Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( x = 0 -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` 0 ) ) |
2 |
1
|
dmeqd |
|- ( x = 0 -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` 0 ) ) |
3 |
2
|
eqeq1d |
|- ( x = 0 -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` 0 ) = dom F ) ) |
4 |
|
fveq2 |
|- ( x = 0 -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` 0 ) ) |
5 |
1
|
reseq1d |
|- ( x = 0 -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` 0 ) |` S ) ) |
6 |
4 5
|
eqeq12d |
|- ( x = 0 -> ( ( ( S Dn ( F |` S ) ) ` x ) = ( ( ( CC Dn F ) ` x ) |` S ) <-> ( ( S Dn ( F |` S ) ) ` 0 ) = ( ( ( CC Dn F ) ` 0 ) |` S ) ) ) |
7 |
3 6
|
imbi12d |
|- ( 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 ) ) ) ) |
8 |
7
|
imbi2d |
|- ( 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 |
|- ( x = n -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` n ) ) |
10 |
9
|
dmeqd |
|- ( x = n -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` n ) ) |
11 |
10
|
eqeq1d |
|- ( x = n -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` n ) = dom F ) ) |
12 |
|
fveq2 |
|- ( x = n -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` n ) ) |
13 |
9
|
reseq1d |
|- ( x = n -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` n ) |` S ) ) |
14 |
12 13
|
eqeq12d |
|- ( x = n -> ( ( ( S Dn ( F |` S ) ) ` x ) = ( ( ( CC Dn F ) ` x ) |` S ) <-> ( ( S Dn ( F |` S ) ) ` n ) = ( ( ( CC Dn F ) ` n ) |` S ) ) ) |
15 |
11 14
|
imbi12d |
|- ( 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 ) ) ) ) |
16 |
15
|
imbi2d |
|- ( 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 |
|- ( x = ( n + 1 ) -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` ( n + 1 ) ) ) |
18 |
17
|
dmeqd |
|- ( x = ( n + 1 ) -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` ( n + 1 ) ) ) |
19 |
18
|
eqeq1d |
|- ( x = ( n + 1 ) -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) |
20 |
|
fveq2 |
|- ( x = ( n + 1 ) -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` ( n + 1 ) ) ) |
21 |
17
|
reseq1d |
|- ( x = ( n + 1 ) -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` ( n + 1 ) ) |` S ) ) |
22 |
20 21
|
eqeq12d |
|- ( 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 ) ) ) |
23 |
19 22
|
imbi12d |
|- ( 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 ) ) ) ) |
24 |
23
|
imbi2d |
|- ( 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 |
|- ( x = N -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` N ) ) |
26 |
25
|
dmeqd |
|- ( x = N -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` N ) ) |
27 |
26
|
eqeq1d |
|- ( x = N -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` N ) = dom F ) ) |
28 |
|
fveq2 |
|- ( x = N -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` N ) ) |
29 |
25
|
reseq1d |
|- ( x = N -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` N ) |` S ) ) |
30 |
28 29
|
eqeq12d |
|- ( x = N -> ( ( ( S Dn ( F |` S ) ) ` x ) = ( ( ( CC Dn F ) ` x ) |` S ) <-> ( ( S Dn ( F |` S ) ) ` N ) = ( ( ( CC Dn F ) ` N ) |` S ) ) ) |
31 |
27 30
|
imbi12d |
|- ( 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 ) ) ) ) |
32 |
31
|
imbi2d |
|- ( 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 |
|- ( S e. { RR , CC } -> S C_ CC ) |
34 |
33
|
adantr |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> S C_ CC ) |
35 |
|
pmresg |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( F |` S ) e. ( CC ^pm S ) ) |
36 |
|
dvn0 |
|- ( ( S C_ CC /\ ( F |` S ) e. ( CC ^pm S ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( F |` S ) ) |
37 |
34 35 36
|
syl2anc |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( F |` S ) ) |
38 |
|
ssidd |
|- ( S e. { RR , CC } -> CC C_ CC ) |
39 |
|
dvn0 |
|- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) ) -> ( ( CC Dn F ) ` 0 ) = F ) |
40 |
38 39
|
sylan |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( CC Dn F ) ` 0 ) = F ) |
41 |
40
|
reseq1d |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( ( CC Dn F ) ` 0 ) |` S ) = ( F |` S ) ) |
42 |
37 41
|
eqtr4d |
|- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( ( ( CC Dn F ) ` 0 ) |` S ) ) |
43 |
42
|
a1d |
|- ( ( 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 ) ) ) |
44 |
|
cnelprrecn |
|- CC e. { RR , CC } |
45 |
|
simplr |
|- ( ( ( 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 ) ) |
46 |
|
simprl |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> n e. NN0 ) |
47 |
|
dvnbss |
|- ( ( CC e. { RR , CC } /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> dom ( ( CC Dn F ) ` n ) C_ dom F ) |
48 |
44 45 46 47
|
mp3an2i |
|- ( ( ( 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 ) |
49 |
|
simprr |
|- ( ( ( 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 ) |
50 |
|
ssidd |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> CC C_ CC ) |
51 |
|
dvnp1 |
|- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> ( ( CC Dn F ) ` ( n + 1 ) ) = ( CC _D ( ( CC Dn F ) ` n ) ) ) |
52 |
50 45 46 51
|
syl3anc |
|- ( ( ( 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 ) ) ) |
53 |
52
|
dmeqd |
|- ( ( ( 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 ) ) ) |
54 |
49 53
|
eqtr3d |
|- ( ( ( 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 ) ) ) |
55 |
|
dvnf |
|- ( ( CC e. { RR , CC } /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> ( ( CC Dn F ) ` n ) : dom ( ( CC Dn F ) ` n ) --> CC ) |
56 |
44 45 46 55
|
mp3an2i |
|- ( ( ( 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 ) |
57 |
|
cnex |
|- CC e. _V |
58 |
57 57
|
elpm2 |
|- ( F e. ( CC ^pm CC ) <-> ( F : dom F --> CC /\ dom F C_ CC ) ) |
59 |
58
|
simprbi |
|- ( F e. ( CC ^pm CC ) -> dom F C_ CC ) |
60 |
45 59
|
syl |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> dom F C_ CC ) |
61 |
48 60
|
sstrd |
|- ( ( ( 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 ) |
62 |
50 56 61
|
dvbss |
|- ( ( ( 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 ) ) |
63 |
54 62
|
eqsstrd |
|- ( ( ( 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 ) ) |
64 |
48 63
|
eqssd |
|- ( ( ( 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 ) |
65 |
64
|
expr |
|- ( ( ( 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 ) ) |
66 |
65
|
imim1d |
|- ( ( ( 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 ) ) ) ) |
67 |
|
oveq2 |
|- ( ( ( S Dn ( F |` S ) ) ` n ) = ( ( ( CC Dn F ) ` n ) |` S ) -> ( S _D ( ( S Dn ( F |` S ) ) ` n ) ) = ( S _D ( ( ( CC Dn F ) ` n ) |` S ) ) ) |
68 |
34
|
adantr |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> S C_ CC ) |
69 |
35
|
adantr |
|- ( ( ( 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 ) ) |
70 |
|
dvnp1 |
|- ( ( 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 ) ) ) |
71 |
68 69 46 70
|
syl3anc |
|- ( ( ( 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 ) ) ) |
72 |
52
|
reseq1d |
|- ( ( ( 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 ) ) |
73 |
|
simpll |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> S e. { RR , CC } ) |
74 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
75 |
74
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
76 |
|
unicntop |
|- CC = U. ( TopOpen ` CCfld ) |
77 |
76
|
ntrss2 |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ dom ( ( CC Dn F ) ` n ) C_ CC ) -> ( ( int ` ( TopOpen ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) C_ dom ( ( CC Dn F ) ` n ) ) |
78 |
75 61 77
|
sylancr |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> ( ( int ` ( TopOpen ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) C_ dom ( ( CC Dn F ) ` n ) ) |
79 |
74
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
80 |
79
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
81 |
50 56 61 80 74
|
dvbssntr |
|- ( ( ( 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 ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) ) |
82 |
54 81
|
eqsstrd |
|- ( ( ( 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 ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) ) |
83 |
48 82
|
sstrd |
|- ( ( ( 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 ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) ) |
84 |
78 83
|
eqssd |
|- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> ( ( int ` ( TopOpen ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) ) |
85 |
76
|
isopn3 |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ dom ( ( CC Dn F ) ` n ) C_ CC ) -> ( dom ( ( CC Dn F ) ` n ) e. ( TopOpen ` CCfld ) <-> ( ( int ` ( TopOpen ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) ) ) |
86 |
75 61 85
|
sylancr |
|- ( ( ( 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 ` CCfld ) <-> ( ( int ` ( TopOpen ` CCfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) ) ) |
87 |
84 86
|
mpbird |
|- ( ( ( 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 ` CCfld ) ) |
88 |
64 54
|
eqtr2d |
|- ( ( ( 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 ) ) |
89 |
74
|
dvres3a |
|- ( ( ( S e. { RR , CC } /\ ( ( CC Dn F ) ` n ) : dom ( ( CC Dn F ) ` n ) --> CC ) /\ ( dom ( ( CC Dn F ) ` n ) e. ( TopOpen ` CCfld ) /\ 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 ) ) |
90 |
73 56 87 88 89
|
syl22anc |
|- ( ( ( 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 ) ) |
91 |
72 90
|
eqtr4d |
|- ( ( ( 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 ) ) ) |
92 |
71 91
|
eqeq12d |
|- ( ( ( 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 ) ) ) ) |
93 |
67 92
|
syl5ibr |
|- ( ( ( 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 ) ) ) |
94 |
66 93
|
animpimp2impd |
|- ( 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 ) ) ) ) ) |
95 |
8 16 24 32 43 94
|
nn0ind |
|- ( 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 ) ) ) ) |
96 |
95
|
com12 |
|- ( ( 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 ) ) ) ) |
97 |
96
|
3impia |
|- ( ( 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 ) ) ) |
98 |
97
|
imp |
|- ( ( ( 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 ) ) |