ioorrnopnxrlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnxrlem 42047

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	⊢ (𝜑 → 𝑋 ∈ Fin)
ioorrnopnxrlem.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
ioorrnopnxrlem.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
ioorrnopnxrlem.f	⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
ioorrnopnxrlem.l	⊢ 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
ioorrnopnxrlem.r	⊢ 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
ioorrnopnxrlem.v	⊢ 𝑉 = X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖))

**Assertion**
Ref	Expression
ioorrnopnxrlem	⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))

Distinct variable groups: 𝑣,𝐴 𝑣,𝐵 𝑖,𝐹,𝑣 𝑖,𝐿 𝑅,𝑖 𝑣,𝑉 𝑖,𝑋,𝑣 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑣) 𝐴(𝑖) 𝐵(𝑖) 𝑅(𝑣) 𝐿(𝑣) 𝑉(𝑖)

**Proof of Theorem ioorrnopnxrlem**
Step	Hyp	Ref	Expression
1		ioorrnopnxrlem.v	. . . 4 ⊢ 𝑉 = X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝑉 = X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
3		ioorrnopnxrlem.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		iftrue 4350	. . . . . . . 8 ⊢ ((𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = ((𝐹‘𝑖) − 1))
5	4	adantl 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = ((𝐹‘𝑖) − 1))
6		ioorrnopnxrlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
7	6	adantr 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
8		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
9		fvixp2 40912	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
10	7, 8, 9	syl2anc 576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
11	10	elioored 41281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
12		1red 10438	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 1 ∈ ℝ)
13	11, 12	resubcld 10867	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) ∈ ℝ)
14	13	adantr 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) ∈ ℝ)
15	5, 14	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
16		iffalse 4353	. . . . . . . 8 ⊢ (¬ (𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
17	16	adantl 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
18		neqne 2969	. . . . . . . . 9 ⊢ (¬ (𝐴‘𝑖) = -∞ → (𝐴‘𝑖) ≠ -∞)
19	18	adantl 474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≠ -∞)
20		ioorrnopnxrlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
21	20	ffvelrnda 6674	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
22	21	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ^*)
23		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ -∞)
24		pnfxr 10492	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → +∞ ∈ ℝ^*)
26	11	rexrd 10488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ^*)
27		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
28	27	ffvelrnda 6674	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
29		ioogtlb 41226	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
30	21, 28, 10, 29	syl3anc 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
31	11	ltpnfd 12331	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < +∞)
32	21, 26, 25, 30, 31	xrlttrd 12367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < +∞)
33	21, 25, 32	xrltned 41079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≠ +∞)
34	33	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ +∞)
35	22, 23, 34	xrred 41087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ)
36	19, 35	syldan 582	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ)
37	17, 36	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
38	15, 37	pm2.61dan 800	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
39		ioorrnopnxrlem.l	. . . . 5 ⊢ 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
40	38, 39	fmptd 6699	. . . 4 ⊢ (𝜑 → 𝐿:𝑋⟶ℝ)
41		iftrue 4350	. . . . . . . 8 ⊢ ((𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
42	41	adantl 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
43	11, 12	readdcld 10467	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 1) ∈ ℝ)
44	43	adantr 473	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) ∈ ℝ)
45	42, 44	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
46		iffalse 4353	. . . . . . . 8 ⊢ (¬ (𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
47	46	adantl 474	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
48		neqne 2969	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑖) = +∞ → (𝐵‘𝑖) ≠ +∞)
49	48	adantl 474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ≠ +∞)
50	28	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ^*)
51		mnfxr 10496	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ ∈ ℝ^*)
53	11	mnfltd 12334	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐹‘𝑖))
54		iooltub 41242	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
55	21, 28, 10, 54	syl3anc 1351	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
56	52, 26, 28, 53, 55	xrlttrd 12367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐵‘𝑖))
57	52, 28, 56	xrgtned 41044	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ≠ -∞)
58	57	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ -∞)
59		simpr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ +∞)
60	50, 58, 59	xrred 41087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ)
61	49, 60	syldan 582	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ)
62	47, 61	eqeltrd 2860	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
63	45, 62	pm2.61dan 800	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
64		ioorrnopnxrlem.r	. . . . 5 ⊢ 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
65	63, 64	fmptd 6699	. . . 4 ⊢ (𝜑 → 𝑅:𝑋⟶ℝ)
66	3, 40, 65	ioorrnopn 42046	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
67	2, 66	eqeltrd 2860	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
68	6	elexd 3429	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69		ixpfn 8263	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹 Fn 𝑋)
70	6, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
71	40	ffvelrnda 6674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ)
72	71	rexrd 10488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ^*)
73	65	ffvelrnda 6674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ)
74	73	rexrd 10488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ^*)
75	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖))))
76	38	elexd 3429	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ V)
77	75, 76	fvmpt2d 6605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
78	77	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
79	78, 5	eqtrd 2808	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = ((𝐹‘𝑖) − 1))
80	11	ltm1d 11371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
81	80	adantr 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
82	79, 81	eqbrtrd 4947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
83	77	adantr 473	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
84	83, 17	eqtrd 2808	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = (𝐴‘𝑖))
85	30	adantr 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐹‘𝑖))
86	84, 85	eqbrtrd 4947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
87	82, 86	pm2.61dan 800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) < (𝐹‘𝑖))
88	11	ltp1d 11369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
89	88	adantr 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
90	64	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖))))
91	63	elexd 3429	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ V)
92	90, 91	fvmpt2d 6605	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
93	92	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
94	93, 42	eqtrd 2808	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = ((𝐹‘𝑖) + 1))
95	94	eqcomd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) = (𝑅‘𝑖))
96	89, 95	breqtrd 4951	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
97	55	adantr 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝐵‘𝑖))
98	92	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
99	98, 47	eqtrd 2808	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = (𝐵‘𝑖))
100	99	eqcomd 2778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = (𝑅‘𝑖))
101	97, 100	breqtrd 4951	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
102	96, 101	pm2.61dan 800	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝑅‘𝑖))
103	72, 74, 11, 87, 102	eliood 41229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
104	103	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
105	68, 70, 104	3jca 1108	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
106		elixp2 8261	. . . . 5 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
107	105, 106	sylibr 226	. . . 4 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
108	107, 1	syl6eleqr 2871	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
109	21	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ^*)
110	72	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) ∈ ℝ^*)
111	15	mnfltd 12334	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
112	111, 5	breqtrd 4951	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < ((𝐹‘𝑖) − 1))
113		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = -∞)
114	113, 79	breq12d 4938	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐴‘𝑖) < (𝐿‘𝑖) ↔ -∞ < ((𝐹‘𝑖) − 1)))
115	112, 114	mpbird 249	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐿‘𝑖))
116	109, 110, 115	xrltled 12358	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
117	84	eqcomd 2778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = (𝐿‘𝑖))
118	36, 117	eqled 10541	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
119	116, 118	pm2.61dan 800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
120	74	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ^*)
121	28	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ^*)
122	44	ltpnfd 12331	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) < +∞)
123		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = +∞)
124	94, 123	breq12d 4938	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝑅‘𝑖) < (𝐵‘𝑖) ↔ ((𝐹‘𝑖) + 1) < +∞))
125	122, 124	mpbird 249	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) < (𝐵‘𝑖))
126	120, 121, 125	xrltled 12358	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
127	73	adantr 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ)
128	127, 99	eqled 10541	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
129	126, 128	pm2.61dan 800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
130		ioossioo 12643	. . . . . . 7 ⊢ ((((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^*) ∧ ((𝐴‘𝑖) ≤ (𝐿‘𝑖) ∧ (𝑅‘𝑖) ≤ (𝐵‘𝑖))) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
131	21, 28, 119, 129, 130	syl22anc 826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
132	131	ralrimiva 3126	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
133		ss2ixp 8270	. . . . 5 ⊢ (∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
134	132, 133	syl 17	. . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
135	2, 134	eqsstrd 3889	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
136	108, 135	jca 504	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
137		eleq2 2848	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
138		sseq1 3876	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
139	137, 138	anbi12d 621	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
140	139	rspcev 3529	. 2 ⊢ ((𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)) ∧ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))) → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
141	67, 136, 140	syl2anc 576	1 ⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 Vcvv 3409 ⊆ wss 3823 ifcif 4344 class class class wbr 4925 ↦ cmpt 5004 Fn wfn 6180 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Xcixp 8257 Fincfn 8304 ℝcr 10332 1c1 10334 + caddc 10336 +∞cpnf 10469 -∞cmnf 10470 ℝ^*cxr 10471 < clt 10472 ≤ cle 10473 − cmin 10668 (,)cioo 12552 TopOpenctopn 16549 ℝ^crrx 23701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-inf2 8896 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-ico 12558 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-clim 14704 df-sum 14902 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-prds 16575 df-pws 16577 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-dvr 19168 df-rnghom 19202 df-drng 19239 df-field 19240 df-subrg 19268 df-abv 19322 df-staf 19350 df-srng 19351 df-lmod 19370 df-lss 19438 df-lmhm 19528 df-lvec 19609 df-sra 19678 df-rgmod 19679 df-psmet 20251 df-xmet 20252 df-met 20253 df-bl 20254 df-mopn 20255 df-cnfld 20260 df-refld 20463 df-phl 20484 df-dsmm 20590 df-frlm 20605 df-top 21218 df-topon 21235 df-topsp 21257 df-bases 21270 df-xms 22645 df-ms 22646 df-nm 22907 df-ngp 22908 df-tng 22909 df-nrg 22910 df-nlm 22911 df-clm 23382 df-cph 23487 df-tcph 23488 df-rrx 23703

This theorem is referenced by: ioorrnopnxr 42048

W3C validator