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Theorem dvnres 19296
Description: Multiple derivative version of dvres3a 19280. (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 5541 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21dmeqd 4897 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  0
) )
32eqeq1d 2304 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  0
)  =  dom  F
) )
4 fveq2 5541 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) ` 
0 ) )
51reseq1d 4970 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 0 )  |`  S ) )
64, 5eqeq12d 2310 . . . . . . 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 311 . . . . . 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 307 . . . . 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 5541 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
109dmeqd 4897 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  n
) )
1110eqeq1d 2304 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
12 fveq2 5541 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  n ) )
139reseq1d 4970 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 n )  |`  S ) )
1412, 13eqeq12d 2310 . . . . . . 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 311 . . . . . 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 307 . . . . 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 5541 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
1817dmeqd 4897 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  (
n  +  1 ) ) )
1918eqeq1d 2304 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F
) )
20 fveq2 5541 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 4970 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 ( n  + 
1 ) )  |`  S ) )
2220, 21eqeq12d 2310 . . . . . . 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 311 . . . . . 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 307 . . . . 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 5541 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
2625dmeqd 4897 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  N
) )
2726eqeq1d 2304 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
) )
28 fveq2 5541 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  N ) )
2925reseq1d 4970 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 N )  |`  S ) )
3028, 29eqeq12d 2310 . . . . . . 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 311 . . . . . 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 307 . . . . 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 19270 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3433adantr 451 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  S  C_  CC )
35 pmresg 6811 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 19289 . . . . . . . 8  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
3734, 35, 36syl2anc 642 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
38 ssid 3210 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 10 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 19289 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
4139, 40sylan 457 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  D n F ) `  0
)  =  F )
4241reseq1d 4970 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  D n F ) `  0
)  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2331 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  D n
( F  |`  S ) ) `  0 )  =  ( ( ( CC  D n F ) `  0 )  |`  S ) )
4443a1d 22 . . . . 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 8834 . . . . . . . . . . . . . 14  |-  CC  e.  _V
4645prid2 3748 . . . . . . . . . . . . 13  |-  CC  e.  { RR ,  CC }
4746a1i 10 . . . . . . . . . . . 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 731 . . . . . . . . . . . 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 732 . . . . . . . . . . . 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 19293 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 733 . . . . . . . . . . . . 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 10 . . . . . . . . . . . . . . 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 19290 . . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . . 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 4897 . . . . . . . . . . . . 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 2330 . . . . . . . . . . . 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 19292 . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . 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 6815 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
6160simprbi 450 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( CC  ^pm  CC )  ->  dom  F  C_  CC )
6248, 61syl 15 . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . 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 19267 . . . . . . . . . . . 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 3225 . . . . . . . . . . 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 3209 . . . . . . . . . 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 598 . . . . . . . . 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 69 . . . . . . . 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 5882 . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 19290 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 4970 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 18309 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 18308 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 16686 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 16810 . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . 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 13354 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2300 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8553, 59, 63, 84, 76dvbssntr 19266 . . . . . . . . . . . . . . . . . 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 3225 . . . . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . . . 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 3209 . . . . . . . . . . . . . . 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 16819 . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 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 2329 . . . . . . . . . . . . . 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 19280 . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . 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 2331 . . . . . . . . . . . 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 2310 . . . . . . . . . . 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 212 . . . . . . . . . 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 598 . . . . . . . . 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 23 . . . . . . . 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 40 . . . . . . 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 424 . . . . . 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 23 . . . . 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 10124 . . . 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 27 . . 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 1148 . 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 418 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   {cpr 3654   dom cdm 4705    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   NN0cn0 9981   ↾t crest 13341   TopOpenctopn 13342  ℂfldccnfld 16393   Topctop 16647   intcnt 16770    _D cdv 19229    D ncdvn 19230
This theorem is referenced by:  cpnres  19302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cnp 16974  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-limc 19232  df-dv 19233  df-dvn 19234
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