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Theorem dvnadd 19805
Description: The  N-th derivative of the  M-th derivative of  F is the same as the  M  +  N-th derivative of  F. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnadd  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  N )  =  ( ( S  D n F ) `
 ( M  +  N ) ) )

Proof of Theorem dvnadd
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  0
) )
2 oveq2 6081 . . . . . . 7  |-  ( n  =  0  ->  ( M  +  n )  =  ( M  + 
0 ) )
32fveq2d 5724 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  0 ) ) )
41, 3eqeq12d 2449 . . . . 5  |-  ( n  =  0  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 n )  =  ( ( S  D n F ) `  ( M  +  n )
)  <->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  0 )  =  ( ( S  D n F ) `
 ( M  + 
0 ) ) ) )
54imbi2d 308 . . . 4  |-  ( n  =  0  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n F ) `  ( M  +  n ) ) )  <->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  0
)  =  ( ( S  D n F ) `  ( M  +  0 ) ) ) ) )
6 fveq2 5720 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  k
) )
7 oveq2 6081 . . . . . . 7  |-  ( n  =  k  ->  ( M  +  n )  =  ( M  +  k ) )
87fveq2d 5724 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  k ) ) )
96, 8eqeq12d 2449 . . . . 5  |-  ( n  =  k  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 n )  =  ( ( S  D n F ) `  ( M  +  n )
)  <->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  k )  =  ( ( S  D n F ) `
 ( M  +  k ) ) ) )
109imbi2d 308 . . . 4  |-  ( n  =  k  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n F ) `  ( M  +  n ) ) )  <->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  k
)  =  ( ( S  D n F ) `  ( M  +  k ) ) ) ) )
11 fveq2 5720 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  (
k  +  1 ) ) )
12 oveq2 6081 . . . . . . 7  |-  ( n  =  ( k  +  1 )  ->  ( M  +  n )  =  ( M  +  ( k  +  1 ) ) )
1312fveq2d 5724 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) )
1411, 13eqeq12d 2449 . . . . 5  |-  ( n  =  ( k  +  1 )  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 n )  =  ( ( S  D n F ) `  ( M  +  n )
)  <->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  ( k  +  1 ) )  =  ( ( S  D n F ) `
 ( M  +  ( k  +  1 ) ) ) ) )
1514imbi2d 308 . . . 4  |-  ( n  =  ( k  +  1 )  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n F ) `  ( M  +  n ) ) )  <->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  (
k  +  1 ) )  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) ) ) )
16 fveq2 5720 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  N
) )
17 oveq2 6081 . . . . . . 7  |-  ( n  =  N  ->  ( M  +  n )  =  ( M  +  N ) )
1817fveq2d 5724 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  N ) ) )
1916, 18eqeq12d 2449 . . . . 5  |-  ( n  =  N  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 n )  =  ( ( S  D n F ) `  ( M  +  n )
)  <->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  N )  =  ( ( S  D n F ) `
 ( M  +  N ) ) ) )
2019imbi2d 308 . . . 4  |-  ( n  =  N  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n F ) `  ( M  +  n ) ) )  <->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  N
)  =  ( ( S  D n F ) `  ( M  +  N ) ) ) ) )
21 recnprss 19781 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2221ad2antrr 707 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  S  C_  CC )
23 ssid 3359 . . . . . . . . . 10  |-  CC  C_  CC
2423a1i 11 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  C_  CC )
25 cnex 9061 . . . . . . . . . . 11  |-  CC  e.  _V
26 elpm2g 7025 . . . . . . . . . . 11  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
2725, 26mpan 652 . . . . . . . . . 10  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
2827simplbda 608 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  dom  F 
C_  S )
2925a1i 11 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  e.  _V )
30 simpl 444 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  { RR ,  CC } )
31 pmss12g 7032 . . . . . . . . 9  |-  ( ( ( CC  C_  CC  /\ 
dom  F  C_  S )  /\  ( CC  e.  _V  /\  S  e.  { RR ,  CC } ) )  ->  ( CC  ^pm 
dom  F )  C_  ( CC  ^pm  S ) )
3224, 28, 29, 30, 31syl22anc 1185 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( CC  ^pm  dom  F )  C_  ( CC  ^pm  S
) )
3332adantr 452 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( CC  ^pm  dom  F )  C_  ( CC  ^pm 
S ) )
34 dvnff 19799 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F ) : NN0 --> ( CC 
^pm  dom  F ) )
3534ffvelrnda 5862 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  M
)  e.  ( CC 
^pm  dom  F ) )
3633, 35sseldd 3341 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  M
)  e.  ( CC 
^pm  S ) )
37 dvn0 19800 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( S  D n F ) `  M
)  e.  ( CC 
^pm  S ) )  ->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  0 )  =  ( ( S  D n F ) `
 M ) )
3822, 36, 37syl2anc 643 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 0 )  =  ( ( S  D n F ) `  M
) )
39 nn0cn 10221 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
4039adantl 453 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  M  e.  CC )
4140addid1d 9256 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
4241fveq2d 5724 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  ( M  +  0 ) )  =  ( ( S  D n F ) `  M ) )
4338, 42eqtr4d 2470 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 0 )  =  ( ( S  D n F ) `  ( M  +  0 ) ) )
44 oveq2 6081 . . . . . . 7  |-  ( ( ( S  D n
( ( S  D n F ) `  M
) ) `  k
)  =  ( ( S  D n F ) `  ( M  +  k ) )  ->  ( S  _D  ( ( S  D n ( ( S  D n F ) `
 M ) ) `
 k ) )  =  ( S  _D  ( ( S  D n F ) `  ( M  +  k )
) ) )
4522adantr 452 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  S  C_  CC )
4636adantr 452 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  D n F ) `  M
)  e.  ( CC 
^pm  S ) )
47 simpr 448 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
48 dvnp1 19801 . . . . . . . . 9  |-  ( ( S  C_  CC  /\  (
( S  D n F ) `  M
)  e.  ( CC 
^pm  S )  /\  k  e.  NN0 )  -> 
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( S  _D  (
( S  D n
( ( S  D n F ) `  M
) ) `  k
) ) )
4945, 46, 47, 48syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  (
k  +  1 ) )  =  ( S  _D  ( ( S  D n ( ( S  D n F ) `  M ) ) `  k ) ) )
5040adantr 452 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  M  e.  CC )
51 nn0cn 10221 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
5251adantl 453 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  CC )
53 ax-1cn 9038 . . . . . . . . . . . 12  |-  1  e.  CC
5453a1i 11 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  CC )
5550, 52, 54addassd 9100 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
5655fveq2d 5724 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  D n F ) `  (
( M  +  k )  +  1 ) )  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) )
57 simpllr 736 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  F  e.  ( CC  ^pm  S
) )
58 nn0addcl 10245 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
5958adantll 695 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  ( M  +  k )  e.  NN0 )
60 dvnp1 19801 . . . . . . . . . 10  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  ( M  +  k )  e. 
NN0 )  ->  (
( S  D n F ) `  (
( M  +  k )  +  1 ) )  =  ( S  _D  ( ( S  D n F ) `
 ( M  +  k ) ) ) )
6145, 57, 59, 60syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  D n F ) `  (
( M  +  k )  +  1 ) )  =  ( S  _D  ( ( S  D n F ) `
 ( M  +  k ) ) ) )
6256, 61eqtr3d 2469 . . . . . . . 8  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  D n F ) `  ( M  +  ( k  +  1 ) ) )  =  ( S  _D  ( ( S  D n F ) `
 ( M  +  k ) ) ) )
6349, 62eqeq12d 2449 . . . . . . 7  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) )  <->  ( S  _D  ( ( S  D n ( ( S  D n F ) `
 M ) ) `
 k ) )  =  ( S  _D  ( ( S  D n F ) `  ( M  +  k )
) ) ) )
6444, 63syl5ibr 213 . . . . . 6  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 k )  =  ( ( S  D n F ) `  ( M  +  k )
)  ->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  ( k  +  1 ) )  =  ( ( S  D n F ) `
 ( M  +  ( k  +  1 ) ) ) ) )
6564expcom 425 . . . . 5  |-  ( k  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( ( S  D n ( ( S  D n F ) `  M ) ) `  k )  =  ( ( S  D n F ) `
 ( M  +  k ) )  -> 
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) ) ) )
6665a2d 24 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  ->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  k )  =  ( ( S  D n F ) `
 ( M  +  k ) ) )  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  (
k  +  1 ) )  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) ) ) )
675, 10, 15, 20, 43, 66nn0ind 10356 . . 3  |-  ( N  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n ( ( S  D n F ) `
 M ) ) `
 N )  =  ( ( S  D n F ) `  ( M  +  N )
) ) )
6867com12 29 . 2  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( ( S  D n ( ( S  D n F ) `
 M ) ) `
 N )  =  ( ( S  D n F ) `  ( M  +  N )
) ) )
6968impr 603 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( S  D n ( ( S  D n F ) `  M ) ) `  N )  =  ( ( S  D n F ) `
 ( M  +  N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   {cpr 3807   dom cdm 4870   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8978   RRcr 8979   0cc0 8980   1c1 8981    + caddc 8983   NN0cn0 10211    _D cdv 19740    D ncdvn 19741
This theorem is referenced by:  dvn2bss  19806  dvtaylp  20276  dvntaylp  20277  dvntaylp0  20278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-icc 10913  df-fz 11034  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-plusg 13532  df-mulr 13533  df-starv 13534  df-tset 13538  df-ple 13539  df-ds 13541  df-unif 13542  df-rest 13640  df-topn 13641  df-topgen 13657  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-fbas 16689  df-fg 16690  df-cnfld 16694  df-top 16953  df-bases 16955  df-topon 16956  df-topsp 16957  df-cld 17073  df-ntr 17074  df-cls 17075  df-nei 17152  df-lp 17190  df-perf 17191  df-cnp 17282  df-haus 17369  df-fil 17868  df-fm 17960  df-flim 17961  df-flf 17962  df-xms 18340  df-ms 18341  df-limc 19743  df-dv 19744  df-dvn 19745
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