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Theorem pserdv 19805
Description: The derivative of a power series on its region of convergence. (Contributed by Mario Carneiro, 31-Mar-2015.)
Hypotheses
Ref Expression
pserf.g  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
pserf.f  |-  F  =  ( y  e.  S  |-> 
sum_ j  e.  NN0  ( ( G `  y ) `  j
) )
pserf.a  |-  ( ph  ->  A : NN0 --> CC )
pserf.r  |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
psercn.s  |-  S  =  ( `' abs " (
0 [,) R ) )
psercn.m  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
pserdv.b  |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) )
Assertion
Ref Expression
pserdv  |-  ( ph  ->  ( CC  _D  F
)  =  ( y  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) ) )
Distinct variable groups:    j, a,
k, n, r, x, y, A    j, M, k, y    B, j, k, x, y    j, G, k, r, y    S, a, j, k, y    F, a    ph, a, j, k, y
Allowed substitution hints:    ph( x, n, r)    B( n, r, a)    R( x, y, j, k, n, r, a)    S( x, n, r)    F( x, y, j, k, n, r)    G( x, n, a)    M( x, n, r, a)

Proof of Theorem pserdv
StepHypRef Expression
1 dvfcn 19258 . . . . 5  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3197 . . . . . . . . 9  |-  CC  C_  CC
32a1i 10 . . . . . . . 8  |-  ( ph  ->  CC  C_  CC )
4 pserf.g . . . . . . . . . 10  |-  G  =  ( x  e.  CC  |->  ( n  e.  NN0  |->  ( ( A `  n )  x.  (
x ^ n ) ) ) )
5 pserf.f . . . . . . . . . 10  |-  F  =  ( y  e.  S  |-> 
sum_ j  e.  NN0  ( ( G `  y ) `  j
) )
6 pserf.a . . . . . . . . . 10  |-  ( ph  ->  A : NN0 --> CC )
7 pserf.r . . . . . . . . . 10  |-  R  =  sup ( { r  e.  RR  |  seq  0 (  +  , 
( G `  r
) )  e.  dom  ~~>  } ,  RR* ,  <  )
8 psercn.s . . . . . . . . . 10  |-  S  =  ( `' abs " (
0 [,) R ) )
9 psercn.m . . . . . . . . . 10  |-  M  =  if ( R  e.  RR ,  ( ( ( abs `  a
)  +  R )  /  2 ) ,  ( ( abs `  a
)  +  1 ) )
104, 5, 6, 7, 8, 9psercn 19802 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S
-cn-> CC ) )
11 cncff 18397 . . . . . . . . 9  |-  ( F  e.  ( S -cn-> CC )  ->  F : S
--> CC )
1210, 11syl 15 . . . . . . . 8  |-  ( ph  ->  F : S --> CC )
13 cnvimass 5033 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
14 absf 11821 . . . . . . . . . . . 12  |-  abs : CC
--> RR
1514fdmi 5394 . . . . . . . . . . 11  |-  dom  abs  =  CC
1613, 15sseqtri 3210 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
178, 16eqsstri 3208 . . . . . . . . 9  |-  S  C_  CC
1817a1i 10 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
193, 12, 18dvbss 19251 . . . . . . 7  |-  ( ph  ->  dom  ( CC  _D  F )  C_  S
)
202a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  CC  C_  CC )
2112adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  F : S --> CC )
2217a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  S  C_  CC )
23 pserdv.b . . . . . . . . . . . . . 14  |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) )
24 cnxmet 18282 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
2524a1i 10 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  ( abs  o.  -  )  e.  ( * Met `  CC ) )
26 0cn 8831 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
2726a1i 10 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  CC )
2818sselda 3180 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
2928abscld 11918 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
304, 5, 6, 7, 8, 9psercnlem1 19801 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
3130simp1d 967 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
3231rpred 10390 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
3329, 32readdcld 8862 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
34 0re 8838 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
3534a1i 10 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
3628absge0d 11926 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
3729, 31ltaddrpd 10419 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
3835, 29, 33, 36, 37lelttrd 8974 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
3933, 38elrpd 10388 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
4039rphalfcld 10402 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
4140rpxrd 10391 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
42 blssm 17968 . . . . . . . . . . . . . . 15  |-  ( ( ( abs  o.  -  )  e.  ( * Met `  CC )  /\  0  e.  CC  /\  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4325, 27, 41, 42syl3anc 1182 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4423, 43syl5eqss 3222 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  B  C_  CC )
45 eqid 2283 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4645cnfldtop 18293 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  Top
4745cnfldtopon 18292 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4847toponunii 16670 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
4948restid 13338 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
5046, 49ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
5150eqcomi 2287 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
5245, 51dvres 19261 . . . . . . . . . . . . 13  |-  ( ( ( CC  C_  CC  /\  F : S --> CC )  /\  ( S  C_  CC  /\  B  C_  CC ) )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
5320, 21, 22, 44, 52syl22anc 1183 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
54 resss 4979 . . . . . . . . . . . . 13  |-  ( ( CC  _D  F )  |`  ( ( int `  ( TopOpen
` fld
) ) `  B
) )  C_  ( CC  _D  F )
5554a1i 10 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
)  |`  ( ( int `  ( TopOpen ` fld ) ) `  B
) )  C_  ( CC  _D  F ) )
5653, 55eqsstrd 3212 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
) )
57 dmss 4878 . . . . . . . . . . 11  |-  ( ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
)  ->  dom  ( CC 
_D  ( F  |`  B ) )  C_  dom  ( CC  _D  F
) )
5856, 57syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) ) 
C_  dom  ( CC  _D  F ) )
594, 5, 6, 7, 8, 9pserdvlem1 19803 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( ( abs `  a )  +  M
)  /  2 )  e.  RR+  /\  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 )  /\  ( ( ( abs `  a )  +  M )  / 
2 )  <  R
) )
604, 5, 6, 7, 8, 59psercnlem2 19800 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  (
a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  /\  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  ( `' abs " ( 0 [,] ( ( ( abs `  a )  +  M
)  /  2 ) ) )  /\  ( `' abs " ( 0 [,] ( ( ( abs `  a )  +  M )  / 
2 ) ) ) 
C_  S ) )
6160simp1d 967 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) ) )
6261, 23syl6eleqr 2374 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  B )
634, 5, 6, 7, 8, 9, 23pserdvlem2 19804 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( y  e.  B  |-> 
sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) ) )
6463dmeqd 4881 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  dom  ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) ) )
65 dmmptg 5170 . . . . . . . . . . . . 13  |-  ( A. y  e.  B  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  x.  ( y ^
k ) )  e. 
_V  ->  dom  ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )  =  B )
66 sumex 12160 . . . . . . . . . . . . . 14  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) )  e.  _V
6766a1i 10 . . . . . . . . . . . . 13  |-  ( y  e.  B  ->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) )  e.  _V )
6865, 67mprg 2612 . . . . . . . . . . . 12  |-  dom  (
y  e.  B  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) ) )  =  B
6964, 68syl6eq 2331 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  B )
7062, 69eleqtrrd 2360 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )
7158, 70sseldd 3181 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  F ) )
7271ex 423 . . . . . . . 8  |-  ( ph  ->  ( a  e.  S  ->  a  e.  dom  ( CC  _D  F ) ) )
7372ssrdv 3185 . . . . . . 7  |-  ( ph  ->  S  C_  dom  ( CC 
_D  F ) )
7419, 73eqssd 3196 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  =  S )
7574feq2d 5380 . . . . 5  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : S --> CC ) )
761, 75mpbii 202 . . . 4  |-  ( ph  ->  ( CC  _D  F
) : S --> CC )
7776feqmptd 5575 . . 3  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  ( ( CC  _D  F ) `
 a ) ) )
78 ffun 5391 . . . . . . 7  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
791, 78mp1i 11 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  Fun  ( CC  _D  F
) )
80 funssfv 5543 . . . . . 6  |-  ( ( Fun  ( CC  _D  F )  /\  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
)  /\  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )  ->  ( ( CC  _D  F ) `  a )  =  ( ( CC  _D  ( F  |`  B ) ) `
 a ) )
8179, 56, 70, 80syl3anc 1182 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  ( ( CC 
_D  ( F  |`  B ) ) `  a ) )
8263fveq1d 5527 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  ( F  |`  B ) ) `
 a )  =  ( ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) ) `  a
) )
83 oveq1 5865 . . . . . . . . 9  |-  ( y  =  a  ->  (
y ^ k )  =  ( a ^
k ) )
8483oveq2d 5874 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8584sumeq2sdv 12177 . . . . . . 7  |-  ( y  =  a  ->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  x.  ( a ^
k ) ) )
86 eqid 2283 . . . . . . 7  |-  ( y  e.  B  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )  =  ( y  e.  B  |-> 
sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) )
87 sumex 12160 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  e.  _V
8885, 86, 87fvmpt 5602 . . . . . 6  |-  ( a  e.  B  ->  (
( y  e.  B  |-> 
sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) ) `  a
)  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8962, 88syl 15 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( y  e.  B  |-> 
sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) ) `  a
)  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
9081, 82, 893eqtrd 2319 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
a ^ k ) ) )
9190mpteq2dva 4106 . . 3  |-  ( ph  ->  ( a  e.  S  |->  ( ( CC  _D  F ) `  a
) )  =  ( a  e.  S  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) ) ) )
9277, 91eqtrd 2315 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) ) )
93 oveq1 5865 . . . . 5  |-  ( a  =  y  ->  (
a ^ k )  =  ( y ^
k ) )
9493oveq2d 5874 . . . 4  |-  ( a  =  y  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )
9594sumeq2sdv 12177 . . 3  |-  ( a  =  y  ->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `
 ( k  +  1 ) ) )  x.  ( y ^
k ) ) )
9695cbvmptv 4111 . 2  |-  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )  =  ( y  e.  S  |-> 
sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
y ^ k ) ) )
9792, 96syl6eq 2331 1  |-  ( ph  ->  ( CC  _D  F
)  =  ( y  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    - cmin 9037    / cdiv 9423   2c2 9795   NN0cn0 9965   RR+crp 10354   [,)cico 10658   [,]cicc 10659    seq cseq 11046   ^cexp 11104   abscabs 11719    ~~> cli 11958   sum_csu 12158   ↾t crest 13325   TopOpenctopn 13326   * Metcxmt 16369   ballcbl 16371  ℂfldccnfld 16377   Topctop 16631   intcnt 16754   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  pserdv2  19806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-ulm 19756
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