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Theorem pserdv 22551
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 22040 . . . . 5  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3516 . . . . . . . . 9  |-  CC  C_  CC
32a1i 11 . . . . . . . 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 22548 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S
-cn-> CC ) )
11 cncff 21125 . . . . . . . . 9  |-  ( F  e.  ( S -cn-> CC )  ->  F : S
--> CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  F : S --> CC )
13 cnvimass 5348 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
14 absf 13119 . . . . . . . . . . . 12  |-  abs : CC
--> RR
1514fdmi 5727 . . . . . . . . . . 11  |-  dom  abs  =  CC
1613, 15sseqtri 3529 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
178, 16eqsstri 3527 . . . . . . . . 9  |-  S  C_  CC
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
193, 12, 18dvbss 22033 . . . . . . 7  |-  ( ph  ->  dom  ( CC  _D  F )  C_  S
)
202a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  CC  C_  CC )
2112adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  F : S --> CC )
2217a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  S  C_  CC )
23 pserdv.b . . . . . . . . . . . . . 14  |-  B  =  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) )
24 cnxmet 21008 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
26 0cnd 9578 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  CC )
2718sselda 3497 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
2827abscld 13216 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
294, 5, 6, 7, 8, 9psercnlem1 22547 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
3029simp1d 1003 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
3130rpred 11245 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
3228, 31readdcld 9612 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
33 0red 9586 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
3427absge0d 13224 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
3528, 30ltaddrpd 11274 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
3633, 28, 32, 34, 35lelttrd 9728 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
3732, 36elrpd 11243 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
3837rphalfcld 11257 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
3938rpxrd 11246 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
40 blssm 20649 . . . . . . . . . . . . . . 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 )
4125, 26, 39, 40syl3anc 1223 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4223, 41syl5eqss 3541 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  B  C_  CC )
43 eqid 2460 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443cnfldtop 21019 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  Top
4543cnfldtopon 21018 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19193 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
4746restid 14678 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4844, 47ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2473 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
5043, 49dvres 22043 . . . . . . . . . . . . 13  |-  ( ( ( CC  C_  CC  /\  F : S --> CC )  /\  ( S  C_  CC  /\  B  C_  CC ) )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
5120, 21, 22, 42, 50syl22anc 1224 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
52 resss 5288 . . . . . . . . . . . 12  |-  ( ( CC  _D  F )  |`  ( ( int `  ( TopOpen
` fld
) ) `  B
) )  C_  ( CC  _D  F )
5351, 52syl6eqss 3547 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
) )
54 dmss 5193 . . . . . . . . . . 11  |-  ( ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
)  ->  dom  ( CC 
_D  ( F  |`  B ) )  C_  dom  ( CC  _D  F
) )
5553, 54syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) ) 
C_  dom  ( CC  _D  F ) )
564, 5, 6, 7, 8, 9pserdvlem1 22549 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( ( abs `  a )  +  M
)  /  2 )  e.  RR+  /\  ( abs `  a )  < 
( ( ( abs `  a )  +  M
)  /  2 )  /\  ( ( ( abs `  a )  +  M )  / 
2 )  <  R
) )
574, 5, 6, 7, 8, 56psercnlem2 22546 . . . . . . . . . . . . 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 ) )
5857simp1d 1003 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) ) )
5958, 23syl6eleqr 2559 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  B )
604, 5, 6, 7, 8, 9, 23pserdvlem2 22550 . . . . . . . . . . . . 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 ) ) ) )
6160dmeqd 5196 . . . . . . . . . . . 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
) ) ) )
62 dmmptg 5495 . . . . . . . . . . . . 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 )
63 sumex 13459 . . . . . . . . . . . . . 14  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) )  e.  _V
6463a1i 11 . . . . . . . . . . . . 13  |-  ( y  e.  B  ->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) )  e.  _V )
6562, 64mprg 2820 . . . . . . . . . . . 12  |-  dom  (
y  e.  B  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) ) )  =  B
6661, 65syl6eq 2517 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  B )
6759, 66eleqtrrd 2551 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )
6855, 67sseldd 3498 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  F ) )
6968ex 434 . . . . . . . 8  |-  ( ph  ->  ( a  e.  S  ->  a  e.  dom  ( CC  _D  F ) ) )
7069ssrdv 3503 . . . . . . 7  |-  ( ph  ->  S  C_  dom  ( CC 
_D  F ) )
7119, 70eqssd 3514 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  =  S )
7271feq2d 5709 . . . . 5  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : S --> CC ) )
731, 72mpbii 211 . . . 4  |-  ( ph  ->  ( CC  _D  F
) : S --> CC )
7473feqmptd 5911 . . 3  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  ( ( CC  _D  F ) `
 a ) ) )
75 ffun 5724 . . . . . . 7  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
761, 75mp1i 12 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  Fun  ( CC  _D  F
) )
77 funssfv 5872 . . . . . 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 ) )
7876, 53, 67, 77syl3anc 1223 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  ( ( CC 
_D  ( F  |`  B ) ) `  a ) )
7960fveq1d 5859 . . . . 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
) )
80 oveq1 6282 . . . . . . . . 9  |-  ( y  =  a  ->  (
y ^ k )  =  ( a ^
k ) )
8180oveq2d 6291 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8281sumeq2sdv 13475 . . . . . . 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 ) ) )
83 eqid 2460 . . . . . . 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 ) ) )
84 sumex 13459 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  e.  _V
8582, 83, 84fvmpt 5941 . . . . . 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
) ) )
8659, 85syl 16 . . . . 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
) ) )
8778, 79, 863eqtrd 2505 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
a ^ k ) ) )
8887mpteq2dva 4526 . . 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 ) ) ) )
8974, 88eqtrd 2501 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) ) )
90 oveq1 6282 . . . . 5  |-  ( a  =  y  ->  (
a ^ k )  =  ( y ^
k ) )
9190oveq2d 6291 . . . 4  |-  ( a  =  y  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )
9291sumeq2sdv 13475 . . 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 ) ) )
9392cbvmptv 4531 . 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 ) ) )
9489, 93syl6eq 2517 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {crab 2811   _Vcvv 3106    C_ wss 3469   ifcif 3932   class class class wbr 4440    |-> cmpt 4498   `'ccnv 4991   dom cdm 4992    |` cres 4994   "cima 4995    o. ccom 4996   Fun wfun 5573   -->wf 5575   ` cfv 5579  (class class class)co 6275   supcsup 7889   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   RR*cxr 9616    < clt 9617    - cmin 9794    / cdiv 10195   2c2 10574   NN0cn0 10784   RR+crp 11209   [,)cico 11520   [,]cicc 11521    seqcseq 12063   ^cexp 12122   abscabs 13017    ~~> cli 13256   sum_csu 13457   ↾t crest 14665   TopOpenctopn 14666   *Metcxmt 18167   ballcbl 18169  ℂfldccnfld 18184   Topctop 19154   intcnt 19277   -cn->ccncf 21108    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-shft 12850  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-limsup 13243  df-clim 13260  df-rlim 13261  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-cmp 19646  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999  df-ulm 22499
This theorem is referenced by:  pserdv2  22552
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