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Theorem dvnres 22770
Description: Multiple derivative version of dvres3a 22754. (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  Dn F ) `  N )  =  dom  F )  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )

Proof of Theorem dvnres
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5881 . . . . . . . . 9  |-  ( x  =  0  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  0 ) )
21dmeqd 5057 . . . . . . . 8  |-  ( x  =  0  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) ` 
0 ) )
32eqeq1d 2431 . . . . . . 7  |-  ( x  =  0  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F ) )
4 fveq2 5881 . . . . . . . 8  |-  ( x  =  0  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) ` 
0 ) )
51reseq1d 5124 . . . . . . . 8  |-  ( x  =  0  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  0 )  |`  S ) )
64, 5eqeq12d 2451 . . . . . . 7  |-  ( x  =  0  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) )
73, 6imbi12d 321 . . . . . 6  |-  ( x  =  0  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) ) )
87imbi2d 317 . . . . 5  |-  ( x  =  0  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) ) ) )
9 fveq2 5881 . . . . . . . . 9  |-  ( x  =  n  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  n ) )
109dmeqd 5057 . . . . . . . 8  |-  ( x  =  n  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  n ) )
1110eqeq1d 2431 . . . . . . 7  |-  ( x  =  n  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  n )  =  dom  F ) )
12 fveq2 5881 . . . . . . . 8  |-  ( x  =  n  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  n ) )
139reseq1d 5124 . . . . . . . 8  |-  ( x  =  n  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )
1412, 13eqeq12d 2451 . . . . . . 7  |-  ( x  =  n  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) )
1511, 14imbi12d 321 . . . . . 6  |-  ( x  =  n  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  n )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) )
1615imbi2d 317 . . . . 5  |-  ( x  =  n  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  n )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) ) )
17 fveq2 5881 . . . . . . . . 9  |-  ( x  =  ( n  + 
1 )  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  ( n  +  1 ) ) )
1817dmeqd 5057 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  ( n  +  1
) ) )
1918eqeq1d 2431 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F ) )
20 fveq2 5881 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) ) )
2117reseq1d 5124 . . . . . . . 8  |-  ( x  =  ( n  + 
1 )  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) )
2220, 21eqeq12d 2451 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) )
2319, 22imbi12d 321 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
2423imbi2d 317 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) ) )
25 fveq2 5881 . . . . . . . . 9  |-  ( x  =  N  ->  (
( CC  Dn
F ) `  x
)  =  ( ( CC  Dn F ) `  N ) )
2625dmeqd 5057 . . . . . . . 8  |-  ( x  =  N  ->  dom  ( ( CC  Dn F ) `  x )  =  dom  ( ( CC  Dn F ) `  N ) )
2726eqeq1d 2431 . . . . . . 7  |-  ( x  =  N  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  <->  dom  ( ( CC  Dn F ) `  N )  =  dom  F ) )
28 fveq2 5881 . . . . . . . 8  |-  ( x  =  N  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( S  Dn ( F  |`  S ) ) `  N ) )
2925reseq1d 5124 . . . . . . . 8  |-  ( x  =  N  ->  (
( ( CC  Dn F ) `  x )  |`  S )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
3028, 29eqeq12d 2451 . . . . . . 7  |-  ( x  =  N  ->  (
( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S )  <-> 
( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) )
3127, 30imbi12d 321 . . . . . 6  |-  ( x  =  N  ->  (
( dom  ( ( CC  Dn F ) `
 x )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) )  <->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
3231imbi2d 317 . . . . 5  |-  ( x  =  N  ->  (
( ( S  e. 
{ RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )
)  ->  ( dom  ( ( CC  Dn F ) `  x )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  x )  =  ( ( ( CC  Dn F ) `  x )  |`  S ) ) )  <->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) ) )
33 recnprss 22744 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
3433adantr 466 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  S  C_  CC )
35 pmresg 7507 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S
) )
36 dvn0 22763 . . . . . . . 8  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
3734, 35, 36syl2anc 665 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( F  |`  S ) )
38 ssid 3489 . . . . . . . . . 10  |-  CC  C_  CC
3938a1i 11 . . . . . . . . 9  |-  ( S  e.  { RR ,  CC }  ->  CC  C_  CC )
40 dvn0 22763 . . . . . . . . 9  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  Dn
F ) `  0
)  =  F )
4139, 40sylan 473 . . . . . . . 8  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( CC  Dn
F ) `  0
)  =  F )
4241reseq1d 5124 . . . . . . 7  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( ( CC  Dn F ) ` 
0 )  |`  S )  =  ( F  |`  S ) )
4337, 42eqtr4d 2473 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( S  Dn
( F  |`  S ) ) `  0 )  =  ( ( ( CC  Dn F ) `  0 )  |`  S ) )
4443a1d 26 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) ` 
0 )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) ` 
0 )  =  ( ( ( CC  Dn F ) ` 
0 )  |`  S ) ) )
45 cnelprrecn 9631 . . . . . . . . . . . . 13  |-  CC  e.  { RR ,  CC }
4645a1i 11 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  e.  { RR ,  CC } )
47 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  F  e.  ( CC  ^pm  CC )
)
48 simprl 762 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  n  e.  NN0 )
49 dvnbss 22767 . . . . . . . . . . . 12  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  dom  F )
5046, 47, 48, 49syl3anc 1264 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  dom  F )
51 simprr 764 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F )
5238a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  CC  C_  CC )
53 dvnp1 22764 . . . . . . . . . . . . . . 15  |-  ( ( CC  C_  CC  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
5452, 47, 48, 53syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
5554dmeqd 5057 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) ) )
5651, 55eqtr3d 2472 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  =  dom  ( CC  _D  (
( CC  Dn
F ) `  n
) ) )
57 dvnf 22766 . . . . . . . . . . . . . 14  |-  ( ( CC  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  n  e.  NN0 )  ->  ( ( CC  Dn F ) `
 n ) : dom  ( ( CC  Dn F ) `
 n ) --> CC )
5846, 47, 48, 57syl3anc 1264 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( CC  Dn F ) `
 n ) : dom  ( ( CC  Dn F ) `
 n ) --> CC )
59 cnex 9619 . . . . . . . . . . . . . . . . 17  |-  CC  e.  _V
6059, 59elpm2 7511 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( CC  ^pm  CC )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  CC ) )
6160simprbi 465 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( CC  ^pm  CC )  ->  dom  F  C_  CC )
6247, 61syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  CC )
6350, 62sstrd 3480 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  CC )
6452, 58, 63dvbss 22741 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
6556, 64eqsstrd 3504 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  dom  ( ( CC  Dn F ) `  n ) )
6650, 65eqssd 3487 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  =  dom  F )
6766expr 618 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  dom  ( ( CC  Dn F ) `  n )  =  dom  F ) )
6867imim1d 78 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  n
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) ) )
69 oveq2 6313 . . . . . . . . . . 11  |-  ( ( ( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  ( S  _D  ( ( S  Dn ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  Dn F ) `  n )  |`  S ) ) )
7034adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  C_  CC )
7135adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( F  |`  S )  e.  ( CC  ^pm  S )
)
72 dvnp1 22764 . . . . . . . . . . . . 13  |-  ( ( S  C_  CC  /\  ( F  |`  S )  e.  ( CC  ^pm  S
)  /\  n  e.  NN0 )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  Dn ( F  |`  S )
) `  n )
) )
7370, 71, 48, 72syl3anc 1264 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( S  _D  ( ( S  Dn ( F  |`  S )
) `  n )
) )
7454reseq1d 5124 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  =  ( ( CC  _D  (
( CC  Dn
F ) `  n
) )  |`  S ) )
75 simpll 758 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  S  e.  { RR ,  CC } )
76 eqid 2429 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776cnfldtop 21719 . . . . . . . . . . . . . . . . 17  |-  ( TopOpen ` fld )  e.  Top
7876cnfldtopon 21718 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
7978toponunii 19882 . . . . . . . . . . . . . . . . . 18  |-  CC  =  U. ( TopOpen ` fld )
8079ntrss2 20007 . . . . . . . . . . . . . . . . 17  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  Dn F ) `
 n )  C_  CC )  ->  ( ( int `  ( TopOpen ` fld )
) `  dom  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
8177, 63, 80sylancr 667 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  C_  dom  ( ( CC  Dn F ) `  n ) )
8279restid 15295 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
TopOpen ` fld )  e.  Top  ->  ( ( TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
8377, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
8483eqcomi 2442 . . . . . . . . . . . . . . . . . . 19  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
8552, 58, 63, 84, 76dvbssntr 22740 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  C_  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8656, 85eqsstrd 3504 . . . . . . . . . . . . . . . . 17  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  F  C_  (
( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8750, 86sstrd 3480 . . . . . . . . . . . . . . . 16  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  C_  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) ) )
8881, 87eqssd 3487 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) )
8979isopn3 20017 . . . . . . . . . . . . . . . 16  |-  ( ( ( TopOpen ` fld )  e.  Top  /\ 
dom  ( ( CC  Dn F ) `
 n )  C_  CC )  ->  ( dom  ( ( CC  Dn F ) `  n )  e.  (
TopOpen ` fld )  <->  ( ( int `  ( TopOpen ` fld ) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )
9077, 63, 89sylancr 667 . . . . . . . . . . . . . . 15  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( dom  (
( CC  Dn
F ) `  n
)  e.  ( TopOpen ` fld )  <->  ( ( int `  ( TopOpen
` fld
) ) `  dom  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )
9188, 90mpbird 235 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( ( CC  Dn F ) `
 n )  e.  ( TopOpen ` fld ) )
9266, 56eqtr2d 2471 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) )
9376dvres3a 22754 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  { RR ,  CC }  /\  ( ( CC  Dn F ) `  n ) : dom  ( ( CC  Dn F ) `  n ) --> CC )  /\  ( dom  (
( CC  Dn
F ) `  n
)  e.  ( TopOpen ` fld )  /\  dom  ( CC  _D  ( ( CC  Dn F ) `  n ) )  =  dom  ( ( CC  Dn F ) `
 n ) ) )  ->  ( S  _D  ( ( ( CC  Dn F ) `
 n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  Dn F ) `  n ) )  |`  S ) )
9475, 58, 91, 92, 93syl22anc 1265 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( S  _D  ( ( ( CC  Dn F ) `
 n )  |`  S ) )  =  ( ( CC  _D  ( ( CC  Dn F ) `  n ) )  |`  S ) )
9574, 94eqtr4d 2473 . . . . . . . . . . . 12  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  =  ( S  _D  ( ( ( CC  Dn
F ) `  n
)  |`  S ) ) )
9673, 95eqeq12d 2451 . . . . . . . . . . 11  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  Dn ( F  |`  S )
) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S )  <->  ( S  _D  ( ( S  Dn ( F  |`  S ) ) `  n ) )  =  ( S  _D  (
( ( CC  Dn F ) `  n )  |`  S ) ) ) )
9769, 96syl5ibr 224 . . . . . . . . . 10  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  ( n  e.  NN0  /\ 
dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F ) )  ->  ( ( ( S  Dn ( F  |`  S )
) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  (
( S  Dn
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) ) )
9897expr 618 . . . . . . . . 9  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S )  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
9998a2d 29 . . . . . . . 8  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  (
n  +  1 ) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
10068, 99syld 45 . . . . . . 7  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  /\  n  e.  NN0 )  -> 
( ( dom  (
( CC  Dn
F ) `  n
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) )
101100expcom 436 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  (
( dom  ( ( CC  Dn F ) `
 n )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) )  -> 
( dom  ( ( CC  Dn F ) `
 ( n  + 
1 ) )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  ( n  +  1 ) )  =  ( ( ( CC  Dn F ) `  ( n  +  1 ) )  |`  S ) ) ) ) )
102101a2d 29 . . . . 5  |-  ( n  e.  NN0  ->  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC ) )  -> 
( dom  ( ( CC  Dn F ) `
 n )  =  dom  F  ->  (
( S  Dn
( F  |`  S ) ) `  n )  =  ( ( ( CC  Dn F ) `  n )  |`  S ) ) )  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  ( n  +  1
) )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  ( n  +  1
) )  =  ( ( ( CC  Dn F ) `  ( n  +  1
) )  |`  S ) ) ) ) )
1038, 16, 24, 32, 44, 102nn0ind 11030 . . . 4  |-  ( N  e.  NN0  ->  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
104103com12 32 . . 3  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC ) )  ->  ( N  e.  NN0  ->  ( dom  ( ( CC  Dn F ) `  N )  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) ) )
1051043impia 1202 . 2  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  CC )  /\  N  e.  NN0 )  ->  ( dom  (
( CC  Dn
F ) `  N
)  =  dom  F  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) ) )
106105imp 430 1  |-  ( ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm 
CC )  /\  N  e.  NN0 )  /\  dom  ( ( CC  Dn F ) `  N )  =  dom  F )  ->  ( ( S  Dn ( F  |`  S ) ) `  N )  =  ( ( ( CC  Dn F ) `  N )  |`  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   {cpr 4004   dom cdm 4854    |` cres 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^pm cpm 7481   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541   NN0cn0 10869   ↾t crest 15282   TopOpenctopn 15283  ℂfldccnfld 18909   Topctop 19852   intcnt 19967    _D cdv 22703    Dncdvn 22704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fi 7931  df-sup 7962  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-fz 11783  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-plusg 15166  df-mulr 15167  df-starv 15168  df-tset 15172  df-ple 15173  df-ds 15175  df-unif 15176  df-rest 15284  df-topn 15285  df-topgen 15305  df-psmet 18901  df-xmet 18902  df-met 18903  df-bl 18904  df-mopn 18905  df-fbas 18906  df-fg 18907  df-cnfld 18910  df-top 19856  df-bases 19857  df-topon 19858  df-topsp 19859  df-cld 19969  df-ntr 19970  df-cls 19971  df-nei 20049  df-lp 20087  df-perf 20088  df-cnp 20179  df-haus 20266  df-fil 20796  df-fm 20888  df-flim 20889  df-flf 20890  df-xms 21270  df-ms 21271  df-limc 22706  df-dv 22707  df-dvn 22708
This theorem is referenced by:  cpnres  22776
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