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Theorem dvnres 19809
Description: Multiple derivative version of dvres3a 19793. (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  D n F ) `  N
)  =  dom  F
)  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )

Proof of Theorem dvnres
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21dmeqd 5064 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  0
) )
32eqeq1d 2443 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  0
)  =  dom  F
) )
4 fveq2 5720 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5137 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 0 )  |`  S ) )
64, 5eqeq12d 2449 . . . . . . 7  |-  ( x  =  0  ->  (
( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S )  <->  ( ( S  D n ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  D n F ) `  0
)  |`  S ) ) )
73, 6imbi12d 312 . . . . . 6  |-  ( x  =  0  ->  (
( dom  ( ( CC  D n F ) `
 x )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  D n F ) `  0
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  D n F ) `  0
)  |`  S ) ) ) )
87imbi2d 308 . . . . 5  |-  ( x  =  0  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  0
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  D n F ) `  0
)  |`  S ) ) ) ) )
9 fveq2 5720 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
109dmeqd 5064 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  n
) )
1110eqeq1d 2443 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
12 fveq2 5720 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  n ) )
139reseq1d 5137 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 n )  |`  S ) )
1412, 13eqeq12d 2449 . . . . . . 7  |-  ( x  =  n  ->  (
( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S )  <->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) ) )
1511, 14imbi12d 312 . . . . . 6  |-  ( x  =  n  ->  (
( dom  ( ( CC  D n F ) `
 x )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  D n F ) `  n
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) ) ) )
1615imbi2d 308 . . . . 5  |-  ( x  =  n  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  n
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) ) ) ) )
17 fveq2 5720 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
1817dmeqd 5064 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  (
n  +  1 ) ) )
1918eqeq1d 2443 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F
) )
20 fveq2 5720 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5137 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 ( n  + 
1 ) )  |`  S ) )
2220, 21eqeq12d 2449 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S )  <->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) )
2319, 22imbi12d 312 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( dom  ( ( CC  D n F ) `
 x )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) )
2423imbi2d 308 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) ) )
25 fveq2 5720 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
2625dmeqd 5064 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  N
) )
2726eqeq1d 2443 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
) )
28 fveq2 5720 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  N ) )
2925reseq1d 5137 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 N )  |`  S ) )
3028, 29eqeq12d 2449 . . . . . . 7  |-  ( x  =  N  ->  (
( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S )  <->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) )
3127, 30imbi12d 312 . . . . . 6  |-  ( x  =  N  ->  (
( dom  ( ( CC  D n F ) `
 x )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  D n F ) `  N
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) ) )
3231imbi2d 308 . . . . 5  |-  ( x  =  N  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  x )  =  ( ( ( CC  D n F ) `  x
)  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  N
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) ) ) )
33 recnprss 19783 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3433adantr 452 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  S  C_  CC )
35 pmresg 7033 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 19802 . . . . . . . 8  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
3734, 35, 36syl2anc 643 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
38 ssid 3359 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 19802 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
4139, 40sylan 458 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
4241reseq1d 5137 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  D n F ) `  0
)  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2470 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( ( ( CC  D n F ) `  0 )  |`  S ) )
4443a1d 23 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  0
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  D n F ) `  0
)  |`  S ) ) )
45 cnex 9063 . . . . . . . . . . . . . 14  |-  CC  e.  _V
4645prid2 3905 . . . . . . . . . . . . 13  |-  CC  e.  { RR ,  CC }
4746a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  e.  { RR ,  CC } )
48 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  F  e.  ( CC  ^pm  CC )
)
49 simprl 733 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  n  e.  NN0 )
50 dvnbss 19806 . . . . . . . . . . . 12  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  dom  ( ( CC  D n F ) `
 n )  C_  dom  F )
5147, 48, 49, 50syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 n )  C_  dom  F )
52 simprr 734 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F )
5338a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  C_  CC )
54 dvnp1 19803 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  D n F ) `  n
) ) )
5553, 48, 49, 54syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  D n F ) `  n
) ) )
5655dmeqd 5064 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  ( CC  _D  ( ( CC  D n F ) `  n
) ) )
5752, 56eqtr3d 2469 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  =  dom  ( CC  _D  (
( CC  D n F ) `  n
) ) )
58 dvnf 19805 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  D n F ) `
 n ) : dom  ( ( CC  D n F ) `
 n ) --> CC )
5947, 48, 49, 58syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  D n F ) `
 n ) : dom  ( ( CC  D n F ) `
 n ) --> CC )
6045, 45elpm2 7037 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
6160simprbi 451 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( CC  ^pm  CC )  ->  dom  F  C_  CC )
6248, 61syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  CC )
6351, 62sstrd 3350 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 n )  C_  CC )
6453, 59, 63dvbss 19780 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  D n F ) `  n
) )  C_  dom  ( ( CC  D n F ) `  n
) )
6557, 64eqsstrd 3374 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  dom  ( ( CC  D n F ) `  n
) )
6651, 65eqssd 3357 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 n )  =  dom  F )
6766expr 599 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F  ->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
6867imim1d 71 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  D n F ) `  n
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) )  ->  ( dom  (
( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) ) ) )
69 oveq2 6081 . . . . . . . . . . 11  |-  ( ( ( S  D n
( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n )  |`  S )  ->  ( S  _D  ( ( S  D n ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  D n F ) `  n
)  |`  S ) ) )
7034adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  C_  CC )
7135adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S )
)
72 dvnp1 19803 . . . . . . . . . . . . 13  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
)  /\  n  e.  NN0 )  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  D n ( F  |`  S )
) `  n )
) )
7370, 71, 49, 72syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  D n ( F  |`  S )
) `  n )
) )
7455reseq1d 5137 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  D n F ) `  ( n  +  1 ) )  |`  S )  =  ( ( CC  _D  (
( CC  D n F ) `  n
) )  |`  S ) )
75 simpll 731 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  e.  { RR ,  CC } )
76 eqid 2435 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 18810 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 18809 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 16989 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 17113 . . . . . . . . . . . . . . . . 17  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  D n F ) `
 n )  C_  CC )  ->  ( ( int `  ( TopOpen ` fld )
) `  dom  ( ( CC  D n F ) `  n ) )  C_  dom  ( ( CC  D n F ) `  n ) )
8177, 63, 80sylancr 645 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  D n F ) `  n
) )  C_  dom  ( ( CC  D n F ) `  n
) )
8279restid 13653 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2439 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8553, 59, 63, 84, 76dvbssntr 19779 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  D n F ) `  n
) )  C_  (
( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  D n F ) `  n
) ) )
8657, 85eqsstrd 3374 . . . . . . . . . . . . . . . . 17  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  (
( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  D n F ) `  n
) ) )
8751, 86sstrd 3350 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 n )  C_  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  D n F ) `  n
) ) )
8881, 87eqssd 3357 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  D n F ) `  n
) )  =  dom  ( ( CC  D n F ) `  n
) )
8979isopn3 17122 . . . . . . . . . . . . . . . 16  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  D n F ) `
 n )  C_  CC )  ->  ( dom  ( ( CC  D n F ) `  n
)  e.  ( TopOpen ` fld )  <->  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  D n F ) `  n
) )  =  dom  ( ( CC  D n F ) `  n
) ) )
9077, 63, 89sylancr 645 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( dom  (
( CC  D n F ) `  n
)  e.  ( TopOpen ` fld )  <->  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  D n F ) `  n
) )  =  dom  ( ( CC  D n F ) `  n
) ) )
9188, 90mpbird 224 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  D n F ) `
 n )  e.  ( TopOpen ` fld ) )
9266, 57eqtr2d 2468 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  D n F ) `  n
) )  =  dom  ( ( CC  D n F ) `  n
) )
9376dvres3a 19793 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  ( ( CC  D n F ) `  n
) : dom  (
( CC  D n F ) `  n
) --> CC )  /\  ( dom  ( ( CC  D n F ) `
 n )  e.  ( TopOpen ` fld )  /\  dom  ( CC  _D  ( ( CC  D n F ) `
 n ) )  =  dom  ( ( CC  D n F ) `  n ) ) )  ->  ( S  _D  ( ( ( CC  D n F ) `  n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  D n F ) `  n
) )  |`  S ) )
9475, 59, 91, 92, 93syl22anc 1185 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( S  _D  ( ( ( CC  D n F ) `
 n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  D n F ) `  n
) )  |`  S ) )
9574, 94eqtr4d 2470 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  D n F ) `  ( n  +  1 ) )  |`  S )  =  ( S  _D  ( ( ( CC  D n F ) `  n
)  |`  S ) ) )
9673, 95eqeq12d 2449 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  D n ( F  |`  S )
) `  ( n  +  1 ) )  =  ( ( ( CC  D n F ) `  ( n  +  1 ) )  |`  S )  <->  ( S  _D  ( ( S  D n ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  D n F ) `  n
)  |`  S ) ) ) )
9769, 96syl5ibr 213 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  D n ( F  |`  S )
) `  n )  =  ( ( ( CC  D n F ) `  n )  |`  S )  ->  (
( S  D n
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  D n F ) `  ( n  +  1 ) )  |`  S ) ) )
9897expr 599 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S )  -> 
( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) )
9998a2d 24 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) )  ->  ( dom  (
( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) )
10068, 99syld 42 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  D n F ) `  n
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n
)  |`  S ) )  ->  ( dom  (
( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) )
101100expcom 425 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( dom  ( ( CC  D n F ) `
 n )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n )  |`  S ) )  -> 
( dom  ( ( CC  D n F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  D n F ) `  ( n  +  1 ) )  |`  S ) ) ) ) )
102101a2d 24 . . . . 5  |-  ( n  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  -> 
( dom  ( ( CC  D n F ) `
 n )  =  dom  F  ->  (
( S  D n
( F  |`  S ) ) `  n )  =  ( ( ( CC  D n F ) `  n )  |`  S ) ) )  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  D n F ) `  (
n  +  1 ) )  |`  S )
) ) ) )
1038, 16, 24, 32, 44, 102nn0ind 10358 . . . 4  |-  ( N  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  D n F ) `  N
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) ) )
104103com12 29 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( N  e.  NN0  ->  ( dom  ( ( CC  D n F ) `  N
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) ) )
1051043impia 1150 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  N  e.  NN0 )  ->  ( dom  (
( CC  D n F ) `  N
)  =  dom  F  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) ) )
106105imp 419 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  D n F ) `  N
)  =  dom  F
)  ->  ( ( S  D n ( F  |`  S ) ) `  N )  =  ( ( ( CC  D n F ) `  N
)  |`  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    C_ wss 3312   {cpr 3807   dom cdm 4870    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985   NN0cn0 10213   ↾t crest 13640   TopOpenctopn 13641  ℂfldccnfld 16695   Topctop 16950   intcnt 17073    _D cdv 19742    D ncdvn 19743
This theorem is referenced by:  cpnres  19815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-icc 10915  df-fz 11036  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-rest 13642  df-topn 13643  df-topgen 13659  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cnp 17284  df-haus 17371  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-limc 19745  df-dv 19746  df-dvn 19747
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