ioorrnopnxrlem - Mathbox for Glauco Siliprandi

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Theorem ioorrnopnxrlem 46262

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 4537	. . . . . . . 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		ioorrnopnxrlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
8		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
9		fvixp2 45142	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
10	7, 8, 9	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
11	10	elioored 45502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
12		1red 11260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 1 ∈ ℝ)
13	11, 12	resubcld 11689	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) ∈ ℝ)
14	13	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) ∈ ℝ)
15	5, 14	eqeltrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
16		iffalse 4540	. . . . . . . 8 ⊢ (¬ (𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
17	16	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
18		neqne 2946	. . . . . . . . 9 ⊢ (¬ (𝐴‘𝑖) = -∞ → (𝐴‘𝑖) ≠ -∞)
19	18	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≠ -∞)
20		ioorrnopnxrlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
21	20	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
22	21	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ^*)
23		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ -∞)
24		pnfxr 11313	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → +∞ ∈ ℝ^*)
26	11	rexrd 11309	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ^*)
27		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
28	27	ffvelcdmda 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
29		ioogtlb 45448	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
30	21, 28, 10, 29	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
31	11	ltpnfd 13161	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < +∞)
32	21, 26, 25, 30, 31	xrlttrd 13198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < +∞)
33	21, 25, 32	xrltned 45307	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≠ +∞)
34	33	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ +∞)
35	22, 23, 34	xrred 45315	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ)
36	19, 35	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ)
37	17, 36	eqeltrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
38	15, 37	pm2.61dan 813	. . . . 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 7134	. . . 4 ⊢ (𝜑 → 𝐿:𝑋⟶ℝ)
41		iftrue 4537	. . . . . . . 8 ⊢ ((𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
42	41	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
43	11, 12	readdcld 11288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 1) ∈ ℝ)
44	43	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) ∈ ℝ)
45	42, 44	eqeltrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
46		iffalse 4540	. . . . . . . 8 ⊢ (¬ (𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
47	46	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
48		neqne 2946	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑖) = +∞ → (𝐵‘𝑖) ≠ +∞)
49	48	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ≠ +∞)
50	28	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ^*)
51		mnfxr 11316	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ ∈ ℝ^*)
53	11	mnfltd 13164	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐹‘𝑖))
54		iooltub 45463	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
55	21, 28, 10, 54	syl3anc 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
56	52, 26, 28, 53, 55	xrlttrd 13198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐵‘𝑖))
57	52, 28, 56	xrgtned 45272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ≠ -∞)
58	57	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ -∞)
59		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ +∞)
60	50, 58, 59	xrred 45315	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ)
61	49, 60	syldan 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ)
62	47, 61	eqeltrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
63	45, 62	pm2.61dan 813	. . . . 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 7134	. . . 4 ⊢ (𝜑 → 𝑅:𝑋⟶ℝ)
66	3, 40, 65	ioorrnopn 46261	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
67	2, 66	eqeltrd 2839	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
68	6	elexd 3502	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69		ixpfn 8942	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹 Fn 𝑋)
70	6, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
71	40	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ)
72	71	rexrd 11309	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ^*)
73	65	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ)
74	73	rexrd 11309	. . . . . . . 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 3502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ V)
77	75, 76	fvmpt2d 7029	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
78	77	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
79	78, 5	eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = ((𝐹‘𝑖) − 1))
80	11	ltm1d 12198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
81	80	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
82	79, 81	eqbrtrd 5170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
83	77	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
84	83, 17	eqtrd 2775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = (𝐴‘𝑖))
85	30	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐹‘𝑖))
86	84, 85	eqbrtrd 5170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
87	82, 86	pm2.61dan 813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) < (𝐹‘𝑖))
88	11	ltp1d 12196	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
89	88	adantr 480	. . . . . . . . . 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 3502	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ V)
92	90, 91	fvmpt2d 7029	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
93	92	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
94	93, 42	eqtrd 2775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = ((𝐹‘𝑖) + 1))
95	94	eqcomd 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) = (𝑅‘𝑖))
96	89, 95	breqtrd 5174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
97	55	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝐵‘𝑖))
98	92	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
99	98, 47	eqtrd 2775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = (𝐵‘𝑖))
100	99	eqcomd 2741	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = (𝑅‘𝑖))
101	97, 100	breqtrd 5174	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
102	96, 101	pm2.61dan 813	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝑅‘𝑖))
103	72, 74, 11, 87, 102	eliood 45451	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
104	103	ralrimiva 3144	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
105	68, 70, 104	3jca 1127	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
106		elixp2 8940	. . . . 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 234	. . . 4 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
108	107, 1	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
109	21	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ^*)
110	72	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) ∈ ℝ^*)
111	15	mnfltd 13164	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
112	111, 5	breqtrd 5174	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < ((𝐹‘𝑖) − 1))
113		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = -∞)
114	113, 79	breq12d 5161	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐴‘𝑖) < (𝐿‘𝑖) ↔ -∞ < ((𝐹‘𝑖) − 1)))
115	112, 114	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐿‘𝑖))
116	109, 110, 115	xrltled 13189	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
117	84	eqcomd 2741	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = (𝐿‘𝑖))
118	36, 117	eqled 11362	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
119	116, 118	pm2.61dan 813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
120	74	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ^*)
121	28	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ^*)
122	44	ltpnfd 13161	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) < +∞)
123		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = +∞)
124	94, 123	breq12d 5161	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝑅‘𝑖) < (𝐵‘𝑖) ↔ ((𝐹‘𝑖) + 1) < +∞))
125	122, 124	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) < (𝐵‘𝑖))
126	120, 121, 125	xrltled 13189	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
127	73	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ)
128	127, 99	eqled 11362	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
129	126, 128	pm2.61dan 813	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
130		ioossioo 13478	. . . . . . 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 839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
132	131	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
133		ss2ixp 8949	. . . . 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 4034	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
136	108, 135	jca 511	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
137		eleq2 2828	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
138		sseq1 4021	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
139	137, 138	anbi12d 632	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
140	139	rspcev 3622	. 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 584	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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 Xcixp 8936 Fincfn 8984 ℝcr 11152 1c1 11154 + caddc 11156 +∞cpnf 11290 -∞cmnf 11291 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 (,)cioo 13384 TopOpenctopn 17468 ℝ^crrx 25431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-abv 20827 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-refld 21641 df-phl 21662 df-dsmm 21770 df-frlm 21785 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-xms 24346 df-ms 24347 df-nm 24611 df-ngp 24612 df-tng 24613 df-nrg 24614 df-nlm 24615 df-clm 25110 df-cph 25216 df-tcph 25217 df-rrx 25433

This theorem is referenced by: ioorrnopnxr 46263

W3C validator