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Theorem binomcxplemdvsum 36088
Description: Lemma for binomcxp 36090. The derivative of the generalized sum in binomcxplemnn0 36082. 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 2564 . . . . . . . 8  |-  F/_ b `' abs
5 nfcv 2564 . . . . . . . . 9  |-  F/_ b
0
6 nfcv 2564 . . . . . . . . 9  |-  F/_ b [,)
7 binomcxplem.r . . . . . . . . . 10  |-  R  =  sup ( { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
8 nfcv 2564 . . . . . . . . . . . . . 14  |-  F/_ b  +
9 nfmpt1 4483 . . . . . . . . . . . . . . . 16  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
101, 9nfcxfr 2562 . . . . . . . . . . . . . . 15  |-  F/_ b S
11 nfcv 2564 . . . . . . . . . . . . . . 15  |-  F/_ b
r
1210, 11nffv 5855 . . . . . . . . . . . . . 14  |-  F/_ b
( S `  r
)
135, 8, 12nfseq 12159 . . . . . . . . . . . . 13  |-  F/_ b  seq 0 (  +  , 
( S `  r
) )
1413nfel1 2580 . . . . . . . . . . . 12  |-  F/ b  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
15 nfcv 2564 . . . . . . . . . . . 12  |-  F/_ b RR
1614, 15nfrab 2988 . . . . . . . . . . 11  |-  F/_ b { r  e.  RR  |  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>  }
17 nfcv 2564 . . . . . . . . . . 11  |-  F/_ b RR*
18 nfcv 2564 . . . . . . . . . . 11  |-  F/_ b  <
1916, 17, 18nfsup 7943 . . . . . . . . . 10  |-  F/_ b sup ( { r  e.  RR  |  seq 0
(  +  ,  ( S `  r ) )  e.  dom  ~~>  } ,  RR* ,  <  )
207, 19nfcxfr 2562 . . . . . . . . 9  |-  F/_ b R
215, 6, 20nfov 6303 . . . . . . . 8  |-  F/_ b
( 0 [,) R
)
224, 21nfima 5164 . . . . . . 7  |-  F/_ b
( `' abs " (
0 [,) R ) )
233, 22nfcxfr 2562 . . . . . 6  |-  F/_ b D
24 nfcv 2564 . . . . . 6  |-  F/_ y D
25 nfcv 2564 . . . . . 6  |-  F/_ y sum_ k  e.  NN0  (
( S `  b
) `  k )
26 nfcv 2564 . . . . . . 7  |-  F/_ b NN0
27 nfcv 2564 . . . . . . . . 9  |-  F/_ b
y
2810, 27nffv 5855 . . . . . . . 8  |-  F/_ b
( S `  y
)
29 nfcv 2564 . . . . . . . 8  |-  F/_ b
m
3028, 29nffv 5855 . . . . . . 7  |-  F/_ b
( ( S `  y ) `  m
)
3126, 30nfsum 13660 . . . . . 6  |-  F/_ b sum_ m  e.  NN0  (
( S `  y
) `  m )
32 simpl 455 . . . . . . . . . 10  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
b  =  y )
3332fveq2d 5852 . . . . . . . . 9  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( S `  b
)  =  ( S `
 y ) )
3433fveq1d 5850 . . . . . . . 8  |-  ( ( b  =  y  /\  k  e.  NN0 )  -> 
( ( S `  b ) `  k
)  =  ( ( S `  y ) `
 k ) )
3534sumeq2dv 13672 . . . . . . 7  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ k  e.  NN0  ( ( S `  y ) `
 k ) )
36 nfcv 2564 . . . . . . . 8  |-  F/_ m
( ( S `  y ) `  k
)
37 nfcv 2564 . . . . . . . . . . . 12  |-  F/_ k CC
38 nfmpt1 4483 . . . . . . . . . . . 12  |-  F/_ k
( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) )
3937, 38nfmpt 4482 . . . . . . . . . . 11  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
401, 39nfcxfr 2562 . . . . . . . . . 10  |-  F/_ k S
41 nfcv 2564 . . . . . . . . . 10  |-  F/_ k
y
4240, 41nffv 5855 . . . . . . . . 9  |-  F/_ k
( S `  y
)
43 nfcv 2564 . . . . . . . . 9  |-  F/_ k
m
4442, 43nffv 5855 . . . . . . . 8  |-  F/_ k
( ( S `  y ) `  m
)
45 fveq2 5848 . . . . . . . 8  |-  ( k  =  m  ->  (
( S `  y
) `  k )  =  ( ( S `
 y ) `  m ) )
4636, 44, 45cbvsumi 13666 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( S `
 y ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m )
4735, 46syl6eq 2459 . . . . . 6  |-  ( b  =  y  ->  sum_ k  e.  NN0  ( ( S `
 b ) `  k )  =  sum_ m  e.  NN0  ( ( S `  y ) `  m ) )
4823, 24, 25, 31, 47cbvmptf 4484 . . . . 5  |-  ( b  e.  D  |->  sum_ k  e.  NN0  ( ( S `
 b ) `  k ) )  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
492, 48eqtri 2431 . . . 4  |-  P  =  ( y  e.  D  |-> 
sum_ m  e.  NN0  ( ( S `  y ) `  m
) )
50 ovex 6305 . . . . . 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 459 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
j  =  k )
5655oveq2d 6293 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  NN0 )  /\  j  =  k )  -> 
( CC𝑐 j )  =  ( CC𝑐 k ) )
57 simpr 459 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  k  e.  NN0 )
58 binomcxp.c . . . . . . . . 9  |-  ( ph  ->  C  e.  CC )
5958adantr 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN0 )  ->  C  e.  CC )
6059, 57bcccl 36072 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( CC𝑐 k
)  e.  CC )
6154, 56, 57, 60fvmptd 5937 . . . . . 6  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  =  ( CC𝑐 k ) )
6261, 60eqeltrd 2490 . . . . 5  |-  ( (
ph  /\  k  e.  NN0 )  ->  ( F `  k )  e.  CC )
6351, 53, 62fmpt2d 6039 . . . 4  |-  ( ph  ->  F : NN0 --> CC )
64 nfcv 2564 . . . . . . 7  |-  F/_ r RR
65 nfcv 2564 . . . . . . 7  |-  F/_ z RR
66 nfv 1728 . . . . . . 7  |-  F/ z  seq 0 (  +  ,  ( S `  r ) )  e. 
dom 
~~>
67 nfcv 2564 . . . . . . . . 9  |-  F/_ r
0
68 nfcv 2564 . . . . . . . . 9  |-  F/_ r  +
69 nfcv 2564 . . . . . . . . . . 11  |-  F/_ r
( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  (
b ^ k ) ) ) )
701, 69nfcxfr 2562 . . . . . . . . . 10  |-  F/_ r S
71 nfcv 2564 . . . . . . . . . 10  |-  F/_ r
z
7270, 71nffv 5855 . . . . . . . . 9  |-  F/_ r
( S `  z
)
7367, 68, 72nfseq 12159 . . . . . . . 8  |-  F/_ r  seq 0 (  +  , 
( S `  z
) )
7473nfel1 2580 . . . . . . 7  |-  F/ r  seq 0 (  +  ,  ( S `  z ) )  e. 
dom 
~~>
75 fveq2 5848 . . . . . . . . 9  |-  ( r  =  z  ->  ( S `  r )  =  ( S `  z ) )
7675seqeq3d 12157 . . . . . . . 8  |-  ( r  =  z  ->  seq 0 (  +  , 
( S `  r
) )  =  seq 0 (  +  , 
( S `  z
) ) )
7776eleq1d 2471 . . . . . . 7  |-  ( r  =  z  ->  (  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  <->  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  ) )
7864, 65, 66, 74, 77cbvrab 3056 . . . . . 6  |-  { r  e.  RR  |  seq 0 (  +  , 
( S `  r
) )  e.  dom  ~~>  }  =  { z  e.  RR  |  seq 0
(  +  ,  ( S `  z ) )  e.  dom  ~~>  }
7978supeq1i 7939 . . . . 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 2431 . . . 4  |-  R  =  sup ( { z  e.  RR  |  seq 0 (  +  , 
( S `  z
) )  e.  dom  ~~>  } ,  RR* ,  <  )
811fveq1i 5849 . . . . . . . . . . . 12  |-  ( S `
 z )  =  ( ( b  e.  CC  |->  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) ) `
 z )
82 seqeq3 12154 . . . . . . . . . . . 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 2479 . . . . . . . . . 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 3047 . . . . . . . 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 7939 . . . . . . 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 2436 . . . . . 6  |-  sup ( { z  e.  RR  |  seq 0 (  +  ,  ( ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) ) `  z ) )  e.  dom  ~~>  } ,  RR* ,  <  )  =  R
8988eleq1i 2479 . . . . 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 6288 . . . . . 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 6287 . . . . 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 2402 . . . . 5  |-  ( ( abs `  x )  +  1 )  =  ( ( abs `  x
)  +  1 )
9389, 91, 92ifbieq12i 3910 . . . 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 6284 . . . . . . . . . . . . . . . . . 18  |-  ( w  =  b  ->  (
w ^ k )  =  ( b ^
k ) )
9594oveq2d 6293 . . . . . . . . . . . . . . . . 17  |-  ( w  =  b  ->  (
( F `  k
)  x.  ( w ^ k ) )  =  ( ( F `
 k )  x.  ( b ^ k
) ) )
9695mpteq2dv 4481 . . . . . . . . . . . . . . . 16  |-  ( w  =  b  ->  (
k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) )  =  ( k  e. 
NN0  |->  ( ( F `
 k )  x.  ( b ^ k
) ) ) )
9796cbvmptv 4486 . . . . . . . . . . . . . . 15  |-  ( w  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( w ^
k ) ) ) )  =  ( b  e.  CC  |->  ( k  e.  NN0  |->  ( ( F `  k )  x.  ( b ^
k ) ) ) )
9897fveq1i 5849 . . . . . . . . . . . . . 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 12154 . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . 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 3047 . . . . . . . . . 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 7939 . . . . . . . . 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 2479 . . . . . . . 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 6288 . . . . . . . . 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 6287 . . . . . . . 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 3910 . . . . . . 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 6288 . . . . . 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 6287 . . . . 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 6288 . . . 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 23115 . . 3  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) ) )
113 cnvimass 5176 . . . . . . . 8  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
1143, 113eqsstri 3471 . . . . . . 7  |-  D  C_  dom  abs
115 absf 13317 . . . . . . . 8  |-  abs : CC
--> RR
116115fdmi 5718 . . . . . . 7  |-  dom  abs  =  CC
117114, 116sseqtri 3473 . . . . . 6  |-  D  C_  CC
118117sseli 3437 . . . . 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 754 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  b  =  y )
122121oveq1d 6292 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  b  =  y
)  /\  k  e.  NN )  ->  ( b ^ ( k  - 
1 ) )  =  ( y ^ (
k  -  1 ) ) )
123122oveq2d 6293 . . . . . . . . . 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 4480 . . . . . . . . 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 459 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  CC )  ->  y  e.  CC )
126 nnex 10581 . . . . . . . . . . 11  |-  NN  e.  _V
127126mptex 6123 . . . . . . . . . 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 5937 . . . . . . . 8  |-  ( (
ph  /\  y  e.  CC )  ->  ( E `
 y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
y ^ ( k  -  1 ) ) ) ) )
130129adantr 463 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  ( E `  y )  =  ( k  e.  NN  |->  ( ( k  x.  ( F `  k ) )  x.  ( y ^ (
k  -  1 ) ) ) ) )
131 simpr 459 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  k  =  n )
132131fveq2d 5852 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( F `  k )  =  ( F `  n ) )
133131, 132oveq12d 6295 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  x.  ( F `  k
) )  =  ( n  x.  ( F `
 n ) ) )
134131oveq1d 6292 . . . . . . . . 9  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( k  -  1 )  =  ( n  -  1 ) )
135134oveq2d 6293 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  /\  k  =  n )  ->  ( y ^ ( k  - 
1 ) )  =  ( y ^ (
n  -  1 ) ) )
136133, 135oveq12d 6295 . . . . . . 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 459 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  n  e.  NN )
138 ovex 6305 . . . . . . . 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 5937 . . . . . 6  |-  ( ( ( ph  /\  y  e.  CC )  /\  n  e.  NN )  ->  (
( E `  y
) `  n )  =  ( ( n  x.  ( F `  n ) )  x.  ( y ^ (
n  -  1 ) ) ) )
141140sumeq2dv 13672 . . . . 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 472 . . . 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 4480 . . 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 2446 . 2  |-  ( ph  ->  ( CC  _D  P
)  =  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) ) )
145 nfcv 2564 . . . 4  |-  F/_ b NN
146 nfmpt1 4483 . . . . . . 7  |-  F/_ b
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
147119, 146nfcxfr 2562 . . . . . 6  |-  F/_ b E
148147, 27nffv 5855 . . . . 5  |-  F/_ b
( E `  y
)
149 nfcv 2564 . . . . 5  |-  F/_ b
n
150148, 149nffv 5855 . . . 4  |-  F/_ b
( ( E `  y ) `  n
)
151145, 150nfsum 13660 . . 3  |-  F/_ b sum_ n  e.  NN  (
( E `  y
) `  n )
152 nfcv 2564 . . 3  |-  F/_ y sum_ k  e.  NN  (
( E `  b
) `  k )
153 simpl 455 . . . . . . 7  |-  ( ( y  =  b  /\  n  e.  NN )  ->  y  =  b )
154153fveq2d 5852 . . . . . 6  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( E `  y
)  =  ( E `
 b ) )
155154fveq1d 5850 . . . . 5  |-  ( ( y  =  b  /\  n  e.  NN )  ->  ( ( E `  y ) `  n
)  =  ( ( E `  b ) `
 n ) )
156155sumeq2dv 13672 . . . 4  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ n  e.  NN  ( ( E `  b ) `
 n ) )
157 nfmpt1 4483 . . . . . . . . 9  |-  F/_ k
( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) )
15837, 157nfmpt 4482 . . . . . . . 8  |-  F/_ k
( b  e.  CC  |->  ( k  e.  NN  |->  ( ( k  x.  ( F `  k
) )  x.  (
b ^ ( k  -  1 ) ) ) ) )
159119, 158nfcxfr 2562 . . . . . . 7  |-  F/_ k E
160 nfcv 2564 . . . . . . 7  |-  F/_ k
b
161159, 160nffv 5855 . . . . . 6  |-  F/_ k
( E `  b
)
162 nfcv 2564 . . . . . 6  |-  F/_ k
n
163161, 162nffv 5855 . . . . 5  |-  F/_ k
( ( E `  b ) `  n
)
164 nfcv 2564 . . . . 5  |-  F/_ n
( ( E `  b ) `  k
)
165 fveq2 5848 . . . . 5  |-  ( n  =  k  ->  (
( E `  b
) `  n )  =  ( ( E `
 b ) `  k ) )
166163, 164, 165cbvsumi 13666 . . . 4  |-  sum_ n  e.  NN  ( ( E `
 b ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
167156, 166syl6eq 2459 . . 3  |-  ( y  =  b  ->  sum_ n  e.  NN  ( ( E `
 y ) `  n )  =  sum_ k  e.  NN  (
( E `  b
) `  k )
)
16824, 23, 151, 152, 167cbvmptf 4484 . 2  |-  ( y  e.  D  |->  sum_ n  e.  NN  ( ( E `
 y ) `  n ) )  =  ( b  e.  D  |-> 
sum_ k  e.  NN  ( ( E `  b ) `  k
) )
169144, 168syl6eq 2459 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 184    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2757   _Vcvv 3058   ifcif 3884   class class class wbr 4394    |-> cmpt 4452   `'ccnv 4821   dom cdm 4822   "cima 4825    o. ccom 4826   ` cfv 5568  (class class class)co 6277   supcsup 7933   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526   RR*cxr 9656    < clt 9657    - cmin 9840    / cdiv 10246   NNcn 10575   2c2 10625   NN0cn0 10835   RR+crp 11264   [,)cico 11583    seqcseq 12149   ^cexp 12208   abscabs 13214    ~~> cli 13454   sum_csu 13655   ballcbl 18723    _D cdv 22557  C𝑐cbcc 36069
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601
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 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6520  df-om 6683  df-1st 6783  df-2nd 6784  df-supp 6902  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-ixp 7507  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-fsupp 7863  df-fi 7904  df-sup 7934  df-oi 7968  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-9 10641  df-10 10642  df-n0 10836  df-z 10905  df-dec 11019  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-ioo 11585  df-ico 11587  df-icc 11588  df-fz 11725  df-fzo 11853  df-fl 11964  df-seq 12150  df-exp 12209  df-fac 12396  df-hash 12451  df-shft 13047  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-limsup 13441  df-clim 13458  df-rlim 13459  df-sum 13656  df-prod 13863  df-fallfac 13950  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-starv 14922  df-sca 14923  df-vsca 14924  df-ip 14925  df-tset 14926  df-ple 14927  df-ds 14929  df-unif 14930  df-hom 14931  df-cco 14932  df-rest 15035  df-topn 15036  df-0g 15054  df-gsum 15055  df-topgen 15056  df-pt 15057  df-prds 15060  df-xrs 15114  df-qtop 15119  df-imas 15120  df-xps 15122  df-mre 15198  df-mrc 15199  df-acs 15201  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-submnd 16289  df-mulg 16382  df-cntz 16677  df-cmn 17122  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-fbas 18734  df-fg 18735  df-cnfld 18739  df-top 19689  df-bases 19691  df-topon 19692  df-topsp 19693  df-cld 19810  df-ntr 19811  df-cls 19812  df-nei 19890  df-lp 19928  df-perf 19929  df-cn 20019  df-cnp 20020  df-haus 20107  df-cmp 20178  df-tx 20353  df-hmeo 20546  df-fil 20637  df-fm 20729  df-flim 20730  df-flf 20731  df-xms 21113  df-ms 21114  df-tms 21115  df-cncf 21672  df-limc 22560  df-dv 22561  df-ulm 23062  df-bcc 36070
This theorem is referenced by:  binomcxplemnotnn0  36089
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