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Theorem binomcxplemdvsum 36774
Description: Lemma for binomcxp 36776. The derivative of the generalized sum in binomcxplemnn0 36768. 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 2612 . . . . . . . 8  |-  F/_ b `' abs
5 nfcv 2612 . . . . . . . . 9  |-  F/_ b
0
6 nfcv 2612 . . . . . . . . 9  |-  F/_ b [,)
7 binomcxplem.r . . . . . . . . . 10  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
8 nfcv 2612 . . . . . . . . . . . . . 14  |-  F/_ b  +
9 nfmpt1 4485 . . . . . . . . . . . . . . . 16  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
101, 9nfcxfr 2610 . . . . . . . . . . . . . . 15  |-  F/_ b S
11 nfcv 2612 . . . . . . . . . . . . . . 15  |-  F/_ b
r
1210, 11nffv 5886 . . . . . . . . . . . . . 14  |-  F/_ b
( S `  r
)
135, 8, 12nfseq 12261 . . . . . . . . . . . . 13  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1413nfel1 2626 . . . . . . . . . . . 12  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
15 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ b RR
1614, 15nfrab 2958 . . . . . . . . . . 11  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
17 nfcv 2612 . . . . . . . . . . 11  |-  F/_ b RR*
18 nfcv 2612 . . . . . . . . . . 11  |-  F/_ b  <
1916, 17, 18nfsup 7983 . . . . . . . . . 10  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
207, 19nfcxfr 2610 . . . . . . . . 9  |-  F/_ b R
215, 6, 20nfov 6334 . . . . . . . 8  |-  F/_ b
( 0 [,) R
)
224, 21nfima 5182 . . . . . . 7  |-  F/_ b
( `' abs " (
0 [,) R ) )
233, 22nfcxfr 2610 . . . . . 6  |-  F/_ b D
24 nfcv 2612 . . . . . 6  |-  F/_ y D
25 nfcv 2612 . . . . . 6  |-  F/_ y sum_ k  e.  NN0  (
( S `  b
) `  k )
26 nfcv 2612 . . . . . . 7  |-  F/_ b NN0
27 nfcv 2612 . . . . . . . . 9  |-  F/_ b
y
2810, 27nffv 5886 . . . . . . . 8  |-  F/_ b
( S `  y
)
29 nfcv 2612 . . . . . . . 8  |-  F/_ b
m
3028, 29nffv 5886 . . . . . . 7  |-  F/_ b
( ( S `  y ) `  m
)
3126, 30nfsum 13834 . . . . . 6  |-  F/_ b sum_ m  e.  NN0  (
( S `  y
) `  m )
32 simpl 464 . . . . . . . . . 10  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
b  =  y )
3332fveq2d 5883 . . . . . . . . 9  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( S `  b
)  =  ( S `
 y ) )
3433fveq1d 5881 . . . . . . . 8  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( ( S `  b ) `  k
)  =  ( ( S `  y ) `
 k ) )
3534sumeq2dv 13846 . . . . . . 7  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ k  e.  NN0  ( ( S `  y ) `
 k ) )
36 nfcv 2612 . . . . . . . 8  |-  F/_ m
( ( S `  y ) `  k
)
37 nfcv 2612 . . . . . . . . . . . 12  |-  F/_ k CC
38 nfmpt1 4485 . . . . . . . . . . . 12  |-  F/_ k
( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) )
3937, 38nfmpt 4484 . . . . . . . . . . 11  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
401, 39nfcxfr 2610 . . . . . . . . . 10  |-  F/_ k S
41 nfcv 2612 . . . . . . . . . 10  |-  F/_ k
y
4240, 41nffv 5886 . . . . . . . . 9  |-  F/_ k
( S `  y
)
43 nfcv 2612 . . . . . . . . 9  |-  F/_ k
m
4442, 43nffv 5886 . . . . . . . 8  |-  F/_ k
( ( S `  y ) `  m
)
45 fveq2 5879 . . . . . . . 8  |-  ( k  =  m  ->  (
( S `  y
) `  k )  =  ( ( S `
 y ) `  m ) )
4636, 44, 45cbvsumi 13840 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( S `
 y ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m )
4735, 46syl6eq 2521 . . . . . 6  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m ) )
4823, 24, 25, 31, 47cbvmptf 4486 . . . . 5  |-  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `
 b ) `  k ) )  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
492, 48eqtri 2493 . . . 4  |-  P  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
50 ovex 6336 . . . . . 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 468 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
j  =  k )
5655oveq2d 6324 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
( CC𝑐 j )  =  ( CC𝑐 k ) )
57 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
58 binomcxp.c . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
5958adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  CC )
6059, 57bcccl 36758 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 k
)  e.  CC )
6154, 56, 57, 60fvmptd 5969 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( CC𝑐 k ) )
6261, 60eqeltrd 2549 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
6351, 53, 62fmpt2d 6069 . . . 4  |-  ( ph  ->  F : NN0 --> CC )
64 nfcv 2612 . . . . . . 7  |-  F/_ r RR
65 nfcv 2612 . . . . . . 7  |-  F/_ z RR
66 nfv 1769 . . . . . . 7  |-  F/ z  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
67 nfcv 2612 . . . . . . . . 9  |-  F/_ r
0
68 nfcv 2612 . . . . . . . . 9  |-  F/_ r  +
69 nfcv 2612 . . . . . . . . . . 11  |-  F/_ r
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
701, 69nfcxfr 2610 . . . . . . . . . 10  |-  F/_ r S
71 nfcv 2612 . . . . . . . . . 10  |-  F/_ r
z
7270, 71nffv 5886 . . . . . . . . 9  |-  F/_ r
( S `  z
)
7367, 68, 72nfseq 12261 . . . . . . . 8  |-  F/_ r  seq 0 (  +  , 
( S `  z
) )
7473nfel1 2626 . . . . . . 7  |-  F/ r  seq 0 (  +  ,  ( S `  z ) )  e. 
dom 
~~>
75 fveq2 5879 . . . . . . . . 9  |-  ( r  =  z  ->  ( S `  r )  =  ( S `  z ) )
7675seqeq3d 12259 . . . . . . . 8  |-  ( r  =  z  ->  seq 0 (  +  , 
( S `  r
) )  =  seq 0 (  +  , 
( S `  z
) ) )
7776eleq1d 2533 . . . . . . 7  |-  ( r  =  z  ->  (  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  ) )
7864, 65, 66, 74, 77cbvrab 3029 . . . . . 6  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  }  =  { z  e.  RR  |  seq 0
(  +  ,  ( S `  z ) )  e.  dom  ~~>  }
7978supeq1i 7979 . . . . 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 2493 . . . 4  |-  R  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  } ,  RR* ,  <  )
811fveq1i 5880 . . . . . . . . . . . 12  |-  ( S `
 z )  =  ( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z )
82 seqeq3 12256 . . . . . . . . . . . 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 2540 . . . . . . . . . 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 3019 . . . . . . . 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 7979 . . . . . . 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 2498 . . . . . 6  |-  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  =  R
8988eleq1i 2540 . . . . 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 6319 . . . . . 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 6318 . . . . 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 2471 . . . . 5  |-  ( ( abs `  x )  +  1 )  =  ( ( abs `  x
)  +  1 )
9389, 91, 92ifbieq12i 3898 . . . 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 6315 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  b  ->  (
w ^ k )  =  ( b ^
k ) )
9594oveq2d 6324 . . . . . . . . . . . . . . . . 17  |-  ( w  =  b  ->  (
( F `  k
)  x.  ( w ^ k ) )  =  ( ( F `
 k )  x.  ( b ^ k
) ) )
9695mpteq2dv 4483 . . . . . . . . . . . . . . . 16  |-  ( w  =  b  ->  (
k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) )  =  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) )
9796cbvmptv 4488 . . . . . . . . . . . . . . 15  |-  ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) )  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) )
9897fveq1i 5880 . . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . . 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 2540 . . . . . . . . . . . 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 3019 . . . . . . . . . 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 7979 . . . . . . . . 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 2540 . . . . . . . 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 6319 . . . . . . . . 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 6318 . . . . . . . 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 3898 . . . . . . 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 6319 . . . . . 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 6318 . . . . 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 6319 . . . 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 23464 . . 3  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) ) )
113 cnvimass 5194 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
1143, 113eqsstri 3448 . . . . . . 7  |-  D  C_  dom  abs
115 absf 13477 . . . . . . . 8  |-  abs : CC
--> RR
116115fdmi 5746 . . . . . . 7  |-  dom  abs  =  CC
117114, 116sseqtri 3450 . . . . . 6  |-  D  C_  CC
118117sseli 3414 . . . . 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 770 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  b  =  y )
122121oveq1d 6323 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  ( b ^ ( k  - 
1 ) )  =  ( y ^ (
k  -  1 ) ) )
123122oveq2d 6324 . . . . . . . . . 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 4482 . . . . . . . . 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 468 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  y  e.  CC )
126 nnex 10637 . . . . . . . . . . 11  |-  NN  e.  _V
127126mptex 6152 . . . . . . . . . 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 5969 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( E `
 y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
y ^ ( k  -  1 ) ) ) ) )
130129adantr 472 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  ( E `  y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k ) )  x.  ( y ^ (
k  -  1 ) ) ) ) )
131 simpr 468 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  k  =  n )
132131fveq2d 5883 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( F `  k )  =  ( F `  n ) )
133131, 132oveq12d 6326 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  x.  ( F `  k
) )  =  ( n  x.  ( F `
 n ) ) )
134131oveq1d 6323 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  -  1 )  =  ( n  -  1 ) )
135134oveq2d 6324 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( y ^ ( k  - 
1 ) )  =  ( y ^ (
n  -  1 ) ) )
136133, 135oveq12d 6326 . . . . . . 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 468 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  n  e.  NN )
138 ovex 6336 . . . . . . . 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 5969 . . . . . 6  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  (
( E `  y
) `  n )  =  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) )
141140sumeq2dv 13846 . . . . 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 482 . . . 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 4482 . . 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 2508 . 2  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) ) )
145 nfcv 2612 . . . 4  |-  F/_ b NN
146 nfmpt1 4485 . . . . . . 7  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
147119, 146nfcxfr 2610 . . . . . 6  |-  F/_ b E
148147, 27nffv 5886 . . . . 5  |-  F/_ b
( E `  y
)
149 nfcv 2612 . . . . 5  |-  F/_ b
n
150148, 149nffv 5886 . . . 4  |-  F/_ b
( ( E `  y ) `  n
)
151145, 150nfsum 13834 . . 3  |-  F/_ b sum_ n  e.  NN  (
( E `  y
) `  n )
152 nfcv 2612 . . 3  |-  F/_ y sum_ k  e.  NN  (
( E `  b
) `  k )
153 simpl 464 . . . . . . 7  |-  ( ( y  =  b  /\  n  e.  NN )  ->  y  =  b )
154153fveq2d 5883 . . . . . 6  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( E `  y
)  =  ( E `
 b ) )
155154fveq1d 5881 . . . . 5  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( ( E `  y ) `  n
)  =  ( ( E `  b ) `
 n ) )
156155sumeq2dv 13846 . . . 4  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ n  e.  NN  ( ( E `  b ) `
 n ) )
157 nfmpt1 4485 . . . . . . . . 9  |-  F/_ k
( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) )
15837, 157nfmpt 4484 . . . . . . . 8  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
159119, 158nfcxfr 2610 . . . . . . 7  |-  F/_ k E
160 nfcv 2612 . . . . . . 7  |-  F/_ k
b
161159, 160nffv 5886 . . . . . 6  |-  F/_ k
( E `  b
)
162 nfcv 2612 . . . . . 6  |-  F/_ k
n
163161, 162nffv 5886 . . . . 5  |-  F/_ k
( ( E `  b ) `  n
)
164 nfcv 2612 . . . . 5  |-  F/_ n
( ( E `  b ) `  k
)
165 fveq2 5879 . . . . 5  |-  ( n  =  k  ->  (
( E `  b
) `  n )  =  ( ( E `
 b ) `  k ) )
166163, 164, 165cbvsumi 13840 . . . 4  |-  sum_ n  e.  NN  ( ( E `
 b ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
167156, 166syl6eq 2521 . . 3  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
)
16824, 23, 151, 152, 167cbvmptf 4486 . 2  |-  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) )  =  ( b  e.  D  |-> 
sum_ k  e.  NN  ( ( E `  b ) `  k
) )
169144, 168syl6eq 2521 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   "cima 4842    o. ccom 4843   ` cfv 5589  (class class class)co 6308   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   RR*cxr 9692    < clt 9693    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   RR+crp 11325   [,)cico 11662    seqcseq 12251   ^cexp 12310   abscabs 13374    ~~> cli 13625   sum_csu 13829   ballcbl 19034    _D cdv 22897  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-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  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-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-ulm 23411  df-bcc 36756
This theorem is referenced by:  binomcxplemnotnn0  36775
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