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Theorem dvnres 19770
Description: Multiple derivative version of dvres3a 19754. (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 5687 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  0 ) )
21dmeqd 5031 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  0
) )
32eqeq1d 2412 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  0
)  =  dom  F
) )
4 fveq2 5687 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5104 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 0 )  |`  S ) )
64, 5eqeq12d 2418 . . . . . . 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 5687 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  n ) )
109dmeqd 5031 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  n
) )
1110eqeq1d 2412 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  n
)  =  dom  F
) )
12 fveq2 5687 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  n ) )
139reseq1d 5104 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 n )  |`  S ) )
1412, 13eqeq12d 2418 . . . . . . 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 5687 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  ( n  +  1 ) ) )
1817dmeqd 5031 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  (
n  +  1 ) ) )
1918eqeq1d 2412 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  (
n  +  1 ) )  =  dom  F
) )
20 fveq2 5687 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5104 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 ( n  + 
1 ) )  |`  S ) )
2220, 21eqeq12d 2418 . . . . . . 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 5687 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  D n F ) `  x
)  =  ( ( CC  D n F ) `  N ) )
2625dmeqd 5031 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  D n F ) `  x
)  =  dom  (
( CC  D n F ) `  N
) )
2726eqeq1d 2412 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  D n F ) `  x
)  =  dom  F  <->  dom  ( ( CC  D n F ) `  N
)  =  dom  F
) )
28 fveq2 5687 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  D n
( F  |`  S ) ) `  x )  =  ( ( S  D n ( F  |`  S ) ) `  N ) )
2925reseq1d 5104 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  D n F ) `  x
)  |`  S )  =  ( ( ( CC  D n F ) `
 N )  |`  S ) )
3028, 29eqeq12d 2418 . . . . . . 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 19744 . . . . . . . . 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 7000 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 19763 . . . . . . . 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 3327 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 19763 . . . . . . . . 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 5104 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  D n F ) `  0
)  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2439 . . . . . 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 9027 . . . . . . . . . . . . . 14  |-  CC  e.  _V
4645prid2 3873 . . . . . . . . . . . . 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 19767 . . . . . . . . . . . 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 19764 . . . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . 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 2438 . . . . . . . . . . . 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 19766 . . . . . . . . . . . . . 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 7004 . . . . . . . . . . . . . . . 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 3318 . . . . . . . . . . . . 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 19741 . . . . . . . . . . . 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 3342 . . . . . . . . . . 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 3325 . . . . . . . . . 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 6048 . . . . . . . . . . 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 19764 . . . . . . . . . . . . 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 5104 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 18771 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 18770 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 16952 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 17076 . . . . . . . . . . . . . . . . 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 13616 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 8 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2408 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8553, 59, 63, 84, 76dvbssntr 19740 . . . . . . . . . . . . . . . . . 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 3342 . . . . . . . . . . . . . . . . 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 3318 . . . . . . . . . . . . . . . 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 3325 . . . . . . . . . . . . . . 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 17085 . . . . . . . . . . . . . . . 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 2437 . . . . . . . . . . . . . 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 19754 . . . . . . . . . . . . . 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 2439 . . . . . . . . . . . 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 2418 . . . . . . . . . . 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 10322 . . . 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 1721    C_ wss 3280   {cpr 3775   dom cdm 4837    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^pm cpm 6978   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949   NN0cn0 10177   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   Topctop 16913   intcnt 17036    _D cdv 19703    D ncdvn 19704
This theorem is referenced by:  cpnres  19776
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cnp 17246  df-haus 17333  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-limc 19706  df-dv 19707  df-dvn 19708
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