Theorem ivthALT 32330

Description: An alternate proof of the Intermediate Value Theorem ivth 23223 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ivthALT	|- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> E. x e. ( A (,) B ) ( F ` x ) = U )

Distinct variable groups:   x, A   x, B   x, D   x, F   x, U

Proof of Theorem ivthALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp31 1097	. . . . . 6 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> F e. ( D -cn-> CC ) )
2		cncff 22696	. . . . . 6 |- ( F e. ( D -cn-> CC ) -> F : D --> CC )
3	1, 2	syl 17	. . . . 5 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> F : D --> CC )
4		ffun 6048	. . . . 5 |- ( F : D --> CC -> Fun F )
5	3, 4	syl 17	. . . 4 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> Fun F )
6	5	3ad2ant3 1084	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> Fun F )
7		iccconn 22633	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR ) -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) e. Conn )
8	7	3adant3 1081	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR /\ U e. RR ) -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) e. Conn )
9	8	3ad2ant1 1082	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) e. Conn )
10		simpr1 1067	. . . . . . . . . . . . . 14 |- ( ( D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> F e. ( D -cn-> CC ) )
11	10, 2	syl 17	. . . . . . . . . . . . 13 |- ( ( D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> F : D --> CC )
12	11	anim2i 593	. . . . . . . . . . . 12 |- ( ( ( A [,] B ) C_ D /\ ( D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( A [,] B ) C_ D /\ F : D --> CC ) )
13	12	3impb 1260	. . . . . . . . . . 11 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> ( ( A [,] B ) C_ D /\ F : D --> CC ) )
14	13	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( A [,] B ) C_ D /\ F : D --> CC ) )
15	4	adantl 482	. . . . . . . . . . 11 |- ( ( ( A [,] B ) C_ D /\ F : D --> CC ) -> Fun F )
16		fdm 6051	. . . . . . . . . . . . 13 |- ( F : D --> CC -> dom F = D )
17	16	sseq2d 3633	. . . . . . . . . . . 12 |- ( F : D --> CC -> ( ( A [,] B ) C_ dom F <-> ( A [,] B ) C_ D ) )
18	17	biimparc 504	. . . . . . . . . . 11 |- ( ( ( A [,] B ) C_ D /\ F : D --> CC ) -> ( A [,] B ) C_ dom F )
19	15, 18	jca 554	. . . . . . . . . 10 |- ( ( ( A [,] B ) C_ D /\ F : D --> CC ) -> ( Fun F /\ ( A [,] B ) C_ dom F ) )
20	14, 19	syl 17	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( Fun F /\ ( A [,] B ) C_ dom F ) )
21		fores 6124	. . . . . . . . 9 |- ( ( Fun F /\ ( A [,] B ) C_ dom F ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> ( F " ( A [,] B ) ) )
22	20, 21	syl 17	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> ( F " ( A [,] B ) ) )
23		retop 22565	. . . . . . . . . 10 |- ( topGen ` ran (,) ) e. Top
24		simp332 1215	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F " ( A [,] B ) ) C_ RR )
25		uniretop 22566	. . . . . . . . . . 11 |- RR = U. ( topGen ` ran (,) )
26	25	restuni 20966	. . . . . . . . . 10 |- ( ( ( topGen ` ran (,) ) e. Top /\ ( F " ( A [,] B ) ) C_ RR ) -> ( F " ( A [,] B ) ) = U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) )
27	23, 24, 26	sylancr 695	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F " ( A [,] B ) ) = U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) )
28		foeq3 6113	. . . . . . . . 9 |- ( ( F " ( A [,] B ) ) = U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) -> ( ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> ( F " ( A [,] B ) ) <-> ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) ) )
29	27, 28	syl 17	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> ( F " ( A [,] B ) ) <-> ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) ) )
30	22, 29	mpbid 222	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) )
31		simp331 1214	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> F e. ( D -cn-> CC ) )
32		ssid 3624	. . . . . . . . . . . . . . 15 |- CC C_ CC
33		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
34		eqid 2622	. . . . . . . . . . . . . . . 16 |- ( ( TopOpen ` ℂfld ) ↾t D ) = ( ( TopOpen ` ℂfld ) ↾t D )
35	33	cnfldtop 22587	. . . . . . . . . . . . . . . . . 18 |- ( TopOpen ` ℂfld ) e. Top
36	33	cnfldtopon 22586	. . . . . . . . . . . . . . . . . . . 20 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
37	36	toponunii 20721	. . . . . . . . . . . . . . . . . . 19 |- CC = U. ( TopOpen ` ℂfld )
38	37	restid 16094	. . . . . . . . . . . . . . . . . 18 |- ( ( TopOpen ` ℂfld ) e. Top -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
39	35, 38	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
40	39	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
41	33, 34, 40	cncfcn 22712	. . . . . . . . . . . . . . 15 |- ( ( D C_ CC /\ CC C_ CC ) -> ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
42	32, 41	mpan2 707	. . . . . . . . . . . . . 14 |- ( D C_ CC -> ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
43	42	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
44	43	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( D -cn-> CC ) = ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
45	31, 44	eleqtrd 2703	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> F e. ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) )
46		simp31 1097	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( A [,] B ) C_ D )
47		simp32 1098	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> D C_ CC )
48		resttopon 20965	. . . . . . . . . . . . . 14 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ D C_ CC ) -> ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D ) )
49	36, 47, 48	sylancr 695	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D ) )
50		toponuni 20719	. . . . . . . . . . . . 13 |- ( ( ( TopOpen ` ℂfld ) ↾t D ) e. (TopOn ` D ) -> D = U. ( ( TopOpen ` ℂfld ) ↾t D ) )
51	49, 50	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> D = U. ( ( TopOpen ` ℂfld ) ↾t D ) )
52	46, 51	sseqtrd 3641	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( A [,] B ) C_ U. ( ( TopOpen ` ℂfld ) ↾t D ) )
53		eqid 2622	. . . . . . . . . . . 12 |- U. ( ( TopOpen ` ℂfld ) ↾t D ) = U. ( ( TopOpen ` ℂfld ) ↾t D )
54	53	cnrest 21089	. . . . . . . . . . 11 |- ( ( F e. ( ( ( TopOpen ` ℂfld ) ↾t D ) Cn ( TopOpen ` ℂfld ) ) /\ ( A [,] B ) C_ U. ( ( TopOpen ` ℂfld ) ↾t D ) ) -> ( F |` ( A [,] B ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) )
55	45, 52, 54	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) e. ( ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) )
56	35	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( TopOpen ` ℂfld ) e. Top )
57		cnex 10017	. . . . . . . . . . . . . 14 |- CC e. _V
58		ssexg 4804	. . . . . . . . . . . . . 14 |- ( ( D C_ CC /\ CC e. _V ) -> D e. _V )
59	47, 57, 58	sylancl 694	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> D e. _V )
60		restabs 20969	. . . . . . . . . . . . 13 |- ( ( ( TopOpen ` ℂfld ) e. Top /\ ( A [,] B ) C_ D /\ D e. _V ) -> ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
61	56, 46, 59, 60	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) = ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) )
62		iccssre 12255	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
63	62	3adant3 1081	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. RR /\ U e. RR ) -> ( A [,] B ) C_ RR )
64	63	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( A [,] B ) C_ RR )
65		eqid 2622	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
66	33, 65	rerest 22607	. . . . . . . . . . . . 13 |- ( ( A [,] B ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
67	64, 66	syl 17	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
68	61, 67	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) = ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) )
69	68	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( ( TopOpen ` ℂfld ) ↾t D ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) = ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) )
70	55, 69	eleqtrd 2703	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) )
71	36	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
72		df-ima 5127	. . . . . . . . . . . 12 |- ( F " ( A [,] B ) ) = ran ( F |` ( A [,] B ) )
73	72	eqimss2i 3660	. . . . . . . . . . 11 |- ran ( F |` ( A [,] B ) ) C_ ( F " ( A [,] B ) )
74	73	a1i 11	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ran ( F |` ( A [,] B ) ) C_ ( F " ( A [,] B ) ) )
75		ax-resscn 9993	. . . . . . . . . . 11 |- RR C_ CC
76	24, 75	syl6ss 3615	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F " ( A [,] B ) ) C_ CC )
77		cnrest2 21090	. . . . . . . . . 10 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ ran ( F |` ( A [,] B ) ) C_ ( F " ( A [,] B ) ) /\ ( F " ( A [,] B ) ) C_ CC ) -> ( ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) ) ) )
78	71, 74, 76, 77	syl3anc 1326	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( TopOpen ` ℂfld ) ) <-> ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) ) ) )
79	70, 78	mpbid 222	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) ) )
80	33, 65	rerest 22607	. . . . . . . . . 10 |- ( ( F " ( A [,] B ) ) C_ RR -> ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) = ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) )
81	24, 80	syl 17	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) = ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) )
82	81	oveq2d 6666	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( TopOpen ` ℂfld ) ↾t ( F " ( A [,] B ) ) ) ) = ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) ) )
83	79, 82	eleqtrd 2703	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) ) )
84		eqid 2622	. . . . . . . 8 |- U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) = U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) )
85	84	cnconn 21225	. . . . . . 7 |- ( ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) e. Conn /\ ( F |` ( A [,] B ) ) : ( A [,] B ) -onto-> U. ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) /\ ( F |` ( A [,] B ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( A [,] B ) ) Cn ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) ) ) -> ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn )
86	9, 30, 83, 85	syl3anc 1326	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn )
87		reconn 22631	. . . . . . . . 9 |- ( ( F " ( A [,] B ) ) C_ RR -> ( ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn <-> A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) ) )
88	87	3ad2ant2 1083	. . . . . . . 8 |- ( ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) -> ( ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn <-> A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) ) )
89	88	3ad2ant3 1084	. . . . . . 7 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> ( ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn <-> A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) ) )
90	89	3ad2ant3 1084	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( topGen ` ran (,) ) ↾t ( F " ( A [,] B ) ) ) e. Conn <-> A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) ) )
91	86, 90	mpbid 222	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) )
92		simp11 1091	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> A e. RR )
93	92	rexrd 10089	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> A e. RR* )
94		simp12 1092	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> B e. RR )
95	94	rexrd 10089	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> B e. RR* )
96		ltle 10126	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR ) -> ( A < B -> A <_ B ) )
97	96	imp 445	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR ) /\ A < B ) -> A <_ B )
98	97	3adantl3 1219	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B ) -> A <_ B )
99	98	3adant3 1081	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> A <_ B )
100		lbicc2 12288	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> A e. ( A [,] B ) )
101	93, 95, 99, 100	syl3anc 1326	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> A e. ( A [,] B ) )
102		funfvima2 6493	. . . . . . 7 |- ( ( Fun F /\ ( A [,] B ) C_ dom F ) -> ( A e. ( A [,] B ) -> ( F ` A ) e. ( F " ( A [,] B ) ) ) )
103	20, 101, 102	sylc 65	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` A ) e. ( F " ( A [,] B ) ) )
104		ubicc2 12289	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
105	93, 95, 99, 104	syl3anc 1326	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> B e. ( A [,] B ) )
106		funfvima2 6493	. . . . . . 7 |- ( ( Fun F /\ ( A [,] B ) C_ dom F ) -> ( B e. ( A [,] B ) -> ( F ` B ) e. ( F " ( A [,] B ) ) ) )
107	20, 105, 106	sylc 65	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` B ) e. ( F " ( A [,] B ) ) )
108		oveq1 6657	. . . . . . . 8 |- ( x = ( F ` A ) -> ( x [,] y ) = ( ( F ` A ) [,] y ) )
109	108	sseq1d 3632	. . . . . . 7 |- ( x = ( F ` A ) -> ( ( x [,] y ) C_ ( F " ( A [,] B ) ) <-> ( ( F ` A ) [,] y ) C_ ( F " ( A [,] B ) ) ) )
110		oveq2 6658	. . . . . . . 8 |- ( y = ( F ` B ) -> ( ( F ` A ) [,] y ) = ( ( F ` A ) [,] ( F ` B ) ) )
111	110	sseq1d 3632	. . . . . . 7 |- ( y = ( F ` B ) -> ( ( ( F ` A ) [,] y ) C_ ( F " ( A [,] B ) ) <-> ( ( F ` A ) [,] ( F ` B ) ) C_ ( F " ( A [,] B ) ) ) )
112	109, 111	rspc2v 3322	. . . . . 6 |- ( ( ( F ` A ) e. ( F " ( A [,] B ) ) /\ ( F ` B ) e. ( F " ( A [,] B ) ) ) -> ( A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) -> ( ( F ` A ) [,] ( F ` B ) ) C_ ( F " ( A [,] B ) ) ) )
113	103, 107, 112	syl2anc 693	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( A. x e. ( F " ( A [,] B ) ) A. y e. ( F " ( A [,] B ) ) ( x [,] y ) C_ ( F " ( A [,] B ) ) -> ( ( F ` A ) [,] ( F ` B ) ) C_ ( F " ( A [,] B ) ) ) )
114	91, 113	mpd 15	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( F ` A ) [,] ( F ` B ) ) C_ ( F " ( A [,] B ) ) )
115		ioossicc 12259	. . . . . . . 8 |- ( ( F ` A ) (,) ( F ` B ) ) C_ ( ( F ` A ) [,] ( F ` B ) )
116	115	sseli 3599	. . . . . . 7 |- ( U e. ( ( F ` A ) (,) ( F ` B ) ) -> U e. ( ( F ` A ) [,] ( F ` B ) ) )
117	116	3ad2ant3 1084	. . . . . 6 |- ( ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) -> U e. ( ( F ` A ) [,] ( F ` B ) ) )
118	117	3ad2ant3 1084	. . . . 5 |- ( ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) -> U e. ( ( F ` A ) [,] ( F ` B ) ) )
119	118	3ad2ant3 1084	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U e. ( ( F ` A ) [,] ( F ` B ) ) )
120	114, 119	sseldd 3604	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U e. ( F " ( A [,] B ) ) )
121		fvelima 6248	. . 3 |- ( ( Fun F /\ U e. ( F " ( A [,] B ) ) ) -> E. x e. ( A [,] B ) ( F ` x ) = U )
122	6, 120, 121	syl2anc 693	. 2 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> E. x e. ( A [,] B ) ( F ` x ) = U )
123		simpl1 1064	. . . . . . . 8 |- ( ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) -> x e. RR* )
124	123	a1i 11	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) -> x e. RR* ) )
125		simprr 796	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> ( F ` x ) = U )
126	24, 103	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` A ) e. RR )
127		simp333 1216	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U e. ( ( F ` A ) (,) ( F ` B ) ) )
128	126	rexrd 10089	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` A ) e. RR* )
129	24, 107	sseldd 3604	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` B ) e. RR )
130	129	rexrd 10089	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` B ) e. RR* )
131		elioo2 12216	. . . . . . . . . . . . . . . . 17 |- ( ( ( F ` A ) e. RR* /\ ( F ` B ) e. RR* ) -> ( U e. ( ( F ` A ) (,) ( F ` B ) ) <-> ( U e. RR /\ ( F ` A ) < U /\ U < ( F ` B ) ) ) )
132	128, 130, 131	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( U e. ( ( F ` A ) (,) ( F ` B ) ) <-> ( U e. RR /\ ( F ` A ) < U /\ U < ( F ` B ) ) ) )
133	127, 132	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( U e. RR /\ ( F ` A ) < U /\ U < ( F ` B ) ) )
134	133	simp2d 1074	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( F ` A ) < U )
135	126, 134	gtned 10172	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U =/= ( F ` A ) )
136	135	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> U =/= ( F ` A ) )
137	125, 136	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> ( F ` x ) =/= ( F ` A ) )
138	137	neneqd 2799	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> -. ( F ` x ) = ( F ` A ) )
139		fveq2 6191	. . . . . . . . . 10 |- ( x = A -> ( F ` x ) = ( F ` A ) )
140	138, 139	nsyl 135	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> -. x = A )
141		simp13 1093	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U e. RR )
142	133	simp3d 1075	. . . . . . . . . . . . . 14 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U < ( F ` B ) )
143	141, 142	ltned 10173	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> U =/= ( F ` B ) )
144	143	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> U =/= ( F ` B ) )
145	125, 144	eqnetrd 2861	. . . . . . . . . . 11 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> ( F ` x ) =/= ( F ` B ) )
146	145	neneqd 2799	. . . . . . . . . 10 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> -. ( F ` x ) = ( F ` B ) )
147		fveq2 6191	. . . . . . . . . 10 |- ( x = B -> ( F ` x ) = ( F ` B ) )
148	146, 147	nsyl 135	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> -. x = B )
149		simprl3 1108	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> ( x = A \/ ( A < x /\ x < B ) \/ x = B ) )
150	140, 148, 149	ecase13d 32307	. . . . . . . 8 |- ( ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) /\ ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) -> ( A < x /\ x < B ) )
151	150	ex 450	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) -> ( A < x /\ x < B ) ) )
152	124, 151	jcad 555	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) -> ( x e. RR* /\ ( A < x /\ x < B ) ) ) )
153		3anass 1042	. . . . . 6 |- ( ( x e. RR* /\ A < x /\ x < B ) <-> ( x e. RR* /\ ( A < x /\ x < B ) ) )
154	152, 153	syl6ibr 242	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) -> ( x e. RR* /\ A < x /\ x < B ) ) )
155		rexr 10085	. . . . . . . . 9 |- ( A e. RR -> A e. RR* )
156		rexr 10085	. . . . . . . . 9 |- ( B e. RR -> B e. RR* )
157		elicc3 32311	. . . . . . . . 9 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) ) )
158	155, 156, 157	syl2an 494	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) ) )
159	158	3adant3 1081	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ U e. RR ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) ) )
160	159	3ad2ant1 1082	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( x e. ( A [,] B ) <-> ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) ) )
161	160	anbi1d 741	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( x e. ( A [,] B ) /\ ( F ` x ) = U ) <-> ( ( x e. RR* /\ A <_ B /\ ( x = A \/ ( A < x /\ x < B ) \/ x = B ) ) /\ ( F ` x ) = U ) ) )
162		elioo1 12215	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( x e. ( A (,) B ) <-> ( x e. RR* /\ A < x /\ x < B ) ) )
163	155, 156, 162	syl2an 494	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( x e. ( A (,) B ) <-> ( x e. RR* /\ A < x /\ x < B ) ) )
164	163	3adant3 1081	. . . . . 6 |- ( ( A e. RR /\ B e. RR /\ U e. RR ) -> ( x e. ( A (,) B ) <-> ( x e. RR* /\ A < x /\ x < B ) ) )
165	164	3ad2ant1 1082	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( x e. ( A (,) B ) <-> ( x e. RR* /\ A < x /\ x < B ) ) )
166	154, 161, 165	3imtr4d 283	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( x e. ( A [,] B ) /\ ( F ` x ) = U ) -> x e. ( A (,) B ) ) )
167		simpr 477	. . . . 5 |- ( ( x e. ( A [,] B ) /\ ( F ` x ) = U ) -> ( F ` x ) = U )
168	167	a1i 11	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( x e. ( A [,] B ) /\ ( F ` x ) = U ) -> ( F ` x ) = U ) )
169	166, 168	jcad 555	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( ( x e. ( A [,] B ) /\ ( F ` x ) = U ) -> ( x e. ( A (,) B ) /\ ( F ` x ) = U ) ) )
170	169	reximdv2 3014	. 2 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> ( E. x e. ( A [,] B ) ( F ` x ) = U -> E. x e. ( A (,) B ) ( F ` x ) = U ) )
171	122, 170	mpd 15	1 |- ( ( ( A e. RR /\ B e. RR /\ U e. RR ) /\ A < B /\ ( ( A [,] B ) C_ D /\ D C_ CC /\ ( F e. ( D -cn-> CC ) /\ ( F " ( A [,] B ) ) C_ RR /\ U e. ( ( F ` A ) (,) ( F ` B ) ) ) ) ) -> E. x e. ( A (,) B ) ( F ` x ) = U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175   [,]cicc 12178   ↾_t crest 16081   TopOpenctopn 16082   topGenctg 16098  ℂ_fldccnfld 19746   Topctop 20698  TopOnctopon 20715   Cn ccn 21028  Conncconn 21214   -cn->ccncf 22679

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-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-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-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-ioo 12179  df-ico 12181  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-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-conn 21215  df-xms 22125  df-ms 22126  df-cncf 22681

This theorem is referenced by: (None)