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Theorem dvnadd 22624
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 5849 . . . . . 6  |-  ( n  =  0  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) ` 
0 ) )
2 oveq2 6286 . . . . . . 7  |-  ( n  =  0  ->  ( M  +  n )  =  ( M  + 
0 ) )
32fveq2d 5853 . . . . . 6  |-  ( n  =  0  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  0 ) ) )
41, 3eqeq12d 2424 . . . . 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 314 . . . 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 5849 . . . . . 6  |-  ( n  =  k  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  k ) )
7 oveq2 6286 . . . . . . 7  |-  ( n  =  k  ->  ( M  +  n )  =  ( M  +  k ) )
87fveq2d 5853 . . . . . 6  |-  ( n  =  k  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  k ) ) )
96, 8eqeq12d 2424 . . . . 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 314 . . . 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 5849 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  ( k  +  1 ) ) )
12 oveq2 6286 . . . . . . 7  |-  ( n  =  ( k  +  1 )  ->  ( M  +  n )  =  ( M  +  ( k  +  1 ) ) )
1312fveq2d 5853 . . . . . 6  |-  ( n  =  ( k  +  1 )  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  ( k  +  1 ) ) ) )
1411, 13eqeq12d 2424 . . . . 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 314 . . . 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 5849 . . . . . 6  |-  ( n  =  N  ->  (
( S  Dn
( ( S  Dn F ) `  M ) ) `  n )  =  ( ( S  Dn
( ( S  Dn F ) `  M ) ) `  N ) )
17 oveq2 6286 . . . . . . 7  |-  ( n  =  N  ->  ( M  +  n )  =  ( M  +  N ) )
1817fveq2d 5853 . . . . . 6  |-  ( n  =  N  ->  (
( S  Dn
F ) `  ( M  +  n )
)  =  ( ( S  Dn F ) `  ( M  +  N ) ) )
1916, 18eqeq12d 2424 . . . . 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 314 . . . 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 22600 . . . . . . 7  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2221ad2antrr 724 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  S  C_  CC )
23 ssid 3461 . . . . . . . . . 10  |-  CC  C_  CC
2423a1i 11 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  CC  C_  CC )
25 cnex 9603 . . . . . . . . . . 11  |-  CC  e.  _V
26 elpm2g 7473 . . . . . . . . . . 11  |-  ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  -> 
( F  e.  ( CC  ^pm  S )  <->  ( F : dom  F --> CC  /\  dom  F  C_  S ) ) )
2725, 26mpan 668 . . . . . . . . . 10  |-  ( S  e.  { RR ,  CC }  ->  ( F  e.  ( CC  ^pm  S
)  <->  ( F : dom  F --> CC  /\  dom  F 
C_  S ) ) )
2827simplbda 622 . . . . . . . . 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 455 . . . . . . . . 9  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  S  e.  { RR ,  CC } )
31 pmss12g 7483 . . . . . . . . 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 1231 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( CC  ^pm  dom  F )  C_  ( CC  ^pm  S
) )
3332adantr 463 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( CC  ^pm  dom  F )  C_  ( CC  ^pm 
S ) )
34 dvnff 22618 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  Dn F ) : NN0 --> ( CC 
^pm  dom  F ) )
3534ffvelrnda 6009 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  M )  e.  ( CC  ^pm  dom  F ) )
3633, 35sseldd 3443 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  M )  e.  ( CC  ^pm  S )
)
37 dvn0 22619 . . . . . 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 659 . . . . 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 10846 . . . . . . . 8  |-  ( M  e.  NN0  ->  M  e.  CC )
4039adantl 464 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  ->  M  e.  CC )
4140addid1d 9814 . . . . . 6  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( M  +  0 )  =  M )
4241fveq2d 5853 . . . . 5  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
S ) )  /\  M  e.  NN0 )  -> 
( ( S  Dn F ) `  ( M  +  0
) )  =  ( ( S  Dn
F ) `  M
) )
4338, 42eqtr4d 2446 . . . 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 6286 . . . . . . 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 463 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  S  C_  CC )
4636adantr 463 . . . . . . . . 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 459 . . . . . . . . 9  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  NN0 )
48 dvnp1 22620 . . . . . . . . 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 1230 . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  M  e.  CC )
51 nn0cn 10846 . . . . . . . . . . . 12  |-  ( k  e.  NN0  ->  k  e.  CC )
5251adantl 464 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  k  e.  CC )
53 1cnd 9642 . . . . . . . . . . 11  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  1  e.  CC )
5450, 52, 53addassd 9648 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  (
( M  +  k )  +  1 )  =  ( M  +  ( k  +  1 ) ) )
5554fveq2d 5853 . . . . . . . . 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 761 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  F  e.  ( CC  ^pm  S
) )
57 nn0addcl 10872 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  k  e.  NN0 )  -> 
( M  +  k )  e.  NN0 )
5857adantll 712 . . . . . . . . . 10  |-  ( ( ( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  S )
)  /\  M  e.  NN0 )  /\  k  e. 
NN0 )  ->  ( M  +  k )  e.  NN0 )
59 dvnp1 22620 . . . . . . . . . 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 1230 . . . . . . . . 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 2445 . . . . . . . 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 2424 . . . . . . 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 433 . . . . 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 10998 . . 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 29 . 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 617 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    C_ wss 3414   {cpr 3974   dom cdm 4823   -->wf 5565   ` cfv 5569  (class class class)co 6278    ^pm cpm 7458   CCcc 9520   RRcr 9521   0cc0 9522   1c1 9523    + caddc 9525   NN0cn0 10836    _D cdv 22559    Dncdvn 22560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fi 7905  df-sup 7935  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-icc 11589  df-fz 11727  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-mulr 14923  df-starv 14924  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-rest 15037  df-topn 15038  df-topgen 15058  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cnp 20022  df-haus 20109  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-limc 22562  df-dv 22563  df-dvn 22564
This theorem is referenced by:  dvn2bss  22625  dvtaylp  23057  dvntaylp  23058  dvntaylp0  23059
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