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Theorem pserdv 21874
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 21363 . . . . 5  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3370 . . . . . . . . 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 21871 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S
-cn-> CC ) )
11 cncff 20449 . . . . . . . . 9  |-  ( F  e.  ( S -cn-> CC )  ->  F : S
--> CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  F : S --> CC )
13 cnvimass 5184 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
14 absf 12817 . . . . . . . . . . . 12  |-  abs : CC
--> RR
1514fdmi 5559 . . . . . . . . . . 11  |-  dom  abs  =  CC
1613, 15sseqtri 3383 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
178, 16eqsstri 3381 . . . . . . . . 9  |-  S  C_  CC
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
193, 12, 18dvbss 21356 . . . . . . 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 20332 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
26 0cnd 9371 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  CC )
2718sselda 3351 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
2827abscld 12914 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
294, 5, 6, 7, 8, 9psercnlem1 21870 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
3029simp1d 1000 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
3130rpred 11019 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
3228, 31readdcld 9405 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
33 0red 9379 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
3427absge0d 12922 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
3528, 30ltaddrpd 11048 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
3633, 28, 32, 34, 35lelttrd 9521 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
3732, 36elrpd 11017 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
3837rphalfcld 11031 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
3938rpxrd 11020 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
40 blssm 19973 . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4223, 41syl5eqss 3395 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  B  C_  CC )
43 eqid 2438 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443cnfldtop 20343 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  Top
4543cnfldtopon 20342 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 18517 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
4746restid 14364 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4844, 47ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2442 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
5043, 49dvres 21366 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
52 resss 5129 . . . . . . . . . . . 12  |-  ( ( CC  _D  F )  |`  ( ( int `  ( TopOpen
` fld
) ) `  B
) )  C_  ( CC  _D  F )
5351, 52syl6eqss 3401 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
) )
54 dmss 5034 . . . . . . . . . . 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 21872 . . . . . . . . . . . . . 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 21869 . . . . . . . . . . . . 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 1000 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) ) )
5958, 23syl6eleqr 2529 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  B )
604, 5, 6, 7, 8, 9, 23pserdvlem2 21873 . . . . . . . . . . . . 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 5037 . . . . . . . . . . . 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 5330 . . . . . . . . . . . . 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 13157 . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . 12  |-  dom  (
y  e.  B  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) ) )  =  B
6661, 65syl6eq 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  B )
6759, 66eleqtrrd 2515 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )
6855, 67sseldd 3352 . . . . . . . . 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 3357 . . . . . . 7  |-  ( ph  ->  S  C_  dom  ( CC 
_D  F ) )
7119, 70eqssd 3368 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  =  S )
7271feq2d 5542 . . . . 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 5739 . . 3  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  ( ( CC  _D  F ) `
 a ) ) )
75 ffun 5556 . . . . . . 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 5700 . . . . . 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 1218 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  ( ( CC 
_D  ( F  |`  B ) ) `  a ) )
7960fveq1d 5688 . . . . 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 6093 . . . . . . . . 9  |-  ( y  =  a  ->  (
y ^ k )  =  ( a ^
k ) )
8180oveq2d 6102 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8281sumeq2sdv 13173 . . . . . . 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 2438 . . . . . . 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 13157 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  e.  _V
8582, 83, 84fvmpt 5769 . . . . . 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 2474 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
a ^ k ) ) )
8887mpteq2dva 4373 . . 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 2470 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) ) )
90 oveq1 6093 . . . . 5  |-  ( a  =  y  ->  (
a ^ k )  =  ( y ^
k ) )
9190oveq2d 6102 . . . 4  |-  ( a  =  y  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )
9291sumeq2sdv 13173 . . 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 4378 . 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 2486 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 1369    e. wcel 1756   {crab 2714   _Vcvv 2967    C_ wss 3323   ifcif 3786   class class class wbr 4287    e. cmpt 4345   `'ccnv 4834   dom cdm 4835    |` cres 4837   "cima 4838    o. ccom 4839   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6086   supcsup 7682   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   RR*cxr 9409    < clt 9410    - cmin 9587    / cdiv 9985   2c2 10363   NN0cn0 10571   RR+crp 10983   [,)cico 11294   [,]cicc 11295    seqcseq 11798   ^cexp 11857   abscabs 12715    ~~> cli 12954   sum_csu 13155   ↾t crest 14351   TopOpenctopn 14352   *Metcxmt 17781   ballcbl 17783  ℂfldccnfld 17798   Topctop 18478   intcnt 18601   -cn->ccncf 20432    _D cdv 21318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322  df-ulm 21822
This theorem is referenced by:  pserdv2  21875
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