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Theorem dvnadd 19684
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 5670 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  0
) )
2 oveq2 6030 . . . . . . 7  |-  ( n  =  0  ->  ( M  +  n )  =  ( M  + 
0 ) )
32fveq2d 5674 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  0 ) ) )
41, 3eqeq12d 2403 . . . . 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 5670 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  k
) )
7 oveq2 6030 . . . . . . 7  |-  ( n  =  k  ->  ( M  +  n )  =  ( M  +  k ) )
87fveq2d 5674 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  k ) ) )
96, 8eqeq12d 2403 . . . . 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 5670 . . . . . 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 6030 . . . . . . 7  |-  ( n  =  ( k  +  1 )  ->  ( M  +  n )  =  ( M  +  ( k  +  1 ) ) )
1312fveq2d 5674 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) )
1411, 13eqeq12d 2403 . . . . 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 5670 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  N
) )
17 oveq2 6030 . . . . . . 7  |-  ( n  =  N  ->  ( M  +  n )  =  ( M  +  N ) )
1817fveq2d 5674 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  N ) ) )
1916, 18eqeq12d 2403 . . . . 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 19660 . . . . . . 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 3312 . . . . . . . . . 10  |-  CC  C_  CC
2423a1i 11 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  C_  CC )
25 cnex 9006 . . . . . . . . . . 11  |-  CC  e.  _V
26 elpm2g 6971 . . . . . . . . . . 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 6978 . . . . . . . . 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 19678 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F ) : NN0 --> ( CC 
^pm  dom  F ) )
3534ffvelrnda 5811 . . . . . . 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 3294 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  M
)  e.  ( CC 
^pm  S ) )
37 dvn0 19679 . . . . . 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 10165 . . . . . . . 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 9200 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
4241fveq2d 5674 . . . . 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 2424 . . . 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 6030 . . . . . . 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 19680 . . . . . . . . 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 10165 . . . . . . . . . . . 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 8983 . . . . . . . . . . . 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 9045 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
5655fveq2d 5674 . . . . . . . . 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 10189 . . . . . . . . . . 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 19680 . . . . . . . . . 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 2423 . . . . . . . 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 2403 . . . . . . 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 10300 . . 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 1649    e. wcel 1717   _Vcvv 2901    C_ wss 3265   {cpr 3760   dom cdm 4820   -->wf 5392   ` cfv 5396  (class class class)co 6022    ^pm cpm 6957   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928   NN0cn0 10155    _D cdv 19619    D ncdvn 19620
This theorem is referenced by:  dvn2bss  19685  dvtaylp  20155  dvntaylp  20156  dvntaylp0  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-icc 10857  df-fz 10978  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-plusg 13471  df-mulr 13472  df-starv 13473  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-rest 13579  df-topn 13580  df-topgen 13596  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-fbas 16625  df-fg 16626  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-lp 17125  df-perf 17126  df-cnp 17216  df-haus 17303  df-fil 17801  df-fm 17893  df-flim 17894  df-flf 17895  df-xms 18261  df-ms 18262  df-limc 19622  df-dv 19623  df-dvn 19624
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