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Theorem dvnadd 22062
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  Dn ( ( S  Dn F ) `  M ) ) `  N )  =  ( ( S  Dn F ) `
 ( M  +  N ) ) )

Proof of Theorem dvnadd
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5859 . . . . . 6  |-  ( n  =  0  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) ` 
0 ) )
2 oveq2 6285 . . . . . . 7  |-  ( n  =  0  ->  ( M  +  n )  =  ( M  + 
0 ) )
32fveq2d 5863 . . . . . 6  |-  ( n  =  0  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  0 ) ) )
41, 3eqeq12d 2484 . . . . 5  |-  ( n  =  0  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 n )  =  ( ( S  Dn F ) `  ( M  +  n
) )  <->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  0 )  =  ( ( S  Dn F ) `
 ( M  + 
0 ) ) ) )
54imbi2d 316 . . . 4  |-  ( n  =  0  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
F ) `  ( M  +  n )
) )  <->  ( (
( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 0 )  =  ( ( S  Dn F ) `  ( M  +  0
) ) ) ) )
6 fveq2 5859 . . . . . 6  |-  ( n  =  k  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  k ) )
7 oveq2 6285 . . . . . . 7  |-  ( n  =  k  ->  ( M  +  n )  =  ( M  +  k ) )
87fveq2d 5863 . . . . . 6  |-  ( n  =  k  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  k ) ) )
96, 8eqeq12d 2484 . . . . 5  |-  ( n  =  k  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 n )  =  ( ( S  Dn F ) `  ( M  +  n
) )  <->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  k )  =  ( ( S  Dn F ) `
 ( M  +  k ) ) ) )
109imbi2d 316 . . . 4  |-  ( n  =  k  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
F ) `  ( M  +  n )
) )  <->  ( (
( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 k )  =  ( ( S  Dn F ) `  ( M  +  k
) ) ) ) )
11 fveq2 5859 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) ) )
12 oveq2 6285 . . . . . . 7  |-  ( n  =  ( k  +  1 )  ->  ( M  +  n )  =  ( M  +  ( k  +  1 ) ) )
1312fveq2d 5863 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  ( k  +  1 ) ) ) )
1411, 13eqeq12d 2484 . . . . 5  |-  ( n  =  ( k  +  1 )  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 n )  =  ( ( S  Dn F ) `  ( M  +  n
) )  <->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) )  =  ( ( S  Dn F ) `
 ( M  +  ( k  +  1 ) ) ) ) )
1514imbi2d 316 . . . 4  |-  ( n  =  ( k  +  1 )  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
F ) `  ( M  +  n )
) )  <->  ( (
( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( ( S  Dn F ) `  ( M  +  (
k  +  1 ) ) ) ) ) )
16 fveq2 5859 . . . . . 6  |-  ( n  =  N  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  N ) )
17 oveq2 6285 . . . . . . 7  |-  ( n  =  N  ->  ( M  +  n )  =  ( M  +  N ) )
1817fveq2d 5863 . . . . . 6  |-  ( n  =  N  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  N ) ) )
1916, 18eqeq12d 2484 . . . . 5  |-  ( n  =  N  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 n )  =  ( ( S  Dn F ) `  ( M  +  n
) )  <->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  N )  =  ( ( S  Dn F ) `
 ( M  +  N ) ) ) )
2019imbi2d 316 . . . 4  |-  ( n  =  N  ->  (
( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
F ) `  ( M  +  n )
) )  <->  ( (
( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 N )  =  ( ( S  Dn F ) `  ( M  +  N
) ) ) ) )
21 recnprss 22038 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2221ad2antrr 725 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  S  C_  CC )
23 ssid 3518 . . . . . . . . . 10  |-  CC  C_  CC
2423a1i 11 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  C_  CC )
25 cnex 9564 . . . . . . . . . . 11  |-  CC  e.  _V
26 elpm2g 7427 . . . . . . . . . . 11  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
2725, 26mpan 670 . . . . . . . . . 10  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
2827simplbda 624 . . . . . . . . 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 457 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  { RR ,  CC } )
31 pmss12g 7437 . . . . . . . . 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 1224 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( CC  ^pm  dom  F )  C_  ( CC  ^pm  S
) )
3332adantr 465 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( CC  ^pm  dom  F )  C_  ( CC  ^pm 
S ) )
34 dvnff 22056 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F ) : NN0 --> ( CC 
^pm  dom  F ) )
3534ffvelrnda 6014 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  M )  e.  ( CC  ^pm  dom  F ) )
3633, 35sseldd 3500 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  M )  e.  ( CC  ^pm  S )
)
37 dvn0 22057 . . . . . 6  |-  ( ( S  C_  CC  /\  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S ) )  ->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  0 )  =  ( ( S  Dn F ) `
 M ) )
3822, 36, 37syl2anc 661 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 0 )  =  ( ( S  Dn F ) `  M ) )
39 nn0cn 10796 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
4039adantl 466 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  M  e.  CC )
4140addid1d 9770 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
4241fveq2d 5863 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  ( M  +  0
) )  =  ( ( S  Dn
F ) `  M
) )
4338, 42eqtr4d 2506 . . . 4  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 0 )  =  ( ( S  Dn F ) `  ( M  +  0
) ) )
44 oveq2 6285 . . . . . . 7  |-  ( ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  k )  =  ( ( S  Dn
F ) `  ( M  +  k )
)  ->  ( S  _D  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 k ) )  =  ( S  _D  ( ( S  Dn F ) `  ( M  +  k
) ) ) )
4522adantr 465 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  S  C_  CC )
4636adantr 465 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S ) )
47 simpr 461 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
48 dvnp1 22058 . . . . . . . . 9  |-  ( ( S  C_  CC  /\  (
( S  Dn
F ) `  M
)  e.  ( CC 
^pm  S )  /\  k  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( S  _D  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  k ) ) )
4945, 46, 47, 48syl3anc 1223 . . . . . . . 8  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) )  =  ( S  _D  ( ( S  Dn ( ( S  Dn
F ) `  M
) ) `  k
) ) )
5040adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  M  e.  CC )
51 nn0cn 10796 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
5251adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  CC )
53 1cnd 9603 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  CC )
5450, 52, 53addassd 9609 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
5554fveq2d 5863 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  Dn
F ) `  (
( M  +  k )  +  1 ) )  =  ( ( S  Dn F ) `  ( M  +  ( k  +  1 ) ) ) )
56 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  F  e.  ( CC  ^pm  S
) )
57 nn0addcl 10822 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
5857adantll 713 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  ( M  +  k )  e.  NN0 )
59 dvnp1 22058 . . . . . . . . . 10  |-  ( ( S  C_  CC  /\  F  e.  ( CC  ^pm  S
)  /\  ( M  +  k )  e. 
NN0 )  ->  (
( S  Dn
F ) `  (
( M  +  k )  +  1 ) )  =  ( S  _D  ( ( S  Dn F ) `
 ( M  +  k ) ) ) )
6045, 56, 58, 59syl3anc 1223 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  Dn
F ) `  (
( M  +  k )  +  1 ) )  =  ( S  _D  ( ( S  Dn F ) `
 ( M  +  k ) ) ) )
6155, 60eqtr3d 2505 . . . . . . . 8  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( S  Dn
F ) `  ( M  +  ( k  +  1 ) ) )  =  ( S  _D  ( ( S  Dn F ) `
 ( M  +  k ) ) ) )
6249, 61eqeq12d 2484 . . . . . . 7  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( ( S  Dn F ) `  ( M  +  (
k  +  1 ) ) )  <->  ( S  _D  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 k ) )  =  ( S  _D  ( ( S  Dn F ) `  ( M  +  k
) ) ) ) )
6344, 62syl5ibr 221 . . . . . 6  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 k )  =  ( ( S  Dn F ) `  ( M  +  k
) )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) )  =  ( ( S  Dn
F ) `  ( M  +  ( k  +  1 ) ) ) ) )
6463expcom 435 . . . . 5  |-  ( k  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  k )  =  ( ( S  Dn F ) `
 ( M  +  k ) )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 ( k  +  1 ) )  =  ( ( S  Dn F ) `  ( M  +  (
k  +  1 ) ) ) ) ) )
6564a2d 26 . . . 4  |-  ( k  e.  NN0  ->  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  ->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  k )  =  ( ( S  Dn F ) `
 ( M  +  k ) ) )  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  /\  M  e.  NN0 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) )  =  ( ( S  Dn
F ) `  ( M  +  ( k  +  1 ) ) ) ) ) )
665, 10, 15, 20, 43, 65nn0ind 10948 . . 3  |-  ( N  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 N )  =  ( ( S  Dn F ) `  ( M  +  N
) ) ) )
6766com12 31 . 2  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( N  e.  NN0  ->  ( ( S  Dn ( ( S  Dn F ) `
 M ) ) `
 N )  =  ( ( S  Dn F ) `  ( M  +  N
) ) ) )
6867impr 619 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( S  Dn ( ( S  Dn F ) `  M ) ) `  N )  =  ( ( S  Dn F ) `
 ( M  +  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3108    C_ wss 3471   {cpr 4024   dom cdm 4994   -->wf 5577   ` cfv 5581  (class class class)co 6277    ^pm cpm 7413   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486   NN0cn0 10786    _D cdv 21997    Dncdvn 21998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fi 7862  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-icc 11527  df-fz 11664  df-seq 12066  df-exp 12125  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-mulr 14560  df-starv 14561  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-rest 14669  df-topn 14670  df-topgen 14690  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cnp 19490  df-haus 19577  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-limc 22000  df-dv 22001  df-dvn 22002
This theorem is referenced by:  dvn2bss  22063  dvtaylp  22494  dvntaylp  22495  dvntaylp0  22496
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