ioorrnopnxrlem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ioorrnopnxrlem		Structured version Visualization version GIF version

Theorem ioorrnopnxrlem 46311

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 4497	. . . . . . . 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 45200	. . . . . . . . . . 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 45554	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
12		1red 11182	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 1 ∈ ℝ)
13	11, 12	resubcld 11613	. . . . . . . 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 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
16		iffalse 4500	. . . . . . . 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 2934	. . . . . . . . 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 7059	. . . . . . . . . 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 11235	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → +∞ ∈ ℝ^*)
26	11	rexrd 11231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ^*)
27		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
28	27	ffvelcdmda 7059	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
29		ioogtlb 45500	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
30	21, 28, 10, 29	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
31	11	ltpnfd 13088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < +∞)
32	21, 26, 25, 30, 31	xrlttrd 13126	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < +∞)
33	21, 25, 32	xrltned 45360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≠ +∞)
34	33	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ +∞)
35	22, 23, 34	xrred 45368	. . . . . . . 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 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ ℝ)
38	15, 37	pm2.61dan 812	. . . . 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 7089	. . . 4 ⊢ (𝜑 → 𝐿:𝑋⟶ℝ)
41		iftrue 4497	. . . . . . . 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 11210	. . . . . . . 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 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
46		iffalse 4500	. . . . . . . 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 2934	. . . . . . . . 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 11238	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ ∈ ℝ^*)
53	11	mnfltd 13091	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐹‘𝑖))
54		iooltub 45515	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
55	21, 28, 10, 54	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
56	52, 26, 28, 53, 55	xrlttrd 13126	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐵‘𝑖))
57	52, 28, 56	xrgtned 45325	. . . . . . . . . 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 45368	. . . . . . . 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 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ ℝ)
63	45, 62	pm2.61dan 812	. . . . 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 7089	. . . 4 ⊢ (𝜑 → 𝑅:𝑋⟶ℝ)
66	3, 40, 65	ioorrnopn 46310	. . 3 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ∈ (TopOpen‘(ℝ^‘𝑋)))
67	2, 66	eqeltrd 2829	. 2 ⊢ (𝜑 → 𝑉 ∈ (TopOpen‘(ℝ^‘𝑋)))
68	6	elexd 3474	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69		ixpfn 8879	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹 Fn 𝑋)
70	6, 69	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
71	40	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ)
72	71	rexrd 11231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ^*)
73	65	ffvelcdmda 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ)
74	73	rexrd 11231	. . . . . . . 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 3474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ V)
77	75, 76	fvmpt2d 6984	. . . . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = ((𝐹‘𝑖) − 1))
80	11	ltm1d 12122	. . . . . . . . . . 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 5132	. . . . . . . . 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 2765	. . . . . . . . . 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 5132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
87	82, 86	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) < (𝐹‘𝑖))
88	11	ltp1d 12120	. . . . . . . . . . 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 3474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ V)
92	90, 91	fvmpt2d 6984	. . . . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = ((𝐹‘𝑖) + 1))
95	94	eqcomd 2736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) = (𝑅‘𝑖))
96	89, 95	breqtrd 5136	. . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = (𝐵‘𝑖))
100	99	eqcomd 2736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = (𝑅‘𝑖))
101	97, 100	breqtrd 5136	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
102	96, 101	pm2.61dan 812	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝑅‘𝑖))
103	72, 74, 11, 87, 102	eliood 45503	. . . . . . 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 1128	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
106		elixp2 8877	. . . . 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 2840	. . 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 13091	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
112	111, 5	breqtrd 5136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < ((𝐹‘𝑖) − 1))
113		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = -∞)
114	113, 79	breq12d 5123	. . . . . . . . . 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 13117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
117	84	eqcomd 2736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = (𝐿‘𝑖))
118	36, 117	eqled 11284	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
119	116, 118	pm2.61dan 812	. . . . . . 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 13088	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) < +∞)
123		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = +∞)
124	94, 123	breq12d 5123	. . . . . . . . . 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 13117	. . . . . . . 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 11284	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
129	126, 128	pm2.61dan 812	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
130		ioossioo 13409	. . . . . . 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 838	. . . . . 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 8886	. . . . 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 3984	. . 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 2818	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
138		sseq1 3975	. . . 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 3591	. 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 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 ifcif 4491 class class class wbr 5110 ↦ cmpt 5191 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 Xcixp 8873 Fincfn 8921 ℝcr 11074 1c1 11076 + caddc 11078 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 (,)cioo 13313 TopOpenctopn 17391 ℝ^crrx 25290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-prds 17417 df-pws 17419 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-abv 20725 df-staf 20755 df-srng 20756 df-lmod 20775 df-lss 20845 df-lmhm 20936 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-refld 21521 df-phl 21542 df-dsmm 21648 df-frlm 21663 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-xms 24215 df-ms 24216 df-nm 24477 df-ngp 24478 df-tng 24479 df-nrg 24480 df-nlm 24481 df-clm 24970 df-cph 25075 df-tcph 25076 df-rrx 25292

This theorem is referenced by: ioorrnopnxr 46312

W3C validator