ioorrnopnxrlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnxrlem 39202

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 4042	. . . . . . . 8 ⊢ ((𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = ((𝐹‘𝑖) − 1))
5	4	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = ((𝐹‘𝑖) − 1))
6		ioossre 12106	. . . . . . . . . 10 ⊢ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ⊆ ℝ
7		ioorrnopnxrlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
8	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
9		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
10		fvixp2 38384	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
11	8, 9, 10	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
12	6, 11	sseldi 3566	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
13		1red 9934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 1 ∈ ℝ)
14	12, 13	resubcld 10337	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) ∈ ℝ)
15	14	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) ∈ ℝ)
16	5, 15	eqeltrd 2688	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
17		iffalse 4045	. . . . . . . 8 ⊢ (¬ (𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
18	17	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
19		neqne 2790	. . . . . . . . 9 ⊢ (¬ (𝐴‘𝑖) = -∞ → (𝐴‘𝑖) ≠ -∞)
20	19	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≠ -∞)
21		ioorrnopnxrlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
22	21	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
23	22	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ^*)
24		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ -∞)
25		pnfxr 9971	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → +∞ ∈ ℝ^*)
27	12	rexrd 9968	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ^*)
28		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
29	28	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
30		ioogtlb 38564	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
31	22, 29, 11, 30	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
32	12	ltpnfd 11831	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < +∞)
33	22, 27, 26, 31, 32	xrlttrd 11866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < +∞)
34	22, 26, 33	xrltned 38514	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≠ +∞)
35	34	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ +∞)
36	23, 24, 35	xrred 38522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ)
37	20, 36	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ)
38	18, 37	eqeltrd 2688	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
39	16, 38	pm2.61dan 828	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
40		ioorrnopnxrlem.l	. . . . 5 ⊢ 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
41	39, 40	fmptd 6292	. . . 4 ⊢ (𝜑 → 𝐿:𝑋⟶ℝ)
42		iftrue 4042	. . . . . . . 8 ⊢ ((𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
43	42	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
44	12, 13	readdcld 9948	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 1) ∈ ℝ)
45	44	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) ∈ ℝ)
46	43, 45	eqeltrd 2688	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
47		iffalse 4045	. . . . . . . 8 ⊢ (¬ (𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
48	47	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
49		neqne 2790	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑖) = +∞ → (𝐵‘𝑖) ≠ +∞)
50	49	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ≠ +∞)
51	29	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ^*)
52		mnfxr 9975	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
53	52	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ ∈ ℝ^*)
54	12	mnfltd 11834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐹‘𝑖))
55		iooltub 38582	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
56	22, 29, 11, 55	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
57	53, 27, 29, 54, 56	xrlttrd 11866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐵‘𝑖))
58	53, 29, 57	xrgtned 38479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ≠ -∞)
59	58	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ -∞)
60		simpr 476	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ +∞)
61	51, 59, 60	xrred 38522	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ)
62	50, 61	syldan 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ)
63	48, 62	eqeltrd 2688	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
64	46, 63	pm2.61dan 828	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
65		ioorrnopnxrlem.r	. . . . 5 ⊢ 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
66	64, 65	fmptd 6292	. . . 4 ⊢ (𝜑 → 𝑅:𝑋⟶ℝ)
67	3, 41, 66	ioorrnopn 39201	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
68	2, 67	eqeltrd 2688	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
69	7	elexd 3187	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
70		ixpfn 7800	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹 Fn 𝑋)
71	7, 70	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
72	41	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ)
73	72	rexrd 9968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ^*)
74	66	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ)
75	74	rexrd 9968	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ^*)
76	40	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 = (𝑖 ∈ 𝑋 ↦ if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖))))
77	39	elexd 3187	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ V)
78	76, 77	fvmpt2d 6202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
79	78	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
80	79, 5	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = ((𝐹‘𝑖) − 1))
81	12	ltm1d 10835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
82	81	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
83	80, 82	eqbrtrd 4605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
84	78	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
85	84, 18	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = (𝐴‘𝑖))
86	31	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐹‘𝑖))
87	85, 86	eqbrtrd 4605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
88	83, 87	pm2.61dan 828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) < (𝐹‘𝑖))
89	12	ltp1d 10833	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
90	89	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
91	65	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 = (𝑖 ∈ 𝑋 ↦ if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖))))
92	64	elexd 3187	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ V)
93	91, 92	fvmpt2d 6202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
94	93	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
95	94, 43	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = ((𝐹‘𝑖) + 1))
96	95	eqcomd 2616	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) = (𝑅‘𝑖))
97	90, 96	breqtrd 4609	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
98	56	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝐵‘𝑖))
99	93	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
100	99, 48	eqtrd 2644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = (𝐵‘𝑖))
101	100	eqcomd 2616	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = (𝑅‘𝑖))
102	98, 101	breqtrd 4609	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
103	97, 102	pm2.61dan 828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝑅‘𝑖))
104	73, 75, 12, 88, 103	eliood 38567	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
105	104	ralrimiva 2949	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
106	69, 71, 105	3jca 1235	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
107		elixp2 7798	. . . . 5 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
108	106, 107	sylibr 223	. . . 4 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
109	108, 1	syl6eleqr 2699	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
110	22	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ^*)
111	73	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) ∈ ℝ^*)
112	16	mnfltd 11834	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
113	112, 5	breqtrd 4609	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < ((𝐹‘𝑖) − 1))
114		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = -∞)
115	114, 80	breq12d 4596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐴‘𝑖) < (𝐿‘𝑖) ↔ -∞ < ((𝐹‘𝑖) − 1)))
116	113, 115	mpbird 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐿‘𝑖))
117	110, 111, 116	xrltled 38427	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
118	85	eqcomd 2616	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = (𝐿‘𝑖))
119	37, 118	eqled 10019	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
120	117, 119	pm2.61dan 828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
121	75	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ^*)
122	29	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ^*)
123	45	ltpnfd 11831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) < +∞)
124		simpr 476	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = +∞)
125	95, 124	breq12d 4596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝑅‘𝑖) < (𝐵‘𝑖) ↔ ((𝐹‘𝑖) + 1) < +∞))
126	123, 125	mpbird 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) < (𝐵‘𝑖))
127	121, 122, 126	xrltled 38427	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
128	74	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ)
129	128, 100	eqled 10019	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
130	127, 129	pm2.61dan 828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
131		ioossioo 12136	. . . . . . 7 ⊢ ((((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^*) ∧ ((𝐴‘𝑖) ≤ (𝐿‘𝑖) ∧ (𝑅‘𝑖) ≤ (𝐵‘𝑖))) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
132	22, 29, 120, 130, 131	syl22anc 1319	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
133	132	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
134		ss2ixp 7807	. . . . 5 ⊢ (∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
135	133, 134	syl 17	. . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
136	2, 135	eqsstrd 3602	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
137	109, 136	jca 553	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
138		eleq2 2677	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
139		sseq1 3589	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
140	138, 139	anbi12d 743	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
141	140	rspcev 3282	. 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))))
142	68, 137, 141	syl2anc 691	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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 Fincfn 7841 ℝcr 9814 1c1 9816 + caddc 9818 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 (,)cioo 12046 TopOpenctopn 15905 ℝ^crrx 22979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-abv 18640 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-refld 19770 df-phl 19790 df-dsmm 19895 df-frlm 19910 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-tng 22199 df-nrg 22200 df-nlm 22201 df-clm 22671 df-cph 22776 df-tch 22777 df-rrx 22981

This theorem is referenced by: ioorrnopnxr 39203

W3C validator