Theorem dvnres 23694

Description: Multiple derivative version of dvres3a 23678. (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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . 9 |- ( x = 0 -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` 0 ) )
2	1	dmeqd 5326	. . . . . . . 8 |- ( x = 0 -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` 0 ) )
3	2	eqeq1d 2624	. . . . . . 7 |- ( x = 0 -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` 0 ) = dom F ) )
4		fveq2 6191	. . . . . . . 8 |- ( x = 0 -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` 0 ) )
5	1	reseq1d 5395	. . . . . . . 8 |- ( x = 0 -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` 0 ) |` S ) )
6	4, 5	eqeq12d 2637	. . . . . . 7 |- ( 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 334	. . . . . 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 ) ) ) )
8	7	imbi2d 330	. . . . 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 6191	. . . . . . . . 9 |- ( x = n -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` n ) )
10	9	dmeqd 5326	. . . . . . . 8 |- ( x = n -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` n ) )
11	10	eqeq1d 2624	. . . . . . 7 |- ( x = n -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` n ) = dom F ) )
12		fveq2 6191	. . . . . . . 8 |- ( x = n -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` n ) )
13	9	reseq1d 5395	. . . . . . . 8 |- ( x = n -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` n ) |` S ) )
14	12, 13	eqeq12d 2637	. . . . . . 7 |- ( 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 334	. . . . . 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 ) ) ) )
16	15	imbi2d 330	. . . . 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 6191	. . . . . . . . 9 |- ( x = ( n + 1 ) -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` ( n + 1 ) ) )
18	17	dmeqd 5326	. . . . . . . 8 |- ( x = ( n + 1 ) -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` ( n + 1 ) ) )
19	18	eqeq1d 2624	. . . . . . 7 |- ( x = ( n + 1 ) -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) )
20		fveq2 6191	. . . . . . . 8 |- ( x = ( n + 1 ) -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` ( n + 1 ) ) )
21	17	reseq1d 5395	. . . . . . . 8 |- ( x = ( n + 1 ) -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` ( n + 1 ) ) |` S ) )
22	20, 21	eqeq12d 2637	. . . . . . 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 ) ) )
23	19, 22	imbi12d 334	. . . . . 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 ) ) ) )
24	23	imbi2d 330	. . . . 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 6191	. . . . . . . . 9 |- ( x = N -> ( ( CC Dn F ) ` x ) = ( ( CC Dn F ) ` N ) )
26	25	dmeqd 5326	. . . . . . . 8 |- ( x = N -> dom ( ( CC Dn F ) ` x ) = dom ( ( CC Dn F ) ` N ) )
27	26	eqeq1d 2624	. . . . . . 7 |- ( x = N -> ( dom ( ( CC Dn F ) ` x ) = dom F <-> dom ( ( CC Dn F ) ` N ) = dom F ) )
28		fveq2 6191	. . . . . . . 8 |- ( x = N -> ( ( S Dn ( F |` S ) ) ` x ) = ( ( S Dn ( F |` S ) ) ` N ) )
29	25	reseq1d 5395	. . . . . . . 8 |- ( x = N -> ( ( ( CC Dn F ) ` x ) |` S ) = ( ( ( CC Dn F ) ` N ) |` S ) )
30	28, 29	eqeq12d 2637	. . . . . . 7 |- ( 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 334	. . . . . 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 ) ) ) )
32	31	imbi2d 330	. . . . 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 23668	. . . . . . . . 9 |- ( S e. { RR , CC } -> S C_ CC )
34	33	adantr 481	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> S C_ CC )
35		pmresg 7885	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( F |` S ) e. ( CC ^pm S ) )
36		dvn0 23687	. . . . . . . 8 |- ( ( S C_ CC /\ ( F |` S ) e. ( CC ^pm S ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( F |` S ) )
37	34, 35, 36	syl2anc 693	. . . . . . 7 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( F |` S ) )
38		ssid 3624	. . . . . . . . . 10 |- CC C_ CC
39	38	a1i 11	. . . . . . . . 9 |- ( S e. { RR , CC } -> CC C_ CC )
40		dvn0 23687	. . . . . . . . 9 |- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) ) -> ( ( CC Dn F ) ` 0 ) = F )
41	39, 40	sylan 488	. . . . . . . 8 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( CC Dn F ) ` 0 ) = F )
42	41	reseq1d 5395	. . . . . . 7 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( ( CC Dn F ) ` 0 ) |` S ) = ( F |` S ) )
43	37, 42	eqtr4d 2659	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) -> ( ( S Dn ( F |` S ) ) ` 0 ) = ( ( ( CC Dn F ) ` 0 ) |` S ) )
44	43	a1d 25	. . . . 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 10029	. . . . . . . . . . . . 13 |- CC e. { RR , CC }
46	45	a1i 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 792	. . . . . . . . . . . 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 794	. . . . . . . . . . . 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 23691	. . . . . . . . . . . 12 |- ( ( CC e. { RR , CC } /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> dom ( ( CC Dn F ) ` n ) C_ dom F )
50	46, 47, 48, 49	syl3anc 1326	. . . . . . . . . . 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 796	. . . . . . . . . . . . 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 )
52	38	a1i 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 23688	. . . . . . . . . . . . . . 15 |- ( ( CC C_ CC /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> ( ( CC Dn F ) ` ( n + 1 ) ) = ( CC _D ( ( CC Dn F ) ` n ) ) )
54	52, 47, 48, 53	syl3anc 1326	. . . . . . . . . . . . . 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 ) ) )
55	54	dmeqd 5326	. . . . . . . . . . . . 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 ) ) )
56	51, 55	eqtr3d 2658	. . . . . . . . . . . 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 23690	. . . . . . . . . . . . . 14 |- ( ( CC e. { RR , CC } /\ F e. ( CC ^pm CC ) /\ n e. NN0 ) -> ( ( CC Dn F ) ` n ) : dom ( ( CC Dn F ) ` n ) --> CC )
58	46, 47, 48, 57	syl3anc 1326	. . . . . . . . . . . . 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 10017	. . . . . . . . . . . . . . . . 17 |- CC e. _V
60	59, 59	elpm2 7889	. . . . . . . . . . . . . . . 16 |- ( F e. ( CC ^pm CC ) <-> ( F : dom F --> CC /\ dom F C_ CC ) )
61	60	simprbi 480	. . . . . . . . . . . . . . 15 |- ( F e. ( CC ^pm CC ) -> dom F C_ CC )
62	47, 61	syl 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 )
63	50, 62	sstrd 3613	. . . . . . . . . . . . 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 )
64	52, 58, 63	dvbss 23665	. . . . . . . . . . . 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 ) )
65	56, 64	eqsstrd 3639	. . . . . . . . . . 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 ) )
66	50, 65	eqssd 3620	. . . . . . . . . 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 )
67	66	expr 643	. . . . . . . . 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 ) )
68	67	imim1d 82	. . . . . . . 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 6658	. . . . . . . . . . 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 ) ) )
70	34	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> S C_ CC )
71	35	adantr 481	. . . . . . . . . . . . 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 23688	. . . . . . . . . . . . 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 ) ) )
73	70, 71, 48, 72	syl3anc 1326	. . . . . . . . . . . 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 ) ) )
74	54	reseq1d 5395	. . . . . . . . . . . . 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 790	. . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
77	76	cnfldtop 22587	. . . . . . . . . . . . . . . . 17 |- ( TopOpen ` ℂfld ) e. Top
78	76	cnfldtopon 22586	. . . . . . . . . . . . . . . . . . 19 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
79	78	toponunii 20721	. . . . . . . . . . . . . . . . . 18 |- CC = U. ( TopOpen ` ℂfld )
80	79	ntrss2 20861	. . . . . . . . . . . . . . . . 17 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ dom ( ( CC Dn F ) ` n ) C_ CC ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) C_ dom ( ( CC Dn F ) ` n ) )
81	77, 63, 80	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) C_ dom ( ( CC Dn F ) ` n ) )
82	79	restid 16094	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
83	77, 82	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
84	83	eqcomi 2631	. . . . . . . . . . . . . . . . . . 19 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
85	52, 58, 63, 84, 76	dvbssntr 23664	. . . . . . . . . . . . . . . . . 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 ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) )
86	56, 85	eqsstrd 3639	. . . . . . . . . . . . . . . . 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 ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) )
87	50, 86	sstrd 3613	. . . . . . . . . . . . . . . 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 ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) )
88	81, 87	eqssd 3620	. . . . . . . . . . . . . . 15 |- ( ( ( S e. { RR , CC } /\ F e. ( CC ^pm CC ) ) /\ ( n e. NN0 /\ dom ( ( CC Dn F ) ` ( n + 1 ) ) = dom F ) ) -> ( ( int ` ( TopOpen ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) )
89	79	isopn3 20870	. . . . . . . . . . . . . . . 16 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ dom ( ( CC Dn F ) ` n ) C_ CC ) -> ( dom ( ( CC Dn F ) ` n ) e. ( TopOpen ` ℂfld ) <-> ( ( int ` ( TopOpen ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) ) )
90	77, 63, 89	sylancr 695	. . . . . . . . . . . . . . 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 ` ℂfld ) <-> ( ( int ` ( TopOpen ` ℂfld ) ) ` dom ( ( CC Dn F ) ` n ) ) = dom ( ( CC Dn F ) ` n ) ) )
91	88, 90	mpbird 247	. . . . . . . . . . . . . 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 ` ℂfld ) )
92	66, 56	eqtr2d 2657	. . . . . . . . . . . . . 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 ) )
93	76	dvres3a 23678	. . . . . . . . . . . . . 14 |- ( ( ( S e. { RR , CC } /\ ( ( CC Dn F ) ` n ) : dom ( ( CC Dn F ) ` n ) --> CC ) /\ ( dom ( ( CC Dn F ) ` n ) e. ( TopOpen ` ℂfld ) /\ 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 ) )
94	75, 58, 91, 92, 93	syl22anc 1327	. . . . . . . . . . . . 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 ) )
95	74, 94	eqtr4d 2659	. . . . . . . . . . . 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 ) ) )
96	73, 95	eqeq12d 2637	. . . . . . . . . . 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 ) ) ) )
97	69, 96	syl5ibr 236	. . . . . . . . . 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 ) ) )
98	97	expr 643	. . . . . . . . 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 ) ) ) )
99	98	a2d 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 ) ) ) )
100	68, 99	syld 47	. . . . . . 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 ) ) ) )
101	100	expcom 451	. . . . . 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 ) ) ) ) )
102	101	a2d 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 ) ) ) ) )
103	8, 16, 24, 32, 44, 102	nn0ind 11472	. . . 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 ) ) ) )
104	103	com12 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 ) ) ) )
105	104	3impia 1261	. 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 ) ) )
106	105	imp 445	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 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {cpr 4179   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ↾_t crest 16081   TopOpenctopn 16082  ℂ_fldccnfld 19746   Topctop 20698   intcnt 20821   _D cdv 23627   Dncdvn 23628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cnp 21032  df-haus 21119  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-limc 23630  df-dv 23631  df-dvn 23632

This theorem is referenced by:  cpnres  23700