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Theorem dvnres 21410
Description: Multiple derivative version of dvres3a 21394. (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 5696 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  0 ) )
21dmeqd 5047 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) ` 
0 ) )
32eqeq1d 2451 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F ) )
4 fveq2 5696 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5114 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  0 )  |`  S ) )
64, 5eqeq12d 2457 . . . . . . 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 5696 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  n ) )
109dmeqd 5047 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  n ) )
1110eqeq1d 2451 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  n )  =  dom  F ) )
12 fveq2 5696 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  n ) )
139reseq1d 5114 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )
1412, 13eqeq12d 2457 . . . . . . 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 5696 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  ( n  +  1 ) ) )
1817dmeqd 5047 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  ( n  +  1
) ) )
1918eqeq1d 2451 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F ) )
20 fveq2 5696 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5114 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) )
2220, 21eqeq12d 2457 . . . . . . 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 5696 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  N ) )
2625dmeqd 5047 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  N ) )
2726eqeq1d 2451 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  N )  =  dom  F ) )
28 fveq2 5696 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  N ) )
2925reseq1d 5114 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
3028, 29eqeq12d 2457 . . . . . . 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 21384 . . . . . . . . 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 7245 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 21403 . . . . . . . 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 3380 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 21403 . . . . . . . . 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 5114 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  Dn F ) ` 
0 )  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2478 . . . . . 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 9380 . . . . . . . . . . . . 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 21407 . . . . . . . . . . . 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 1218 . . . . . . . . . . 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 21404 . . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . . 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 5047 . . . . . . . . . . . . 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 2477 . . . . . . . . . . . 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 21406 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 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 9368 . . . . . . . . . . . . . . . . 17  |-  CC  e.  _V
6059, 59elpm2 7249 . . . . . . . . . . . . . . . 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 3371 . . . . . . . . . . . . 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 21381 . . . . . . . . . . . 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 3395 . . . . . . . . . . 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 3378 . . . . . . . . . 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 6104 . . . . . . . . . . 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 21404 . . . . . . . . . . . . 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 1218 . . . . . . . . . . . 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 5114 . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 20368 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 20367 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 18542 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 18666 . . . . . . . . . . . . . . . . 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 14377 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2447 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8552, 58, 63, 84, 76dvbssntr 21380 . . . . . . . . . . . . . . . . . 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 3395 . . . . . . . . . . . . . . . . 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 3371 . . . . . . . . . . . . . . . 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 3378 . . . . . . . . . . . . . . 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 18675 . . . . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . . 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 21394 . . . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 2478 . . . . . . . . . . . 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 2457 . . . . . . . . . . 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 10743 . . . 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 1184 . 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 965    = wceq 1369    e. wcel 1756    C_ wss 3333   {cpr 3884   dom cdm 4845    |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096    ^pm cpm 7220   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290   NN0cn0 10584   ↾t crest 14364   TopOpenctopn 14365  ℂfldccnfld 17823   Topctop 18503   intcnt 18626    _D cdv 21343    Dncdvn 21344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-icc 11312  df-fz 11443  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-plusg 14256  df-mulr 14257  df-starv 14258  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cnp 18837  df-haus 18924  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-limc 21346  df-dv 21347  df-dvn 21348
This theorem is referenced by:  cpnres  21416
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