MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pserdv Structured version   Unicode version

Theorem pserdv 22689
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 22178 . . . . 5  |-  ( CC 
_D  F ) : dom  ( CC  _D  F ) --> CC
2 ssid 3505 . . . . . . . . 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 22686 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( S
-cn-> CC ) )
11 cncff 21263 . . . . . . . . 9  |-  ( F  e.  ( S -cn-> CC )  ->  F : S
--> CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  F : S --> CC )
13 cnvimass 5343 . . . . . . . . . . 11  |-  ( `' abs " ( 0 [,) R ) ) 
C_  dom  abs
14 absf 13144 . . . . . . . . . . . 12  |-  abs : CC
--> RR
1514fdmi 5722 . . . . . . . . . . 11  |-  dom  abs  =  CC
1613, 15sseqtri 3518 . . . . . . . . . 10  |-  ( `' abs " ( 0 [,) R ) ) 
C_  CC
178, 16eqsstri 3516 . . . . . . . . 9  |-  S  C_  CC
1817a1i 11 . . . . . . . 8  |-  ( ph  ->  S  C_  CC )
193, 12, 18dvbss 22171 . . . . . . 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 21146 . . . . . . . . . . . . . . . 16  |-  ( abs 
o.  -  )  e.  ( *Met `  CC )
2524a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  ( abs  o.  -  )  e.  ( *Met `  CC ) )
26 0cnd 9587 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  CC )
2718sselda 3486 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  CC )
2827abscld 13241 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  e.  RR )
294, 5, 6, 7, 8, 9psercnlem1 22685 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  a  e.  S )  ->  ( M  e.  RR+  /\  ( abs `  a )  < 
M  /\  M  <  R ) )
3029simp1d 1007 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR+ )
3130rpred 11260 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  M  e.  RR )
3228, 31readdcld 9621 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR )
33 0red 9595 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  e.  RR )
3427absge0d 13249 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  0  <_  ( abs `  a
) )
3528, 30ltaddrpd 11289 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  a  e.  S )  ->  ( abs `  a )  < 
( ( abs `  a
)  +  M ) )
3633, 28, 32, 34, 35lelttrd 9738 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  a  e.  S )  ->  0  <  ( ( abs `  a
)  +  M ) )
3732, 36elrpd 11258 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  a  e.  S )  ->  (
( abs `  a
)  +  M )  e.  RR+ )
3837rphalfcld 11272 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e.  RR+ )
3938rpxrd 11261 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  a  e.  S )  ->  (
( ( abs `  a
)  +  M )  /  2 )  e. 
RR* )
40 blssm 20787 . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  a  e.  S )  ->  (
0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a )  +  M
)  /  2 ) )  C_  CC )
4223, 41syl5eqss 3530 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  e.  S )  ->  B  C_  CC )
43 eqid 2441 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4443cnfldtop 21157 . . . . . . . . . . . . . . . 16  |-  ( TopOpen ` fld )  e.  Top
4543cnfldtopon 21156 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19300 . . . . . . . . . . . . . . . . 17  |-  CC  =  U. ( TopOpen ` fld )
4746restid 14703 . . . . . . . . . . . . . . . 16  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4844, 47ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2454 . . . . . . . . . . . . . 14  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
5043, 49dvres 22181 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  =  ( ( CC  _D  F )  |`  (
( int `  ( TopOpen
` fld
) ) `  B
) ) )
52 resss 5283 . . . . . . . . . . . 12  |-  ( ( CC  _D  F )  |`  ( ( int `  ( TopOpen
` fld
) ) `  B
) )  C_  ( CC  _D  F )
5351, 52syl6eqss 3536 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  ( CC  _D  ( F  |`  B ) )  C_  ( CC  _D  F
) )
54 dmss 5188 . . . . . . . . . . 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 22687 . . . . . . . . . . . . . 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 22684 . . . . . . . . . . . . 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 1007 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  ( 0 ( ball `  ( abs  o.  -  ) ) ( ( ( abs `  a
)  +  M )  /  2 ) ) )
5958, 23syl6eleqr 2540 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  B )
604, 5, 6, 7, 8, 9, 23pserdvlem2 22688 . . . . . . . . . . . . 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 5191 . . . . . . . . . . . 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 5490 . . . . . . . . . . . . 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 13484 . . . . . . . . . . . . . 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 2804 . . . . . . . . . . . 12  |-  dom  (
y  e.  B  |->  sum_ k  e.  NN0  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) ) )  =  B
6661, 65syl6eq 2498 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  S )  ->  dom  ( CC  _D  ( F  |`  B ) )  =  B )
6759, 66eleqtrrd 2532 . . . . . . . . . 10  |-  ( (
ph  /\  a  e.  S )  ->  a  e.  dom  ( CC  _D  ( F  |`  B ) ) )
6855, 67sseldd 3487 . . . . . . . . 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 3492 . . . . . . 7  |-  ( ph  ->  S  C_  dom  ( CC 
_D  F ) )
7119, 70eqssd 3503 . . . . . 6  |-  ( ph  ->  dom  ( CC  _D  F )  =  S )
7271feq2d 5704 . . . . 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 5907 . . 3  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  ( ( CC  _D  F ) `
 a ) ) )
75 ffun 5719 . . . . . . 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 5867 . . . . . 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 1227 . . . . 5  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  ( ( CC 
_D  ( F  |`  B ) ) `  a ) )
7960fveq1d 5854 . . . . 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 6284 . . . . . . . . 9  |-  ( y  =  a  ->  (
y ^ k )  =  ( a ^
k ) )
8180oveq2d 6293 . . . . . . . 8  |-  ( y  =  a  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) )
8281sumeq2sdv 13500 . . . . . . 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 2441 . . . . . . 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 13484 . . . . . . 7  |-  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) )  e.  _V
8582, 83, 84fvmpt 5937 . . . . . 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 2486 . . . 4  |-  ( (
ph  /\  a  e.  S )  ->  (
( CC  _D  F
) `  a )  =  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  (
k  +  1 ) ) )  x.  (
a ^ k ) ) )
8887mpteq2dva 4519 . . 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 2482 . 2  |-  ( ph  ->  ( CC  _D  F
)  =  ( a  e.  S  |->  sum_ k  e.  NN0  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k
) ) ) )
90 oveq1 6284 . . . . 5  |-  ( a  =  y  ->  (
a ^ k )  =  ( y ^
k ) )
9190oveq2d 6293 . . . 4  |-  ( a  =  y  ->  (
( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( a ^ k ) )  =  ( ( ( k  +  1 )  x.  ( A `  ( k  +  1 ) ) )  x.  ( y ^ k
) ) )
9291sumeq2sdv 13500 . . 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 4524 . 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 2498 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 1381    e. wcel 1802   {crab 2795   _Vcvv 3093    C_ wss 3458   ifcif 3922   class class class wbr 4433    |-> cmpt 4491   `'ccnv 4984   dom cdm 4985    |` cres 4987   "cima 4988    o. ccom 4989   Fun wfun 5568   -->wf 5570   ` cfv 5574  (class class class)co 6277   supcsup 7898   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495   RR*cxr 9625    < clt 9626    - cmin 9805    / cdiv 10207   2c2 10586   NN0cn0 10796   RR+crp 11224   [,)cico 11535   [,]cicc 11536    seqcseq 12081   ^cexp 12140   abscabs 13041    ~~> cli 13281   sum_csu 13482   ↾t crest 14690   TopOpenctopn 14691   *Metcxmt 18271   ballcbl 18273  ℂfldccnfld 18288   Topctop 19261   intcnt 19384   -cn->ccncf 21246    _D cdv 22133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-shft 12874  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-cmp 19753  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-ulm 22637
This theorem is referenced by:  pserdv2  22690
  Copyright terms: Public domain W3C validator