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Theorem pserdv 23368
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 22847 . . . . 5  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3483 . . . . . . . . 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 23365 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S
-cn-> CC ) )
11 cncff 21909 . . . . . . . . 9  |-  ( F  e.  ( S -cn-> CC )  ->  F : S
--> CC )
1210, 11syl 17 . . . . . . . 8  |-  ( ph  ->  F : S --> CC )
13 cnvimass 5203 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
14 absf 13386 . . . . . . . . . . . 12  |-  abs : CC
--> RR
1514fdmi 5747 . . . . . . . . . . 11  |-  dom  abs  =  CC
1613, 15sseqtri 3496 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
178, 16eqsstri 3494 . . . . . . . . 9  |-  S  C_  CC
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
193, 12, 18dvbss 22840 . . . . . . 7  |-  ( ph  ->  dom  ( CC  _D  F )  C_  S
)
202a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  CC  C_  CC )
2112adantr 466 . . . . . . . . . . . . 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 21777 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
26 0cnd 9636 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  CC )
2718sselda 3464 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
2827abscld 13483 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
294, 5, 6, 7, 8, 9psercnlem1 23364 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
3029simp1d 1017 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
3130rpred 11341 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
3228, 31readdcld 9670 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
33 0red 9644 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
3427absge0d 13491 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
3528, 30ltaddrpd 11371 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
3633, 28, 32, 34, 35lelttrd 9793 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
3732, 36elrpd 11338 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
3837rphalfcld 11353 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
3938rpxrd 11342 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
40 blssm 21417 . . . . . . . . . . . . . . 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 1264 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4223, 41syl5eqss 3508 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  B  C_  CC )
43 eqid 2422 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443cnfldtop 21788 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  Top
4543cnfldtopon 21787 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19931 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
4746restid 15317 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4844, 47ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2435 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
5043, 49dvres 22850 . . . . . . . . . . . . 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 1265 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
52 resss 5143 . . . . . . . . . . . 12  |-  ( ( CC  _D  F )  |`  ( ( int `  ( TopOpen
` fld
) ) `  B
) )  C_  ( CC  _D  F )
5351, 52syl6eqss 3514 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
) )
54 dmss 5049 . . . . . . . . . . 11  |-  ( ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
)  ->  dom  ( CC 
_D  ( F  |`  B ) )  C_  dom  ( CC  _D  F
) )
5553, 54syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) ) 
C_  dom  ( CC  _D  F ) )
564, 5, 6, 7, 8, 9pserdvlem1 23366 . . . . . . . . . . . . . 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 23363 . . . . . . . . . . . . 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 1017 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) ) )
5958, 23syl6eleqr 2521 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  B )
604, 5, 6, 7, 8, 9, 23pserdvlem2 23367 . . . . . . . . . . . . 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 5052 . . . . . . . . . . . 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 5347 . . . . . . . . . . . . 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 13739 . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . 12  |-  dom  (
y  e.  B  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) ) )  =  B
6661, 65syl6eq 2479 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  B )
6759, 66eleqtrrd 2513 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )
6855, 67sseldd 3465 . . . . . . . . 9  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  F ) )
6968ex 435 . . . . . . . 8  |-  ( ph  ->  ( a  e.  S  ->  a  e.  dom  ( CC  _D  F ) ) )
7069ssrdv 3470 . . . . . . 7  |-  ( ph  ->  S  C_  dom  ( CC 
_D  F ) )
7119, 70eqssd 3481 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  =  S )
7271feq2d 5729 . . . . 5  |-  ( ph  ->  ( ( CC  _D  F ) : dom  ( CC  _D  F
) --> CC  <->  ( CC  _D  F ) : S --> CC ) )
731, 72mpbii 214 . . . 4  |-  ( ph  ->  ( CC  _D  F
) : S --> CC )
7473feqmptd 5930 . . 3  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  ( ( CC  _D  F ) `
 a ) ) )
75 ffun 5744 . . . . . . 7  |-  ( ( CC  _D  F ) : dom  ( CC 
_D  F ) --> CC 
->  Fun  ( CC  _D  F ) )
761, 75mp1i 13 . . . . . 6  |-  ( (
ph  /\  a  e.  S )  ->  Fun  ( CC  _D  F
) )
77 funssfv 5892 . . . . . 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 1264 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  ( ( CC 
_D  ( F  |`  B ) ) `  a ) )
7960fveq1d 5879 . . . . 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 6308 . . . . . . . . 9  |-  ( y  =  a  ->  (
y ^ k )  =  ( a ^
k ) )
8180oveq2d 6317 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8281sumeq2sdv 13755 . . . . . . 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 2422 . . . . . . 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 13739 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  e.  _V
8582, 83, 84fvmpt 5960 . . . . . 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 17 . . . . 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 2467 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
a ^ k ) ) )
8887mpteq2dva 4507 . . 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 2463 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) ) )
90 oveq1 6308 . . . . 5  |-  ( a  =  y  ->  (
a ^ k )  =  ( y ^
k ) )
9190oveq2d 6317 . . . 4  |-  ( a  =  y  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )
9291sumeq2sdv 13755 . . 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 4513 . 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 2479 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 370    = wceq 1437    e. wcel 1868   {crab 2779   _Vcvv 3081    C_ wss 3436   ifcif 3909   class class class wbr 4420    |-> cmpt 4479   `'ccnv 4848   dom cdm 4849    |` cres 4851   "cima 4852    o. ccom 4853   Fun wfun 5591   -->wf 5593   ` cfv 5597  (class class class)co 6301   supcsup 7956   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544   RR*cxr 9674    < clt 9675    - cmin 9860    / cdiv 10269   2c2 10659   NN0cn0 10869   RR+crp 11302   [,)cico 11637   [,]cicc 11638    seqcseq 12212   ^cexp 12271   abscabs 13283    ~~> cli 13533   sum_csu 13737   ↾t crest 15304   TopOpenctopn 15305   *Metcxmt 18940   ballcbl 18942  ℂfldccnfld 18955   Topctop 19901   intcnt 20016   -cn->ccncf 21892    _D cdv 22802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-seq 12213  df-exp 12272  df-hash 12515  df-shft 13116  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-limsup 13511  df-clim 13537  df-rlim 13538  df-sum 13738  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-starv 15190  df-sca 15191  df-vsca 15192  df-ip 15193  df-tset 15194  df-ple 15195  df-ds 15197  df-unif 15198  df-hom 15199  df-cco 15200  df-rest 15306  df-topn 15307  df-0g 15325  df-gsum 15326  df-topgen 15327  df-pt 15328  df-prds 15331  df-xrs 15385  df-qtop 15391  df-imas 15392  df-xps 15395  df-mre 15477  df-mrc 15478  df-acs 15480  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-mulg 16661  df-cntz 16956  df-cmn 17417  df-psmet 18947  df-xmet 18948  df-met 18949  df-bl 18950  df-mopn 18951  df-fbas 18952  df-fg 18953  df-cnfld 18956  df-top 19905  df-bases 19906  df-topon 19907  df-topsp 19908  df-cld 20018  df-ntr 20019  df-cls 20020  df-nei 20098  df-lp 20136  df-perf 20137  df-cn 20227  df-cnp 20228  df-haus 20315  df-cmp 20386  df-tx 20561  df-hmeo 20754  df-fil 20845  df-fm 20937  df-flim 20938  df-flf 20939  df-xms 21319  df-ms 21320  df-tms 21321  df-cncf 21894  df-limc 22805  df-dv 22806  df-ulm 23316
This theorem is referenced by:  pserdv2  23369
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