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Theorem dvnadd 19278
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 5525 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  0
) )
2 oveq2 5866 . . . . . . 7  |-  ( n  =  0  ->  ( M  +  n )  =  ( M  + 
0 ) )
32fveq2d 5529 . . . . . 6  |-  ( n  =  0  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  0 ) ) )
41, 3eqeq12d 2297 . . . . 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 307 . . . 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 5525 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  k
) )
7 oveq2 5866 . . . . . . 7  |-  ( n  =  k  ->  ( M  +  n )  =  ( M  +  k ) )
87fveq2d 5529 . . . . . 6  |-  ( n  =  k  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  k ) ) )
96, 8eqeq12d 2297 . . . . 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 307 . . . 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 5525 . . . . . 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 5866 . . . . . . 7  |-  ( n  =  ( k  +  1 )  ->  ( M  +  n )  =  ( M  +  ( k  +  1 ) ) )
1312fveq2d 5529 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  ( k  +  1 ) ) ) )
1411, 13eqeq12d 2297 . . . . 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 307 . . . 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 5525 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n
( ( S  D n F ) `  M
) ) `  n
)  =  ( ( S  D n ( ( S  D n F ) `  M
) ) `  N
) )
17 oveq2 5866 . . . . . . 7  |-  ( n  =  N  ->  ( M  +  n )  =  ( M  +  N ) )
1817fveq2d 5529 . . . . . 6  |-  ( n  =  N  ->  (
( S  D n F ) `  ( M  +  n )
)  =  ( ( S  D n F ) `  ( M  +  N ) ) )
1916, 18eqeq12d 2297 . . . . 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 307 . . . 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 19254 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2221ad2antrr 706 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  S  C_  CC )
23 ssid 3197 . . . . . . . . . 10  |-  CC  C_  CC
2423a1i 10 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  C_  CC )
25 cnex 8818 . . . . . . . . . . 11  |-  CC  e.  _V
26 elpm2g 6787 . . . . . . . . . . 11  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
2725, 26mpan 651 . . . . . . . . . 10  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
2827simplbda 607 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  dom  F 
C_  S )
2925a1i 10 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  e.  _V )
30 simpl 443 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  { RR ,  CC } )
31 pmss12g 6794 . . . . . . . . 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 1183 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( CC  ^pm  dom  F )  C_  ( CC  ^pm  S
) )
3332adantr 451 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( CC  ^pm  dom  F )  C_  ( CC  ^pm 
S ) )
34 dvnff 19272 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  D n F ) : NN0 --> ( CC 
^pm  dom  F ) )
35 ffvelrn 5663 . . . . . . . 8  |-  ( ( ( S  D n F ) : NN0 --> ( CC  ^pm  dom  F )  /\  M  e.  NN0 )  ->  ( ( S  D n F ) `
 M )  e.  ( CC  ^pm  dom  F ) )
3634, 35sylan 457 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  M
)  e.  ( CC 
^pm  dom  F ) )
3733, 36sseldd 3181 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  M
)  e.  ( CC 
^pm  S ) )
38 dvn0 19273 . . . . . 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 ) )
3922, 37, 38syl2anc 642 . . . . 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
) )
40 nn0cn 9975 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
4140adantl 452 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  M  e.  CC )
4241addid1d 9012 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
4342fveq2d 5529 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  D n F ) `  ( M  +  0 ) )  =  ( ( S  D n F ) `  M ) )
4439, 43eqtr4d 2318 . . . 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 ) ) )
45 oveq2 5866 . . . . . . 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 )
) ) )
4622adantr 451 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  S  C_  CC )
4737adantr 451 . . . . . . . . 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 ) )
48 simpr 447 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
49 dvnp1 19274 . . . . . . . . 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
) ) )
5046, 47, 48, 49syl3anc 1182 . . . . . . . 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 ) ) )
5141adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  M  e.  CC )
52 nn0cn 9975 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
5352adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  CC )
54 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
5554a1i 10 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  CC )
5651, 53, 55addassd 8857 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
5756fveq2d 5529 . . . . . . . . 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 ) ) ) )
58 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  F  e.  ( CC  ^pm  S
) )
59 nn0addcl 9999 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
6059adantll 694 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  ( M  +  k )  e.  NN0 )
61 dvnp1 19274 . . . . . . . . . 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 ) ) ) )
6246, 58, 60, 61syl3anc 1182 . . . . . . . . 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 ) ) ) )
6357, 62eqtr3d 2317 . . . . . . . 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 ) ) ) )
6450, 63eqeq12d 2297 . . . . . . 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 )
) ) ) )
6545, 64syl5ibr 212 . . . . . 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 ) ) ) ) )
6665expcom 424 . . . . 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 ) ) ) ) ) )
6766a2d 23 . . . 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 ) ) ) ) ) )
685, 10, 15, 20, 44, 67nn0ind 10108 . . 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 )
) ) )
6968com12 27 . 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 )
) ) )
7069impr 602 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  dvn2bss  19279  dvtaylp  19749  dvntaylp  19750  dvntaylp0  19751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-limc 19216  df-dv 19217  df-dvn 19218
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