Theorem ioorrnopnxrlem 40526

Description: Given a point F that belongs to an indexed product of (possibly unbounded) open intervals, then F belongs to an open product of bounded open intervals that's a subset of the original indexed product. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
ioorrnopnxrlem.x	|- ( ph -> X e. Fin )
ioorrnopnxrlem.a	|- ( ph -> A : X --> RR* )
ioorrnopnxrlem.b	|- ( ph -> B : X --> RR* )
ioorrnopnxrlem.f	|- ( ph -> F e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
ioorrnopnxrlem.l	|- L = ( i e. X |-> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
ioorrnopnxrlem.r	|- R = ( i e. X |-> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) )
ioorrnopnxrlem.v	|- V = X_ i e. X ( ( L ` i ) (,) ( R ` i ) )

Assertion

Ref	Expression
ioorrnopnxrlem	|- ( ph -> E. v e. ( TopOpen ` (ℝ^ ` X ) ) ( F e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) )

Distinct variable groups:   v, A   v, B   i, F, v   i, L   R, i   v, V   i, X, v   ph, i

Allowed substitution hints:   ph( v)   A( i)   B( i)   R( v)   L( v)   V( i)

Proof of Theorem ioorrnopnxrlem

Step	Hyp	Ref	Expression
1		ioorrnopnxrlem.v	. . . 4 |- V = X_ i e. X ( ( L ` i ) (,) ( R ` i ) )
2	1	a1i 11	. . 3 |- ( ph -> V = X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) )
3		ioorrnopnxrlem.x	. . . 4 |- ( ph -> X e. Fin )
4		iftrue 4092	. . . . . . . 8 |- ( ( A ` i ) = -oo -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) = ( ( F ` i ) - 1 ) )
5	4	adantl 482	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) = ( ( F ` i ) - 1 ) )
6		ioorrnopnxrlem.f	. . . . . . . . . . . 12 |- ( ph -> F e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
7	6	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> F e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
8		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> i e. X )
9		fvixp2 39389	. . . . . . . . . . 11 |- ( ( F e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) /\ i e. X ) -> ( F ` i ) e. ( ( A ` i ) (,) ( B ` i ) ) )
10	7, 8, 9	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ i e. X ) -> ( F ` i ) e. ( ( A ` i ) (,) ( B ` i ) ) )
11	10	elioored 39776	. . . . . . . . 9 |- ( ( ph /\ i e. X ) -> ( F ` i ) e. RR )
12		1red 10055	. . . . . . . . 9 |- ( ( ph /\ i e. X ) -> 1 e. RR )
13	11, 12	resubcld 10458	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( ( F ` i ) - 1 ) e. RR )
14	13	adantr 481	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( ( F ` i ) - 1 ) e. RR )
15	5, 14	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) e. RR )
16		iffalse 4095	. . . . . . . 8 |- ( -. ( A ` i ) = -oo -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) = ( A ` i ) )
17	16	adantl 482	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) = ( A ` i ) )
18		neqne 2802	. . . . . . . . 9 |- ( -. ( A ` i ) = -oo -> ( A ` i ) =/= -oo )
19	18	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( A ` i ) =/= -oo )
20		ioorrnopnxrlem.a	. . . . . . . . . . 11 |- ( ph -> A : X --> RR* )
21	20	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ i e. X ) -> ( A ` i ) e. RR* )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) =/= -oo ) -> ( A ` i ) e. RR* )
23		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) =/= -oo ) -> ( A ` i ) =/= -oo )
24		pnfxr 10092	. . . . . . . . . . . 12 |- +oo e. RR*
25	24	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> +oo e. RR* )
26	11	rexrd 10089	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> ( F ` i ) e. RR* )
27		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 |- ( ph -> B : X --> RR* )
28	27	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. X ) -> ( B ` i ) e. RR* )
29		ioogtlb 39717	. . . . . . . . . . . . 13 |- ( ( ( A ` i ) e. RR* /\ ( B ` i ) e. RR* /\ ( F ` i ) e. ( ( A ` i ) (,) ( B ` i ) ) ) -> ( A ` i ) < ( F ` i ) )
30	21, 28, 10, 29	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> ( A ` i ) < ( F ` i ) )
31	11	ltpnfd 11955	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> ( F ` i ) < +oo )
32	21, 26, 25, 30, 31	xrlttrd 11990	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> ( A ` i ) < +oo )
33	21, 25, 32	xrltned 39573	. . . . . . . . . 10 |- ( ( ph /\ i e. X ) -> ( A ` i ) =/= +oo )
34	33	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) =/= -oo ) -> ( A ` i ) =/= +oo )
35	22, 23, 34	xrred 39581	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) =/= -oo ) -> ( A ` i ) e. RR )
36	19, 35	syldan 487	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( A ` i ) e. RR )
37	17, 36	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) e. RR )
38	15, 37	pm2.61dan 832	. . . . 5 |- ( ( ph /\ i e. X ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) e. RR )
39		ioorrnopnxrlem.l	. . . . 5 |- L = ( i e. X |-> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
40	38, 39	fmptd 6385	. . . 4 |- ( ph -> L : X --> RR )
41		iftrue 4092	. . . . . . . 8 |- ( ( B ` i ) = +oo -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) = ( ( F ` i ) + 1 ) )
42	41	adantl 482	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) = ( ( F ` i ) + 1 ) )
43	11, 12	readdcld 10069	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( ( F ` i ) + 1 ) e. RR )
44	43	adantr 481	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( ( F ` i ) + 1 ) e. RR )
45	42, 44	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) e. RR )
46		iffalse 4095	. . . . . . . 8 |- ( -. ( B ` i ) = +oo -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) = ( B ` i ) )
47	46	adantl 482	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) = ( B ` i ) )
48		neqne 2802	. . . . . . . . 9 |- ( -. ( B ` i ) = +oo -> ( B ` i ) =/= +oo )
49	48	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( B ` i ) =/= +oo )
50	28	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) =/= +oo ) -> ( B ` i ) e. RR* )
51		mnfxr 10096	. . . . . . . . . . . 12 |- -oo e. RR*
52	51	a1i 11	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> -oo e. RR* )
53	11	mnfltd 11958	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> -oo < ( F ` i ) )
54		iooltub 39735	. . . . . . . . . . . . 13 |- ( ( ( A ` i ) e. RR* /\ ( B ` i ) e. RR* /\ ( F ` i ) e. ( ( A ` i ) (,) ( B ` i ) ) ) -> ( F ` i ) < ( B ` i ) )
55	21, 28, 10, 54	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> ( F ` i ) < ( B ` i ) )
56	52, 26, 28, 53, 55	xrlttrd 11990	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> -oo < ( B ` i ) )
57	52, 28, 56	xrgtned 39538	. . . . . . . . . 10 |- ( ( ph /\ i e. X ) -> ( B ` i ) =/= -oo )
58	57	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) =/= +oo ) -> ( B ` i ) =/= -oo )
59		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) =/= +oo ) -> ( B ` i ) =/= +oo )
60	50, 58, 59	xrred 39581	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) =/= +oo ) -> ( B ` i ) e. RR )
61	49, 60	syldan 487	. . . . . . 7 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( B ` i ) e. RR )
62	47, 61	eqeltrd 2701	. . . . . 6 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) e. RR )
63	45, 62	pm2.61dan 832	. . . . 5 |- ( ( ph /\ i e. X ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) e. RR )
64		ioorrnopnxrlem.r	. . . . 5 |- R = ( i e. X |-> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) )
65	63, 64	fmptd 6385	. . . 4 |- ( ph -> R : X --> RR )
66	3, 40, 65	ioorrnopn 40525	. . 3 |- ( ph -> X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) e. ( TopOpen ` (ℝ^ ` X ) ) )
67	2, 66	eqeltrd 2701	. 2 |- ( ph -> V e. ( TopOpen ` (ℝ^ ` X ) ) )
68	6	elexd 3214	. . . . . 6 |- ( ph -> F e. _V )
69		ixpfn 7914	. . . . . . 7 |- ( F e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) -> F Fn X )
70	6, 69	syl 17	. . . . . 6 |- ( ph -> F Fn X )
71	40	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ i e. X ) -> ( L ` i ) e. RR )
72	71	rexrd 10089	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( L ` i ) e. RR* )
73	65	ffvelrnda 6359	. . . . . . . . 9 |- ( ( ph /\ i e. X ) -> ( R ` i ) e. RR )
74	73	rexrd 10089	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( R ` i ) e. RR* )
75	39	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> L = ( i e. X |-> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) ) )
76	38	elexd 3214	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. X ) -> if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) e. _V )
77	75, 76	fvmpt2d 6293	. . . . . . . . . . . 12 |- ( ( ph /\ i e. X ) -> ( L ` i ) = if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
78	77	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( L ` i ) = if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
79	78, 5	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( L ` i ) = ( ( F ` i ) - 1 ) )
80	11	ltm1d 10956	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> ( ( F ` i ) - 1 ) < ( F ` i ) )
81	80	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( ( F ` i ) - 1 ) < ( F ` i ) )
82	79, 81	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( L ` i ) < ( F ` i ) )
83	77	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( L ` i ) = if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
84	83, 17	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( L ` i ) = ( A ` i ) )
85	30	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( A ` i ) < ( F ` i ) )
86	84, 85	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( L ` i ) < ( F ` i ) )
87	82, 86	pm2.61dan 832	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( L ` i ) < ( F ` i ) )
88	11	ltp1d 10954	. . . . . . . . . . 11 |- ( ( ph /\ i e. X ) -> ( F ` i ) < ( ( F ` i ) + 1 ) )
89	88	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( F ` i ) < ( ( F ` i ) + 1 ) )
90	64	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> R = ( i e. X |-> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) ) )
91	63	elexd 3214	. . . . . . . . . . . . . 14 |- ( ( ph /\ i e. X ) -> if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) e. _V )
92	90, 91	fvmpt2d 6293	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. X ) -> ( R ` i ) = if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) )
93	92	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( R ` i ) = if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) )
94	93, 42	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( R ` i ) = ( ( F ` i ) + 1 ) )
95	94	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( ( F ` i ) + 1 ) = ( R ` i ) )
96	89, 95	breqtrd 4679	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( F ` i ) < ( R ` i ) )
97	55	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( F ` i ) < ( B ` i ) )
98	92	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( R ` i ) = if ( ( B ` i ) = +oo , ( ( F ` i ) + 1 ) , ( B ` i ) ) )
99	98, 47	eqtrd 2656	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( R ` i ) = ( B ` i ) )
100	99	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( B ` i ) = ( R ` i ) )
101	97, 100	breqtrd 4679	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( F ` i ) < ( R ` i ) )
102	96, 101	pm2.61dan 832	. . . . . . . 8 |- ( ( ph /\ i e. X ) -> ( F ` i ) < ( R ` i ) )
103	72, 74, 11, 87, 102	eliood 39720	. . . . . . 7 |- ( ( ph /\ i e. X ) -> ( F ` i ) e. ( ( L ` i ) (,) ( R ` i ) ) )
104	103	ralrimiva 2966	. . . . . 6 |- ( ph -> A. i e. X ( F ` i ) e. ( ( L ` i ) (,) ( R ` i ) ) )
105	68, 70, 104	3jca 1242	. . . . 5 |- ( ph -> ( F e. _V /\ F Fn X /\ A. i e. X ( F ` i ) e. ( ( L ` i ) (,) ( R ` i ) ) ) )
106		elixp2 7912	. . . . 5 |- ( F e. X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) <-> ( F e. _V /\ F Fn X /\ A. i e. X ( F ` i ) e. ( ( L ` i ) (,) ( R ` i ) ) ) )
107	105, 106	sylibr 224	. . . 4 |- ( ph -> F e. X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) )
108	107, 1	syl6eleqr 2712	. . 3 |- ( ph -> F e. V )
109	21	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( A ` i ) e. RR* )
110	72	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( L ` i ) e. RR* )
111	15	mnfltd 11958	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> -oo < if ( ( A ` i ) = -oo , ( ( F ` i ) - 1 ) , ( A ` i ) ) )
112	111, 5	breqtrd 4679	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> -oo < ( ( F ` i ) - 1 ) )
113		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( A ` i ) = -oo )
114	113, 79	breq12d 4666	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( ( A ` i ) < ( L ` i ) <-> -oo < ( ( F ` i ) - 1 ) ) )
115	112, 114	mpbird 247	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( A ` i ) < ( L ` i ) )
116	109, 110, 115	xrltled 39486	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ ( A ` i ) = -oo ) -> ( A ` i ) <_ ( L ` i ) )
117	84	eqcomd 2628	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( A ` i ) = ( L ` i ) )
118	36, 117	eqled 10140	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ -. ( A ` i ) = -oo ) -> ( A ` i ) <_ ( L ` i ) )
119	116, 118	pm2.61dan 832	. . . . . . 7 |- ( ( ph /\ i e. X ) -> ( A ` i ) <_ ( L ` i ) )
120	74	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( R ` i ) e. RR* )
121	28	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( B ` i ) e. RR* )
122	44	ltpnfd 11955	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( ( F ` i ) + 1 ) < +oo )
123		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( B ` i ) = +oo )
124	94, 123	breq12d 4666	. . . . . . . . . 10 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( ( R ` i ) < ( B ` i ) <-> ( ( F ` i ) + 1 ) < +oo ) )
125	122, 124	mpbird 247	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( R ` i ) < ( B ` i ) )
126	120, 121, 125	xrltled 39486	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ ( B ` i ) = +oo ) -> ( R ` i ) <_ ( B ` i ) )
127	73	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( R ` i ) e. RR )
128	127, 99	eqled 10140	. . . . . . . 8 |- ( ( ( ph /\ i e. X ) /\ -. ( B ` i ) = +oo ) -> ( R ` i ) <_ ( B ` i ) )
129	126, 128	pm2.61dan 832	. . . . . . 7 |- ( ( ph /\ i e. X ) -> ( R ` i ) <_ ( B ` i ) )
130		ioossioo 12265	. . . . . . 7 |- ( ( ( ( A ` i ) e. RR* /\ ( B ` i ) e. RR* ) /\ ( ( A ` i ) <_ ( L ` i ) /\ ( R ` i ) <_ ( B ` i ) ) ) -> ( ( L ` i ) (,) ( R ` i ) ) C_ ( ( A ` i ) (,) ( B ` i ) ) )
131	21, 28, 119, 129, 130	syl22anc 1327	. . . . . 6 |- ( ( ph /\ i e. X ) -> ( ( L ` i ) (,) ( R ` i ) ) C_ ( ( A ` i ) (,) ( B ` i ) ) )
132	131	ralrimiva 2966	. . . . 5 |- ( ph -> A. i e. X ( ( L ` i ) (,) ( R ` i ) ) C_ ( ( A ` i ) (,) ( B ` i ) ) )
133		ss2ixp 7921	. . . . 5 |- ( A. i e. X ( ( L ` i ) (,) ( R ` i ) ) C_ ( ( A ` i ) (,) ( B ` i ) ) -> X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
134	132, 133	syl 17	. . . 4 |- ( ph -> X_ i e. X ( ( L ` i ) (,) ( R ` i ) ) C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
135	2, 134	eqsstrd 3639	. . 3 |- ( ph -> V C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) )
136	108, 135	jca 554	. 2 |- ( ph -> ( F e. V /\ V C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) )
137		eleq2 2690	. . . 4 |- ( v = V -> ( F e. v <-> F e. V ) )
138		sseq1 3626	. . . 4 |- ( v = V -> ( v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) <-> V C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) )
139	137, 138	anbi12d 747	. . 3 |- ( v = V -> ( ( F e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) <-> ( F e. V /\ V C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) )
140	139	rspcev 3309	. 2 |- ( ( V e. ( TopOpen ` (ℝ^ ` X ) ) /\ ( F e. V /\ V C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) -> E. v e. ( TopOpen ` (ℝ^ ` X ) ) ( F e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) )
141	67, 136, 140	syl2anc 693	1 |- ( ph -> E. v e. ( TopOpen ` (ℝ^ ` X ) ) ( F e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   Fincfn 7955   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   (,)cioo 12175   TopOpenctopn 16082  ℝ^crrx 23171

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-fal 1489  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-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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-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-clim 14219  df-sum 14417  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-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-rnghom 18715  df-drng 18749  df-field 18750  df-subrg 18778  df-abv 18817  df-staf 18845  df-srng 18846  df-lmod 18865  df-lss 18933  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-refld 19951  df-phl 19971  df-dsmm 20076  df-frlm 20091  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-tng 22389  df-nrg 22390  df-nlm 22391  df-clm 22863  df-cph 22968  df-tch 22969  df-rrx 23173

This theorem is referenced by:  ioorrnopnxr  40527