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Theorem dvnres 19280
Description: Multiple derivative version of dvres3a 19264. (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 5525 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21dmeqd 4881 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  0
) )
32eqeq1d 2291 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  0
)  =  dom  F
) )
4 fveq2 5525 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) ` 
0 ) )
51reseq1d 4954 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 0 )  |`  S ) )
64, 5eqeq12d 2297 . . . . . . 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 5525 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
109dmeqd 4881 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  n
) )
1110eqeq1d 2291 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
12 fveq2 5525 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  n ) )
139reseq1d 4954 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 n )  |`  S ) )
1412, 13eqeq12d 2297 . . . . . . 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 5525 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
1817dmeqd 4881 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  (
n  +  1 ) ) )
1918eqeq1d 2291 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F
) )
20 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 4954 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 ( n  + 
1 ) )  |`  S ) )
2220, 21eqeq12d 2297 . . . . . . 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 5525 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
2625dmeqd 4881 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  N
) )
2726eqeq1d 2291 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
) )
28 fveq2 5525 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  N ) )
2925reseq1d 4954 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 N )  |`  S ) )
3028, 29eqeq12d 2297 . . . . . . 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 19254 . . . . . . . . 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 6795 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 19273 . . . . . . . 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 3197 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 10 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 19273 . . . . . . . . 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 4954 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  D n F ) `  0
)  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2318 . . . . . 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 8818 . . . . . . . . . . . . . 14  |-  CC  e.  _V
4645prid2 3735 . . . . . . . . . . . . 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 19277 . . . . . . . . . . . 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 19274 . . . . . . . . . . . . . . 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 4881 . . . . . . . . . . . . 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 2317 . . . . . . . . . . . 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 19276 . . . . . . . . . . . . . 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 6799 . . . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . 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 19251 . . . . . . . . . . . 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 3212 . . . . . . . . . . 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 3196 . . . . . . . . . 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 5866 . . . . . . . . . . 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 19274 . . . . . . . . . . . . 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 4954 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 18293 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 18292 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 16670 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 16794 . . . . . . . . . . . . . . . . 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 13338 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2287 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8553, 59, 63, 84, 76dvbssntr 19250 . . . . . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . . . . . 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 3189 . . . . . . . . . . . . . . . 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 3196 . . . . . . . . . . . . . . 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 16803 . . . . . . . . . . . . . . . 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 2316 . . . . . . . . . . . . . 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 19264 . . . . . . . . . . . . . 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 2318 . . . . . . . . . . . 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 2297 . . . . . . . . . . 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 10108 . . . 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 1623    e. wcel 1684    C_ wss 3152   {cpr 3641   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^pm cpm 6773   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   NN0cn0 9965   ↾t crest 13325   TopOpenctopn 13326  ℂfldccnfld 16377   Topctop 16631   intcnt 16754    _D cdv 19213    D ncdvn 19214
This theorem is referenced by:  cpnres  19286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cnp 16958  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-limc 19216  df-dv 19217  df-dvn 19218
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