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Theorem dvnres 19685
Description: Multiple derivative version of dvres3a 19669. (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 5669 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21dmeqd 5013 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  0
) )
32eqeq1d 2396 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  0
)  =  dom  F
) )
4 fveq2 5669 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5086 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 0 )  |`  S ) )
64, 5eqeq12d 2402 . . . . . . 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 5669 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
109dmeqd 5013 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  n
) )
1110eqeq1d 2396 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
12 fveq2 5669 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  n ) )
139reseq1d 5086 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 n )  |`  S ) )
1412, 13eqeq12d 2402 . . . . . . 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 5669 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
1817dmeqd 5013 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  (
n  +  1 ) ) )
1918eqeq1d 2396 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F
) )
20 fveq2 5669 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5086 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 ( n  + 
1 ) )  |`  S ) )
2220, 21eqeq12d 2402 . . . . . . 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 5669 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
2625dmeqd 5013 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  N
) )
2726eqeq1d 2396 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
) )
28 fveq2 5669 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  N ) )
2925reseq1d 5086 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 N )  |`  S ) )
3028, 29eqeq12d 2402 . . . . . . 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 19659 . . . . . . . . 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 6978 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 19678 . . . . . . . 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 3311 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 19678 . . . . . . . . 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 5086 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  D n F ) `  0
)  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2423 . . . . . 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 9005 . . . . . . . . . . . . . 14  |-  CC  e.  _V
4645prid2 3857 . . . . . . . . . . . . 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 19682 . . . . . . . . . . . 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 19679 . . . . . . . . . . . . . . 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 5013 . . . . . . . . . . . . 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 2422 . . . . . . . . . . . 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 19681 . . . . . . . . . . . . . 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 6982 . . . . . . . . . . . . . . . 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 3302 . . . . . . . . . . . . 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 19656 . . . . . . . . . . . 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 3326 . . . . . . . . . . 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 3309 . . . . . . . . . 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 6029 . . . . . . . . . . 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 19679 . . . . . . . . . . . . 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 5086 . . . . . . . . . . . . 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 2388 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 18690 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 18689 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 16921 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 17045 . . . . . . . . . . . . . . . . 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 13589 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2392 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8553, 59, 63, 84, 76dvbssntr 19655 . . . . . . . . . . . . . . . . . 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 3326 . . . . . . . . . . . . . . . . 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 3302 . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . . . 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 17054 . . . . . . . . . . . . . . . 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 2421 . . . . . . . . . . . . . 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 19669 . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . 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 2402 . . . . . . . . . . 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 10299 . . . 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 1649    e. wcel 1717    C_ wss 3264   {cpr 3759   dom cdm 4819    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021    ^pm cpm 6956   CCcc 8922   RRcr 8923   0cc0 8924   1c1 8925    + caddc 8927   NN0cn0 10154   ↾t crest 13576   TopOpenctopn 13577  ℂfldccnfld 16627   Topctop 16882   intcnt 17005    _D cdv 19618    D ncdvn 19619
This theorem is referenced by:  cpnres  19691
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-icc 10856  df-fz 10977  df-seq 11252  df-exp 11311  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-plusg 13470  df-mulr 13471  df-starv 13472  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-rest 13578  df-topn 13579  df-topgen 13595  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cnp 17215  df-haus 17302  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-limc 19621  df-dv 19622  df-dvn 19623
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