Theorem dvcnvrelem1 23780

Description: Lemma for dvcnvre 23782. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
dvcnvre.f	|- ( ph -> F e. ( X -cn-> RR ) )
dvcnvre.d	|- ( ph -> dom ( RR _D F ) = X )
dvcnvre.z	|- ( ph -> -. 0 e. ran ( RR _D F ) )
dvcnvre.1	|- ( ph -> F : X -1-1-onto-> Y )
dvcnvre.c	|- ( ph -> C e. X )
dvcnvre.r	|- ( ph -> R e. RR+ )
dvcnvre.s	|- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ X )

Assertion

Ref	Expression
dvcnvrelem1	|- ( ph -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )

Proof of Theorem dvcnvrelem1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcnvre.d	. . . . . 6 |- ( ph -> dom ( RR _D F ) = X )
2		dvbsss 23666	. . . . . 6 |- dom ( RR _D F ) C_ RR
3	1, 2	syl6eqssr 3656	. . . . 5 |- ( ph -> X C_ RR )
4		dvcnvre.c	. . . . 5 |- ( ph -> C e. X )
5	3, 4	sseldd 3604	. . . 4 |- ( ph -> C e. RR )
6		dvcnvre.r	. . . . 5 |- ( ph -> R e. RR+ )
7	6	rpred 11872	. . . 4 |- ( ph -> R e. RR )
8	5, 7	resubcld 10458	. . 3 |- ( ph -> ( C - R ) e. RR )
9	5, 7	readdcld 10069	. . 3 |- ( ph -> ( C + R ) e. RR )
10	5, 6	ltsubrpd 11904	. . . . 5 |- ( ph -> ( C - R ) < C )
11	5, 6	ltaddrpd 11905	. . . . 5 |- ( ph -> C < ( C + R ) )
12	8, 5, 9, 10, 11	lttrd 10198	. . . 4 |- ( ph -> ( C - R ) < ( C + R ) )
13	8, 9, 12	ltled 10185	. . 3 |- ( ph -> ( C - R ) <_ ( C + R ) )
14		dvcnvre.s	. . . 4 |- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ X )
15		dvcnvre.f	. . . 4 |- ( ph -> F e. ( X -cn-> RR ) )
16		rescncf 22700	. . . 4 |- ( ( ( C - R ) [,] ( C + R ) ) C_ X -> ( F e. ( X -cn-> RR ) -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> RR ) ) )
17	14, 15, 16	sylc 65	. . 3 |- ( ph -> ( F |` ( ( C - R ) [,] ( C + R ) ) ) e. ( ( ( C - R ) [,] ( C + R ) ) -cn-> RR ) )
18	8, 9, 13, 17	evthicc2 23229	. 2 |- ( ph -> E. x e. RR E. y e. RR ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) )
19		cncff 22696	. . . . . . . . 9 |- ( F e. ( X -cn-> RR ) -> F : X --> RR )
20	15, 19	syl 17	. . . . . . . 8 |- ( ph -> F : X --> RR )
21	20, 4	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( F ` C ) e. RR )
22	21	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` C ) e. RR )
23	8	rexrd 10089	. . . . . . . . . . . 12 |- ( ph -> ( C - R ) e. RR* )
24	9	rexrd 10089	. . . . . . . . . . . 12 |- ( ph -> ( C + R ) e. RR* )
25		lbicc2 12288	. . . . . . . . . . . 12 |- ( ( ( C - R ) e. RR* /\ ( C + R ) e. RR* /\ ( C - R ) <_ ( C + R ) ) -> ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) )
26	23, 24, 13, 25	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) )
27	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) )
28	8, 5, 10	ltled 10185	. . . . . . . . . . . 12 |- ( ph -> ( C - R ) <_ C )
29	5, 9, 11	ltled 10185	. . . . . . . . . . . 12 |- ( ph -> C <_ ( C + R ) )
30		elicc2 12238	. . . . . . . . . . . . 13 |- ( ( ( C - R ) e. RR /\ ( C + R ) e. RR ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) <-> ( C e. RR /\ ( C - R ) <_ C /\ C <_ ( C + R ) ) ) )
31	8, 9, 30	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( C e. ( ( C - R ) [,] ( C + R ) ) <-> ( C e. RR /\ ( C - R ) <_ C /\ C <_ ( C + R ) ) ) )
32	5, 28, 29, 31	mpbir3and 1245	. . . . . . . . . . 11 |- ( ph -> C e. ( ( C - R ) [,] ( C + R ) ) )
33	32	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> C e. ( ( C - R ) [,] ( C + R ) ) )
34	10	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( C - R ) < C )
35		isorel 6576	. . . . . . . . . . . . 13 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) /\ C e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) < C <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
36	35	biimpd 219	. . . . . . . . . . . 12 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) /\ C e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
37	36	exp32 631	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) ) ) )
38	37	com4l 92	. . . . . . . . . 10 |- ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) ) ) )
39	27, 33, 34, 38	syl3c 66	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
40		fvres 6207	. . . . . . . . . . 11 |- ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) = ( F ` ( C - R ) ) )
41	27, 40	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) = ( F ` ( C - R ) ) )
42		fvres 6207	. . . . . . . . . . 11 |- ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) = ( F ` C ) )
43	33, 42	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) = ( F ` C ) )
44	41, 43	breq12d 4666	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) <-> ( F ` ( C - R ) ) < ( F ` C ) ) )
45	39, 44	sylibd 229	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` ( C - R ) ) < ( F ` C ) ) )
46	20	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> F : X --> RR )
47		ffun 6048	. . . . . . . . . . . . . . 15 |- ( F : X --> RR -> Fun F )
48	46, 47	syl 17	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> Fun F )
49	14	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( C - R ) [,] ( C + R ) ) C_ X )
50		fdm 6051	. . . . . . . . . . . . . . . 16 |- ( F : X --> RR -> dom F = X )
51	46, 50	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> dom F = X )
52	49, 51	sseqtr4d 3642	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( C - R ) [,] ( C + R ) ) C_ dom F )
53		funfvima2 6493	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( ( C - R ) [,] ( C + R ) ) C_ dom F ) -> ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` ( C - R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
54	48, 52, 53	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` ( C - R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
55	27, 54	mpd 15	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C - R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) )
56		df-ima 5127	. . . . . . . . . . . . 13 |- ( F " ( ( C - R ) [,] ( C + R ) ) ) = ran ( F |` ( ( C - R ) [,] ( C + R ) ) )
57		simprr 796	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) )
58	56, 57	syl5eq 2668	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F " ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) )
59	55, 58	eleqtrd 2703	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C - R ) ) e. ( x [,] y ) )
60		elicc2 12238	. . . . . . . . . . . 12 |- ( ( x e. RR /\ y e. RR ) -> ( ( F ` ( C - R ) ) e. ( x [,] y ) <-> ( ( F ` ( C - R ) ) e. RR /\ x <_ ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) <_ y ) ) )
61	60	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C - R ) ) e. ( x [,] y ) <-> ( ( F ` ( C - R ) ) e. RR /\ x <_ ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) <_ y ) ) )
62	59, 61	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C - R ) ) e. RR /\ x <_ ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) <_ y ) )
63	62	simp2d 1074	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> x <_ ( F ` ( C - R ) ) )
64		simprll 802	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> x e. RR )
65	14, 26	sseldd 3604	. . . . . . . . . . . 12 |- ( ph -> ( C - R ) e. X )
66	20, 65	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ph -> ( F ` ( C - R ) ) e. RR )
67	66	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C - R ) ) e. RR )
68		lelttr 10128	. . . . . . . . . 10 |- ( ( x e. RR /\ ( F ` ( C - R ) ) e. RR /\ ( F ` C ) e. RR ) -> ( ( x <_ ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) < ( F ` C ) ) -> x < ( F ` C ) ) )
69	64, 67, 22, 68	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( x <_ ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) < ( F ` C ) ) -> x < ( F ` C ) ) )
70	63, 69	mpand 711	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C - R ) ) < ( F ` C ) -> x < ( F ` C ) ) )
71	45, 70	syld 47	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> x < ( F ` C ) ) )
72		ubicc2 12289	. . . . . . . . . . . 12 |- ( ( ( C - R ) e. RR* /\ ( C + R ) e. RR* /\ ( C - R ) <_ ( C + R ) ) -> ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) )
73	23, 24, 13, 72	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) )
74	73	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) )
75	11	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> C < ( C + R ) )
76		isorel 6576	. . . . . . . . . . . . 13 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( C e. ( ( C - R ) [,] ( C + R ) ) /\ ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C < ( C + R ) <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
77	76	biimpd 219	. . . . . . . . . . . 12 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( C e. ( ( C - R ) [,] ( C + R ) ) /\ ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
78	77	exp32 631	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) ) ) )
79	78	com4l 92	. . . . . . . . . 10 |- ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) ) ) )
80	33, 74, 75, 79	syl3c 66	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
81		fvex 6201	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) e. _V
82		fvex 6201	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) e. _V
83	81, 82	brcnv 5305	. . . . . . . . . 10 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) )
84		fvres 6207	. . . . . . . . . . . 12 |- ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) = ( F ` ( C + R ) ) )
85	74, 84	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) = ( F ` ( C + R ) ) )
86	85, 43	breq12d 4666	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) <-> ( F ` ( C + R ) ) < ( F ` C ) ) )
87	83, 86	syl5bb 272	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) <-> ( F ` ( C + R ) ) < ( F ` C ) ) )
88	80, 87	sylibd 229	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` ( C + R ) ) < ( F ` C ) ) )
89		funfvima2 6493	. . . . . . . . . . . . . 14 |- ( ( Fun F /\ ( ( C - R ) [,] ( C + R ) ) C_ dom F ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` ( C + R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
90	48, 52, 89	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( F ` ( C + R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
91	74, 90	mpd 15	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C + R ) ) e. ( F " ( ( C - R ) [,] ( C + R ) ) ) )
92	91, 58	eleqtrd 2703	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C + R ) ) e. ( x [,] y ) )
93		elicc2 12238	. . . . . . . . . . . 12 |- ( ( x e. RR /\ y e. RR ) -> ( ( F ` ( C + R ) ) e. ( x [,] y ) <-> ( ( F ` ( C + R ) ) e. RR /\ x <_ ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) <_ y ) ) )
94	93	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C + R ) ) e. ( x [,] y ) <-> ( ( F ` ( C + R ) ) e. RR /\ x <_ ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) <_ y ) ) )
95	92, 94	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C + R ) ) e. RR /\ x <_ ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) <_ y ) )
96	95	simp2d 1074	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> x <_ ( F ` ( C + R ) ) )
97	14, 73	sseldd 3604	. . . . . . . . . . . 12 |- ( ph -> ( C + R ) e. X )
98	20, 97	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ph -> ( F ` ( C + R ) ) e. RR )
99	98	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C + R ) ) e. RR )
100		lelttr 10128	. . . . . . . . . 10 |- ( ( x e. RR /\ ( F ` ( C + R ) ) e. RR /\ ( F ` C ) e. RR ) -> ( ( x <_ ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) < ( F ` C ) ) -> x < ( F ` C ) ) )
101	64, 99, 22, 100	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( x <_ ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) < ( F ` C ) ) -> x < ( F ` C ) ) )
102	96, 101	mpand 711	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` ( C + R ) ) < ( F ` C ) -> x < ( F ` C ) ) )
103	88, 102	syld 47	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> x < ( F ` C ) ) )
104		ax-resscn 9993	. . . . . . . . . . . . . 14 |- RR C_ CC
105	104	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR C_ CC )
106		fss 6056	. . . . . . . . . . . . . 14 |- ( ( F : X --> RR /\ RR C_ CC ) -> F : X --> CC )
107	20, 104, 106	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> F : X --> CC )
108	14, 3	sstrd 3613	. . . . . . . . . . . . 13 |- ( ph -> ( ( C - R ) [,] ( C + R ) ) C_ RR )
109		eqid 2622	. . . . . . . . . . . . . 14 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
110	109	tgioo2 22606	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( ( TopOpen ` ℂfld ) ↾t RR )
111	109, 110	dvres 23675	. . . . . . . . . . . . 13 |- ( ( ( RR C_ CC /\ F : X --> CC ) /\ ( X C_ RR /\ ( ( C - R ) [,] ( C + R ) ) C_ RR ) ) -> ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( C - R ) [,] ( C + R ) ) ) ) )
112	105, 107, 3, 108, 111	syl22anc 1327	. . . . . . . . . . . 12 |- ( ph -> ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( C - R ) [,] ( C + R ) ) ) ) )
113		iccntr 22624	. . . . . . . . . . . . . 14 |- ( ( ( C - R ) e. RR /\ ( C + R ) e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( ( C - R ) [,] ( C + R ) ) ) = ( ( C - R ) (,) ( C + R ) ) )
114	8, 9, 113	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( ( int ` ( topGen ` ran (,) ) ) ` ( ( C - R ) [,] ( C + R ) ) ) = ( ( C - R ) (,) ( C + R ) ) )
115	114	reseq2d 5396	. . . . . . . . . . . 12 |- ( ph -> ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) )
116	112, 115	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) )
117	116	dmeqd 5326	. . . . . . . . . 10 |- ( ph -> dom ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) = dom ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) )
118		dmres 5419	. . . . . . . . . . 11 |- dom ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) = ( ( ( C - R ) (,) ( C + R ) ) i^i dom ( RR _D F ) )
119		ioossicc 12259	. . . . . . . . . . . . . 14 |- ( ( C - R ) (,) ( C + R ) ) C_ ( ( C - R ) [,] ( C + R ) )
120	119, 14	syl5ss 3614	. . . . . . . . . . . . 13 |- ( ph -> ( ( C - R ) (,) ( C + R ) ) C_ X )
121	120, 1	sseqtr4d 3642	. . . . . . . . . . . 12 |- ( ph -> ( ( C - R ) (,) ( C + R ) ) C_ dom ( RR _D F ) )
122		df-ss 3588	. . . . . . . . . . . 12 |- ( ( ( C - R ) (,) ( C + R ) ) C_ dom ( RR _D F ) <-> ( ( ( C - R ) (,) ( C + R ) ) i^i dom ( RR _D F ) ) = ( ( C - R ) (,) ( C + R ) ) )
123	121, 122	sylib 208	. . . . . . . . . . 11 |- ( ph -> ( ( ( C - R ) (,) ( C + R ) ) i^i dom ( RR _D F ) ) = ( ( C - R ) (,) ( C + R ) ) )
124	118, 123	syl5eq 2668	. . . . . . . . . 10 |- ( ph -> dom ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) = ( ( C - R ) (,) ( C + R ) ) )
125	117, 124	eqtrd 2656	. . . . . . . . 9 |- ( ph -> dom ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( C - R ) (,) ( C + R ) ) )
126		resss 5422	. . . . . . . . . . . 12 |- ( ( RR _D F ) |` ( ( C - R ) (,) ( C + R ) ) ) C_ ( RR _D F )
127	116, 126	syl6eqss 3655	. . . . . . . . . . 11 |- ( ph -> ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( RR _D F ) )
128		rnss 5354	. . . . . . . . . . 11 |- ( ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) C_ ( RR _D F ) -> ran ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) C_ ran ( RR _D F ) )
129	127, 128	syl 17	. . . . . . . . . 10 |- ( ph -> ran ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) C_ ran ( RR _D F ) )
130		dvcnvre.z	. . . . . . . . . 10 |- ( ph -> -. 0 e. ran ( RR _D F ) )
131	129, 130	ssneldd 3606	. . . . . . . . 9 |- ( ph -> -. 0 e. ran ( RR _D ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) )
132	8, 9, 17, 125, 131	dvne0 23774	. . . . . . . 8 |- ( ph -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) \/ ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) ) )
133	132	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) \/ ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) ) )
134	71, 103, 133	mpjaod 396	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> x < ( F ` C ) )
135		isorel 6576	. . . . . . . . . . . . 13 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( C e. ( ( C - R ) [,] ( C + R ) ) /\ ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C < ( C + R ) <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
136	135	biimpd 219	. . . . . . . . . . . 12 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( C e. ( ( C - R ) [,] ( C + R ) ) /\ ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
137	136	exp32 631	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) ) ) )
138	137	com4l 92	. . . . . . . . . 10 |- ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C + R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C < ( C + R ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) ) ) )
139	33, 74, 75, 138	syl3c 66	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) ) )
140	43, 85	breq12d 4666	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C + R ) ) <-> ( F ` C ) < ( F ` ( C + R ) ) ) )
141	139, 140	sylibd 229	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` C ) < ( F ` ( C + R ) ) ) )
142	95	simp3d 1075	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C + R ) ) <_ y )
143		simprlr 803	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> y e. RR )
144		ltletr 10129	. . . . . . . . . 10 |- ( ( ( F ` C ) e. RR /\ ( F ` ( C + R ) ) e. RR /\ y e. RR ) -> ( ( ( F ` C ) < ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) <_ y ) -> ( F ` C ) < y ) )
145	22, 99, 143, 144	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F ` C ) < ( F ` ( C + R ) ) /\ ( F ` ( C + R ) ) <_ y ) -> ( F ` C ) < y ) )
146	142, 145	mpan2d 710	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` C ) < ( F ` ( C + R ) ) -> ( F ` C ) < y ) )
147	141, 146	syld 47	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` C ) < y ) )
148		isorel 6576	. . . . . . . . . . . . 13 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) /\ C e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) < C <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
149	148	biimpd 219	. . . . . . . . . . . 12 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) /\ ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) /\ C e. ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
150	149	exp32 631	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) ) ) )
151	150	com4l 92	. . . . . . . . . 10 |- ( ( C - R ) e. ( ( C - R ) [,] ( C + R ) ) -> ( C e. ( ( C - R ) [,] ( C + R ) ) -> ( ( C - R ) < C -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) ) ) )
152	27, 33, 34, 151	syl3c 66	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) ) )
153		fvex 6201	. . . . . . . . . . 11 |- ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) e. _V
154	153, 81	brcnv 5305	. . . . . . . . . 10 |- ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) <-> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) )
155	43, 41	breq12d 4666	. . . . . . . . . 10 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) <-> ( F ` C ) < ( F ` ( C - R ) ) ) )
156	154, 155	syl5bb 272	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` ( C - R ) ) `' < ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) ` C ) <-> ( F ` C ) < ( F ` ( C - R ) ) ) )
157	152, 156	sylibd 229	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` C ) < ( F ` ( C - R ) ) ) )
158	62	simp3d 1075	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` ( C - R ) ) <_ y )
159		ltletr 10129	. . . . . . . . . 10 |- ( ( ( F ` C ) e. RR /\ ( F ` ( C - R ) ) e. RR /\ y e. RR ) -> ( ( ( F ` C ) < ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) <_ y ) -> ( F ` C ) < y ) )
160	22, 67, 143, 159	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( ( F ` C ) < ( F ` ( C - R ) ) /\ ( F ` ( C - R ) ) <_ y ) -> ( F ` C ) < y ) )
161	158, 160	mpan2d 710	. . . . . . . 8 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` C ) < ( F ` ( C - R ) ) -> ( F ` C ) < y ) )
162	157, 161	syld 47	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F |` ( ( C - R ) [,] ( C + R ) ) ) Isom < , `' < ( ( ( C - R ) [,] ( C + R ) ) , ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) ) -> ( F ` C ) < y ) )
163	147, 162, 133	mpjaod 396	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` C ) < y )
164	64	rexrd 10089	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> x e. RR* )
165	143	rexrd 10089	. . . . . . 7 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> y e. RR* )
166		elioo2 12216	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( ( F ` C ) e. ( x (,) y ) <-> ( ( F ` C ) e. RR /\ x < ( F ` C ) /\ ( F ` C ) < y ) ) )
167	164, 165, 166	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( F ` C ) e. ( x (,) y ) <-> ( ( F ` C ) e. RR /\ x < ( F ` C ) /\ ( F ` C ) < y ) ) )
168	22, 134, 163, 167	mpbir3and 1245	. . . . 5 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` C ) e. ( x (,) y ) )
169	58	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) )
170		iccntr 22624	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
171	170	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( x [,] y ) ) = ( x (,) y ) )
172	169, 171	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) = ( x (,) y ) )
173	168, 172	eleqtrrd 2704	. . . 4 |- ( ( ph /\ ( ( x e. RR /\ y e. RR ) /\ ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) ) ) -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )
174	173	expr 643	. . 3 |- ( ( ph /\ ( x e. RR /\ y e. RR ) ) -> ( ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) ) )
175	174	rexlimdvva 3038	. 2 |- ( ph -> ( E. x e. RR E. y e. RR ran ( F |` ( ( C - R ) [,] ( C + R ) ) ) = ( x [,] y ) -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) ) )
176	18, 175	mpd 15	1 |- ( ph -> ( F ` C ) e. ( ( int ` ( topGen ` ran (,) ) ) ` ( F " ( ( C - R ) [,] ( C + R ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   (,)cioo 12175   [,]cicc 12178   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   intcnt 20821   -cn->ccncf 22679   _D cdv 23627

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  ax-addf 10015  ax-mulf 10016

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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  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-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  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-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvcnvrelem2  23781