S `  y
) `  k )  =  ( ( S `
 y ) `  m ) )
4636, 44, 45cbvsumi 13763 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( S `
 y ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m )
4735, 46syl6eq 2501 . . . . . 6  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m ) )
4823, 24, 25, 31, 47cbvmptf 4493 . . . . 5  |-  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `
 b ) `  k ) )  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
492, 48eqtri 2473 . . . 4  |-  P  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
50 ovex 6318 . . . . . 6  |-  ( CC𝑐 j )  e.  _V
5150a1i 11 . . . . 5  |-  ( (
ph  /\  j  e.  NN0 )  ->  ( CC𝑐 j
)  e.  _V )
52 binomcxplem.f . . . . . 6  |-  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) )
5352a1i 11 . . . . 5  |-  ( ph  ->  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) ) )
5452a1i 11 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  F  =  ( j  e.  NN0  |->  ( CC𝑐 j ) ) )
55 simpr 463 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
j  =  k )
5655oveq2d 6306 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
( CC𝑐 j )  =  ( CC𝑐 k ) )
57 simpr 463 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
58 binomcxp.c . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
5958adantr 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  CC )
6059, 57bcccl 36688 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 k
)  e.  CC )
6154, 56, 57, 60fvmptd 5954 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( CC𝑐 k ) )
6261, 60eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
6351, 53, 62fmpt2d 6053 . . . 4  |-  ( ph  ->  F : NN0 --> CC )
64 nfcv 2592 . . . . . . 7  |-  F/_ r RR
65 nfcv 2592 . . . . . . 7  |-  F/_ z RR
66 nfv 1761 . . . . . . 7  |-  F/ z  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
67 nfcv 2592 . . . . . . . . 9  |-  F/_ r
0
68 nfcv 2592 . . . . . . . . 9  |-  F/_ r  +
69 nfcv 2592 . . . . . . . . . . 11  |-  F/_ r
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
701, 69nfcxfr 2590 . . . . . . . . . 10  |-  F/_ r S
71 nfcv 2592 . . . . . . . . . 10  |-  F/_ r
z
7270, 71nffv 5872 . . . . . . . . 9  |-  F/_ r
( S `  z
)
7367, 68, 72nfseq 12223 . . . . . . . 8  |-  F/_ r  seq 0 (  +  , 
( S `  z
) )
7473nfel1 2606 . . . . . . 7  |-  F/ r  seq 0 (  +  ,  ( S `  z ) )  e. 
dom 
~~>
75 fveq2 5865 . . . . . . . . 9  |-  ( r  =  z  ->  ( S `  r )  =  ( S `  z ) )
7675seqeq3d 12221 . . . . . . . 8  |-  ( r  =  z  ->  seq 0 (  +  , 
( S `  r
) )  =  seq 0 (  +  , 
( S `  z
) ) )
7776eleq1d 2513 . . . . . . 7  |-  ( r  =  z  ->  (  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  ) )
7864, 65, 66, 74, 77cbvrab 3043 . . . . . 6  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  }  =  { z  e.  RR  |  seq 0
(  +  ,  ( S `  z ) )  e.  dom  ~~>  }
7978supeq1i 7961 . . . . 5  |-  sup ( { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  } ,  RR* ,  <  )  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  } ,  RR* ,  <  )
807, 79eqtri 2473 . . . 4  |-  R  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  } ,  RR* ,  <  )
811fveq1i 5866 . . . . . . . . . . . 12  |-  ( S `
 z )  =  ( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z )
82 seqeq3 12218 . . . . . . . . . . . 12  |-  ( ( S `  z )  =  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z )  ->  seq 0 (  +  ,  ( S `  z ) )  =  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) ) )
8381, 82ax-mp 5 . . . . . . . . . . 11  |-  seq 0
(  +  ,  ( S `  z ) )  =  seq 0
(  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z ) )
8483eleq1i 2520 . . . . . . . . . 10  |-  (  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  )
8584a1i 11 . . . . . . . . 9  |-  ( z  e.  RR  ->  (  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  ) )
8685rabbiia 3033 . . . . . . . 8  |-  { z  e.  RR  |  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  }  =  { z  e.  RR  |  seq 0
(  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  }
8786supeq1i 7961 . . . . . . 7  |-  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( S `  z ) )  e. 
dom 
~~>  } ,  RR* ,  <  )  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )
887, 79, 873eqtrri 2478 . . . . . 6  |-  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  =  R
8988eleq1i 2520 . . . . 5  |-  ( sup ( { z  e.  RR  |  seq 0
(  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  } ,  RR* ,  <  )  e.  RR  <->  R  e.  RR )
9088oveq2i 6301 . . . . . 6  |-  ( ( abs `  x )  +  sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  =  ( ( abs `  x )  +  R
)
9190oveq1i 6300 . . . . 5  |-  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 )  =  ( ( ( abs `  x )  +  R
)  /  2 )
92 eqid 2451 . . . . 5  |-  ( ( abs `  x )  +  1 )  =  ( ( abs `  x
)  +  1 )
9389, 91, 92ifbieq12i 3907 . . . 4  |-  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) )  =  if ( R  e.  RR , 
( ( ( abs `  x )  +  R
)  /  2 ) ,  ( ( abs `  x )  +  1 ) )
94 oveq1 6297 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  b  ->  (
w ^ k )  =  ( b ^
k ) )
9594oveq2d 6306 . . . . . . . . . . . . . . . . 17  |-  ( w  =  b  ->  (
( F `  k
)  x.  ( w ^ k ) )  =  ( ( F `
 k )  x.  ( b ^ k
) ) )
9695mpteq2dv 4490 . . . . . . . . . . . . . . . 16  |-  ( w  =  b  ->  (
k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) )  =  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) )
9796cbvmptv 4495 . . . . . . . . . . . . . . 15  |-  ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) )  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) )
9897fveq1i 5866 . . . . . . . . . . . . . 14  |-  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z )  =  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z )
99 seqeq3 12218 . . . . . . . . . . . . . 14  |-  ( ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
w ^ k ) ) ) ) `  z )  =  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z )  ->  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  =  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) ) )
10098, 99ax-mp 5 . . . . . . . . . . . . 13  |-  seq 0
(  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
w ^ k ) ) ) ) `  z ) )  =  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )
101100eleq1i 2520 . . . . . . . . . . . 12  |-  (  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  <->  seq 0
(  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  )
102101a1i 11 . . . . . . . . . . 11  |-  ( z  e.  RR  ->  (  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  <->  seq 0
(  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  ) )
103102rabbiia 3033 . . . . . . . . . 10  |-  { z  e.  RR  |  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  }  =  { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  }
104103supeq1i 7961 . . . . . . . . 9  |-  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )
105104eleq1i 2520 . . . . . . . 8  |-  ( sup ( { z  e.  RR  |  seq 0
(  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
w ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  } ,  RR* ,  <  )  e.  RR  <->  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR )
106104oveq2i 6301 . . . . . . . . 9  |-  ( ( abs `  x )  +  sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  =  ( ( abs `  x )  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )
107106oveq1i 6300 . . . . . . . 8  |-  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 )  =  ( ( ( abs `  x )  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 )
108105, 107, 92ifbieq12i 3907 . . . . . . 7  |-  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) )  =  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) )
109108oveq2i 6301 . . . . . 6  |-  ( ( abs `  x )  +  if ( sup ( { z  e.  RR  |  seq 0
(  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
w ^ k ) ) ) ) `  z ) )  e. 
dom 
~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )  =  ( ( abs `  x
)  +  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )
110109oveq1i 6300 . . . . 5  |-  ( ( ( abs `  x
)  +  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )  /  2
)  =  ( ( ( abs `  x
)  +  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )  /  2
)
111110oveq2i 6301 . . . 4  |-  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  x )  +  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( w  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( w ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )  /  2
) )  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  x )  +  if ( sup ( { z  e.  RR  |  seq 0 (  +  , 
( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  e.  RR ,  ( ( ( abs `  x
)  +  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  ) )  /  2 ) ,  ( ( abs `  x
)  +  1 ) ) )  /  2
) )
1121, 49, 63, 80, 3, 93, 111pserdv2 23385 . . 3  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) ) )
113 cnvimass 5188 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
1143, 113eqsstri 3462 . . . . . . 7  |-  D  C_  dom  abs
115 absf 13400 . . . . . . . 8  |-  abs : CC
--> RR
116115fdmi 5734 . . . . . . 7  |-  dom  abs  =  CC
117114, 116sseqtri 3464 . . . . . 6  |-  D  C_  CC
118117sseli 3428 . . . . 5  |-  ( y  e.  D  ->  y  e.  CC )
119 binomcxplem.e . . . . . . . . . 10  |-  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
120119a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  E  =  ( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) ) )
121 simplr 762 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  b  =  y )
122121oveq1d 6305 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  ( b ^ ( k  - 
1 ) )  =  ( y ^ (
k  -  1 ) ) )
123122oveq2d 6306 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  ( ( k  x.  ( F `
 k ) )  x.  ( b ^
( k  -  1 ) ) )  =  ( ( k  x.  ( F `  k
) )  x.  (
y ^ ( k  -  1 ) ) ) )
124123mpteq2dva 4489 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  CC )  /\  b  =  y )  -> 
( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k )
)  x.  ( y ^ ( k  - 
1 ) ) ) ) )
125 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  y  e.  CC )
126 nnex 10615 . . . . . . . . . . 11  |-  NN  e.  _V
127126mptex 6136 . . . . . . . . . 10  |-  ( k  e.  NN  |->  ( ( k  x.  ( F `
 k ) )  x.  ( y ^
( k  -  1 ) ) ) )  e.  _V
128127a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  ( k  e.  NN  |->  ( ( k  x.  ( F `
 k ) )  x.  ( y ^
( k  -  1 ) ) ) )  e.  _V )
129120, 124, 125, 128fvmptd 5954 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( E `
 y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
y ^ ( k  -  1 ) ) ) ) )
130129adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  ( E `  y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k ) )  x.  ( y ^ (
k  -  1 ) ) ) ) )
131 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  k  =  n )
132131fveq2d 5869 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( F `  k )  =  ( F `  n ) )
133131, 132oveq12d 6308 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  x.  ( F `  k
) )  =  ( n  x.  ( F `
 n ) ) )
134131oveq1d 6305 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  -  1 )  =  ( n  -  1 ) )
135134oveq2d 6306 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( y ^ ( k  - 
1 ) )  =  ( y ^ (
n  -  1 ) ) )
136133, 135oveq12d 6308 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( (
k  x.  ( F `
 k ) )  x.  ( y ^
( k  -  1 ) ) )  =  ( ( n  x.  ( F `  n
) )  x.  (
y ^ ( n  -  1 ) ) ) )
137 simpr 463 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  n  e.  NN )
138 ovex 6318 . . . . . . . 8  |-  ( ( n  x.  ( F `
 n ) )  x.  ( y ^
( n  -  1 ) ) )  e. 
_V
139138a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  (
( n  x.  ( F `  n )
)  x.  ( y ^ ( n  - 
1 ) ) )  e.  _V )
140130, 136, 137, 139fvmptd 5954 . . . . . 6  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  (
( E `  y
) `  n )  =  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) )
141140sumeq2dv 13769 . . . . 5  |-  ( (
ph  /\  y  e.  CC )  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ n  e.  NN  ( ( n  x.  ( F `
 n ) )  x.  ( y ^
( n  -  1 ) ) ) )
142118, 141sylan2 477 . . . 4  |-  ( (
ph  /\  y  e.  D )  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ n  e.  NN  ( ( n  x.  ( F `
 n ) )  x.  ( y ^
( n  -  1 ) ) ) )
143142mpteq2dva 4489 . . 3  |-  ( ph  ->  ( y  e.  D  |-> 
sum_ n  e.  NN  ( ( E `  y ) `  n
) )  =  ( y  e.  D  |->  sum_
n  e.  NN  (
( n  x.  ( F `  n )
)  x.  ( y ^ ( n  - 
1 ) ) ) ) )
144112, 143eqtr4d 2488 . 2  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) ) )
145 nfcv 2592 . . . 4  |-  F/_ b NN
146 nfmpt1 4492 . . . . . . 7  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
147119, 146nfcxfr 2590 . . . . . 6  |-  F/_ b E
148147, 27nffv 5872 . . . . 5  |-  F/_ b
( E `  y
)
149 nfcv 2592 . . . . 5  |-  F/_ b
n
150148, 149nffv 5872 . . . 4  |-  F/_ b
( ( E `  y ) `  n
)
151145, 150nfsum 13757 . . 3  |-  F/_ b sum_ n  e.  NN  (
( E `  y
) `  n )
152 nfcv 2592 . . 3  |-  F/_ y sum_ k  e.  NN  (
( E `  b
) `  k )
153 simpl 459 . . . . . . 7  |-  ( ( y  =  b  /\  n  e.  NN )  ->  y  =  b )
154153fveq2d 5869 . . . . . 6  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( E `  y
)  =  ( E `
 b ) )
155154fveq1d 5867 . . . . 5  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( ( E `  y ) `  n
)  =  ( ( E `  b ) `
 n ) )
156155sumeq2dv 13769 . . . 4  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ n  e.  NN  ( ( E `  b ) `
 n ) )
157 nfmpt1 4492 . . . . . . . . 9  |-  F/_ k
( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) )
15837, 157nfmpt 4491 . . . . . . . 8  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
159119, 158nfcxfr 2590 . . . . . . 7  |-  F/_ k E
160 nfcv 2592 . . . . . . 7  |-  F/_ k
b
161159, 160nffv 5872 . . . . . 6  |-  F/_ k
( E `  b
)
162 nfcv 2592 . . . . . 6  |-  F/_ k
n
163161, 162nffv 5872 . . . . 5  |-  F/_ k
( ( E `  b ) `  n
)
164 nfcv 2592 . . . . 5  |-  F/_ n
( ( E `  b ) `  k
)
165 fveq2 5865 . . . . 5  |-  ( n  =  k  ->  (
( E `  b
) `  n )  =  ( ( E `
 b ) `  k ) )
166163, 164, 165cbvsumi 13763 . . . 4  |-  sum_ n  e.  NN  ( ( E `
 b ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
167156, 166syl6eq 2501 . . 3  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
)
16824, 23, 151, 152, 167cbvmptf 4493 . 2  |-  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) )  =  ( b  e.  D  |-> 
sum_ k  e.  NN  ( ( E `  b ) `  k
) )
169144, 168syl6eq 2501 1  |-  ( ph  ->  ( CC  _D  P
)  =  ( b  e.  D  |->  sum_ k  e.  NN  ( ( E `
 b ) `  k ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   {crab 2741   _Vcvv 3045   ifcif 3881   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   dom cdm 4834   "cima 4837    o. ccom 4838   ` cfv 5582  (class class class)co 6290   supcsup 7954   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   RR*cxr 9674    < clt 9675    - cmin 9860    / cdiv 10269   NNcn 10609   2c2 10659   NN0cn0 10869   RR+crp 11302   [,)cico 11637    seqcseq 12213   ^cexp 12272   abscabs 13297    ~~> cli 13548   sum_csu 13752   ballcbl 18957    _D cdv 22818  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-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-fac 12460  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-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-ulm 23332  df-bcc 36686
This theorem is referenced by:  binomcxplemnotnn0  36705
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