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Theorem dvnres 22202
Description: Multiple derivative version of dvres3a 22186. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnres  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  Dn F ) `  N )  =  dom  F )  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )

Proof of Theorem dvnres
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  0 ) )
21dmeqd 5211 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) ` 
0 ) )
32eqeq1d 2469 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F ) )
4 fveq2 5872 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5278 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  0 )  |`  S ) )
64, 5eqeq12d 2489 . . . . . . 7  |-  ( x  =  0  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) )
73, 6imbi12d 320 . . . . . 6  |-  ( x  =  0  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) ) )
87imbi2d 316 . . . . 5  |-  ( x  =  0  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) ) ) )
9 fveq2 5872 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  n ) )
109dmeqd 5211 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  n ) )
1110eqeq1d 2469 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  n )  =  dom  F ) )
12 fveq2 5872 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  n ) )
139reseq1d 5278 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )
1412, 13eqeq12d 2489 . . . . . . 7  |-  ( x  =  n  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) )
1511, 14imbi12d 320 . . . . . 6  |-  ( x  =  n  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  n )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) )
1615imbi2d 316 . . . . 5  |-  ( x  =  n  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  n )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) ) )
17 fveq2 5872 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  ( n  +  1 ) ) )
1817dmeqd 5211 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  ( n  +  1
) ) )
1918eqeq1d 2469 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F ) )
20 fveq2 5872 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5278 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) )
2220, 21eqeq12d 2489 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) )
2319, 22imbi12d 320 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
2423imbi2d 316 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) ) )
25 fveq2 5872 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  N ) )
2625dmeqd 5211 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  N ) )
2726eqeq1d 2469 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  N )  =  dom  F ) )
28 fveq2 5872 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  N ) )
2925reseq1d 5278 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
3028, 29eqeq12d 2489 . . . . . . 7  |-  ( x  =  N  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) )
3127, 30imbi12d 320 . . . . . 6  |-  ( x  =  N  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
3231imbi2d 316 . . . . 5  |-  ( x  =  N  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) ) )
33 recnprss 22176 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3433adantr 465 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  S  C_  CC )
35 pmresg 7458 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 22195 . . . . . . . 8  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
3734, 35, 36syl2anc 661 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
38 ssid 3528 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 22195 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  Dn
F ) `  0
)  =  F )
4139, 40sylan 471 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  Dn
F ) `  0
)  =  F )
4241reseq1d 5278 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  Dn F ) ` 
0 )  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2511 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( ( ( CC  Dn F ) `  0 )  |`  S ) )
4443a1d 25 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) )
45 cnelprrecn 9597 . . . . . . . . . . . . 13  |-  CC  e.  { RR ,  CC }
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  e.  { RR ,  CC } )
47 simplr 754 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  F  e.  ( CC  ^pm  CC )
)
48 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  n  e.  NN0 )
49 dvnbss 22199 . . . . . . . . . . . 12  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  dom  F )
5046, 47, 48, 49syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  dom  F )
51 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F )
5238a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  C_  CC )
53 dvnp1 22196 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
5452, 47, 48, 53syl3anc 1228 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
5554dmeqd 5211 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) ) )
5651, 55eqtr3d 2510 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  =  dom  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
57 dvnf 22198 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  Dn F ) `
 n ) : dom  ( ( CC  Dn F ) `
 n ) --> CC )
5846, 47, 48, 57syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  Dn F ) `
 n ) : dom  ( ( CC  Dn F ) `
 n ) --> CC )
59 cnex 9585 . . . . . . . . . . . . . . . . 17  |-  CC  e.  _V
6059, 59elpm2 7462 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
6160simprbi 464 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( CC  ^pm  CC )  ->  dom  F  C_  CC )
6247, 61syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  CC )
6350, 62sstrd 3519 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  CC )
6452, 58, 63dvbss 22173 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
6556, 64eqsstrd 3543 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  dom  ( ( CC  Dn F ) `  n ) )
6650, 65eqssd 3526 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  =  dom  F )
6766expr 615 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  dom  ( ( CC  Dn F ) `  n )  =  dom  F ) )
6867imim1d 75 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  n
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) )
69 oveq2 6303 . . . . . . . . . . 11  |-  ( ( ( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  ( S  _D  ( ( S  Dn ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  Dn F ) `  n )  |`  S ) ) )
7034adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  C_  CC )
7135adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S )
)
72 dvnp1 22196 . . . . . . . . . . . . 13  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
)  /\  n  e.  NN0 )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  Dn ( F  |`  S )
) `  n )
) )
7370, 71, 48, 72syl3anc 1228 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  Dn ( F  |`  S )
) `  n )
) )
7454reseq1d 5278 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  =  ( ( CC  _D  (
( CC  Dn
F ) `  n
) )  |`  S ) )
75 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  e.  { RR ,  CC } )
76 eqid 2467 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 21159 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 21158 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 19302 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 19426 . . . . . . . . . . . . . . . . 17  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  Dn F ) `
 n )  C_  CC )  ->  ( ( int `  ( TopOpen ` fld )
) `  dom  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
8177, 63, 80sylancr 663 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
8279restid 14706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2480 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8552, 58, 63, 84, 76dvbssntr 22172 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  C_  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8656, 85eqsstrd 3543 . . . . . . . . . . . . . . . . 17  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  (
( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8750, 86sstrd 3519 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8881, 87eqssd 3526 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) )
8979isopn3 19435 . . . . . . . . . . . . . . . 16  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  Dn F ) `
 n )  C_  CC )  ->  ( dom  ( ( CC  Dn F ) `  n )  e.  (
TopOpen ` fld )  <->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )
9077, 63, 89sylancr 663 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( dom  (
( CC  Dn
F ) `  n
)  e.  ( TopOpen ` fld )  <->  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )
9188, 90mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  e.  ( TopOpen ` fld ) )
9266, 56eqtr2d 2509 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) )
9376dvres3a 22186 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  ( ( CC  Dn F ) `  n ) : dom  ( ( CC  Dn F ) `  n ) --> CC )  /\  ( dom  (
( CC  Dn
F ) `  n
)  e.  ( TopOpen ` fld )  /\  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )  ->  ( S  _D  ( ( ( CC  Dn F ) `
 n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  Dn F ) `  n ) )  |`  S ) )
9475, 58, 91, 92, 93syl22anc 1229 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( S  _D  ( ( ( CC  Dn F ) `
 n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  Dn F ) `  n ) )  |`  S ) )
9574, 94eqtr4d 2511 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  =  ( S  _D  ( ( ( CC  Dn
F ) `  n
)  |`  S ) ) )
9673, 95eqeq12d 2489 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  Dn ( F  |`  S )
) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  <->  ( S  _D  ( ( S  Dn ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  Dn F ) `  n )  |`  S ) ) ) )
9769, 96syl5ibr 221 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  Dn ( F  |`  S )
) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  (
( S  Dn
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) ) )
9897expr 615 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
9998a2d 26 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
10068, 99syld 44 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  n
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
101100expcom 435 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( dom  ( ( CC  Dn F ) `
 n )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) ) ) ) )
102101a2d 26 . . . . 5  |-  ( n  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  -> 
( dom  ( ( CC  Dn F ) `
 n )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) )  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) ) )
1038, 16, 24, 32, 44, 102nn0ind 10969 . . . 4  |-  ( N  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
104103com12 31 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( N  e.  NN0  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
1051043impia 1193 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  N  e.  NN0 )  ->  ( dom  (
( CC  Dn
F ) `  N
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) )
106105imp 429 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  Dn F ) `  N )  =  dom  F )  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3481   {cpr 4035   dom cdm 5005    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^pm cpm 7433   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    + caddc 9507   NN0cn0 10807   ↾t crest 14693   TopOpenctopn 14694  ℂfldccnfld 18290   Topctop 19263   intcnt 19386    _D cdv 22135    Dncdvn 22136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fi 7883  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-fz 11685  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-rest 14695  df-topn 14696  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cnp 19597  df-haus 19684  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-limc 22138  df-dv 22139  df-dvn 22140
This theorem is referenced by:  cpnres  22208
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