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 47410

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 4487	. . . . . . . 8 ⊢ ((𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = ((𝐹‘𝑖) − 1))
5	4	adantl 487	. . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
8		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 𝑖 ∈ 𝑋)
9		fvixp2 46307	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
10	7, 8, 9	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
11	10	elioored 46655	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ)
12		1red 11313	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → 1 ∈ ℝ)
13	11, 12	resubcld 11748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) ∈ ℝ)
14	13	adantr 486	. . . . . . 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 4490	. . . . . . . 8 ⊢ (¬ (𝐴‘𝑖) = -∞ → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
17	16	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) = (𝐴‘𝑖))
18		neqne 2963	. . . . . . . . 9 ⊢ (¬ (𝐴‘𝑖) = -∞ → (𝐴‘𝑖) ≠ -∞)
19	18	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≠ -∞)
20		ioorrnopnxrlem.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
21	20	ffvelcdmda 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ∈ ℝ^*)
22	21	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ^*)
23		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ -∞)
24		pnfxr 11367	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → +∞ ∈ ℝ^*)
26	11	rexrd 11363	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ℝ^*)
27		ioorrnopnxrlem.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
28	27	ffvelcdmda 7085	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ∈ ℝ^*)
29		ioogtlb 46601	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐴‘𝑖) < (𝐹‘𝑖))
30	21, 28, 10, 29	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < (𝐹‘𝑖))
31	11	ltpnfd 13254	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < +∞)
32	21, 26, 25, 30, 31	xrlttrd 13292	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) < +∞)
33	21, 25, 32	xrltned 46463	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≠ +∞)
34	33	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ≠ +∞)
35	22, 23, 34	xrred 46470	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) ≠ -∞) → (𝐴‘𝑖) ∈ ℝ)
36	19, 35	syldan 603	. . . . . . 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 825	. . . . 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 7115	. . . 4 ⊢ (𝜑 → 𝐿:𝑋⟶ℝ)
41		iftrue 4487	. . . . . . . 8 ⊢ ((𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
42	41	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = ((𝐹‘𝑖) + 1))
43	11, 12	readdcld 11342	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) + 1) ∈ ℝ)
44	43	adantr 486	. . . . . . 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 4490	. . . . . . . 8 ⊢ (¬ (𝐵‘𝑖) = +∞ → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
47	46	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) = (𝐵‘𝑖))
48		neqne 2963	. . . . . . . . 9 ⊢ (¬ (𝐵‘𝑖) = +∞ → (𝐵‘𝑖) ≠ +∞)
49	48	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ≠ +∞)
50	28	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ^*)
51		mnfxr 11370	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
52	51	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ ∈ ℝ^*)
53	11	mnfltd 13257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐹‘𝑖))
54		iooltub 46616	. . . . . . . . . . . . 13 ⊢ (((𝐴‘𝑖) ∈ ℝ^* ∧ (𝐵‘𝑖) ∈ ℝ^* ∧ (𝐹‘𝑖) ∈ ((𝐴‘𝑖)(,)(𝐵‘𝑖))) → (𝐹‘𝑖) < (𝐵‘𝑖))
55	21, 28, 10, 54	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝐵‘𝑖))
56	52, 26, 28, 53, 55	xrlttrd 13292	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → -∞ < (𝐵‘𝑖))
57	52, 28, 56	xrgtned 13297	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐵‘𝑖) ≠ -∞)
58	57	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ -∞)
59		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ≠ +∞)
60	50, 58, 59	xrred 46470	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) ≠ +∞) → (𝐵‘𝑖) ∈ ℝ)
61	49, 60	syldan 603	. . . . . . 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 825	. . . . 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 7115	. . . 4 ⊢ (𝜑 → 𝑅:𝑋⟶ℝ)
66	3, 40, 65	ioorrnopn 47409	. . 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 3473	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V)
69		ixpfn 8935	. . . . . . 7 ⊢ (𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) → 𝐹 Fn 𝑋)
70	6, 69	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝑋)
71	40	ffvelcdmda 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ)
72	71	rexrd 11363	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) ∈ ℝ^*)
73	65	ffvelcdmda 7085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ∈ ℝ)
74	73	rexrd 11363	. . . . . . . 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 3473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)) ∈ V)
77	75, 76	fvmpt2d 7008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
78	77	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
79	78, 5	eqtrd 2795	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = ((𝐹‘𝑖) − 1))
80	11	ltm1d 12253	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
81	80	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐹‘𝑖) − 1) < (𝐹‘𝑖))
82	79, 81	eqbrtrd 5126	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
83	77	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
84	83, 17	eqtrd 2795	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) = (𝐴‘𝑖))
85	30	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐹‘𝑖))
86	84, 85	eqbrtrd 5126	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) < (𝐹‘𝑖))
87	82, 86	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐿‘𝑖) < (𝐹‘𝑖))
88	11	ltp1d 12251	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < ((𝐹‘𝑖) + 1))
89	88	adantr 486	. . . . . . . . . 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 3473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)) ∈ V)
92	90, 91	fvmpt2d 7008	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
93	92	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
94	93, 42	eqtrd 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = ((𝐹‘𝑖) + 1))
95	94	eqcomd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) = (𝑅‘𝑖))
96	89, 95	breqtrd 5130	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
97	55	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝐵‘𝑖))
98	92	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = if((𝐵‘𝑖) = +∞, ((𝐹‘𝑖) + 1), (𝐵‘𝑖)))
99	98, 47	eqtrd 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) = (𝐵‘𝑖))
100	99	eqcomd 2766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = (𝑅‘𝑖))
101	97, 100	breqtrd 5130	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝐹‘𝑖) < (𝑅‘𝑖))
102	96, 101	pm2.61dan 825	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) < (𝑅‘𝑖))
103	72, 74, 11, 87, 102	eliood 46604	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
104	103	ralrimiva 3154	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
105	68, 70, 104	3jca 1146	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑖 ∈ 𝑋 (𝐹‘𝑖) ∈ ((𝐿‘𝑖)(,)(𝑅‘𝑖))))
106		elixp2 8933	. . . . 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 237	. . . 4 ⊢ (𝜑 → 𝐹 ∈ X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)))
108	107, 1	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
109	21	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ∈ ℝ^*)
110	72	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐿‘𝑖) ∈ ℝ^*)
111	15	mnfltd 13257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < if((𝐴‘𝑖) = -∞, ((𝐹‘𝑖) − 1), (𝐴‘𝑖)))
112	111, 5	breqtrd 5130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → -∞ < ((𝐹‘𝑖) − 1))
113		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = -∞)
114	113, 79	breq12d 5115	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → ((𝐴‘𝑖) < (𝐿‘𝑖) ↔ -∞ < ((𝐹‘𝑖) − 1)))
115	112, 114	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) < (𝐿‘𝑖))
116	109, 110, 115	xrltled 13283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
117	84	eqcomd 2766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) = (𝐿‘𝑖))
118	36, 117	eqled 11417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐴‘𝑖) = -∞) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
119	116, 118	pm2.61dan 825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝐴‘𝑖) ≤ (𝐿‘𝑖))
120	74	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ^*)
121	28	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) ∈ ℝ^*)
122	44	ltpnfd 13254	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝐹‘𝑖) + 1) < +∞)
123		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝐵‘𝑖) = +∞)
124	94, 123	breq12d 5115	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → ((𝑅‘𝑖) < (𝐵‘𝑖) ↔ ((𝐹‘𝑖) + 1) < +∞))
125	122, 124	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) < (𝐵‘𝑖))
126	120, 121, 125	xrltled 13283	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
127	73	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ∈ ℝ)
128	127, 99	eqled 11417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑋) ∧ ¬ (𝐵‘𝑖) = +∞) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
129	126, 128	pm2.61dan 825	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → (𝑅‘𝑖) ≤ (𝐵‘𝑖))
130		ioossioo 13576	. . . . . . 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 852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋) → ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
132	131	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
133		ss2ixp 8942	. . . . 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 18	. . . 4 ⊢ (𝜑 → X𝑖 ∈ 𝑋 ((𝐿‘𝑖)(,)(𝑅‘𝑖)) ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
135	2, 134	eqsstrd 3942	. . 3 ⊢ (𝜑 → 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))
136	108, 135	jca 521	. 2 ⊢ (𝜑 → (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
137		eleq2 2849	. . . 4 ⊢ (𝑣 = 𝑉 → (𝐹 ∈ 𝑣 ↔ 𝐹 ∈ 𝑉))
138		sseq1 3933	. . . 4 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)) ↔ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))
139	137, 138	anbi12d 644	. . 3 ⊢ (𝑣 = 𝑉 → ((𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))) ↔ (𝐹 ∈ 𝑉 ∧ 𝑉 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖)))))
140	139	rspcev 3576	. 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 596	1 ⊢ (𝜑 → ∃𝑣 ∈ (TopOpen‘(ℝ^‘𝑋))(𝐹 ∈ 𝑣 ∧ 𝑣 ⊆ X𝑖 ∈ 𝑋 ((𝐴‘𝑖)(,)(𝐵‘𝑖))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3895 ifcif 4481 class class class wbr 5102 ↦ cmpt 5185 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 Xcixp 8929 Fincfn 8977 ℝcr 11203 1c1 11205 + caddc 11207 +∞cpnf 11344 -∞cmnf 11345 ℝ^*cxr 11346 < clt 11347 ≤ cle 11348 − cmin 11545 (,)cioo 13480 TopOpenctopn 17598 ℝ^crrx 25710

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283 ax-mulf 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-tpos 8247 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-map 8853 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ico 13486 df-fz 13644 df-fzo 13793 df-seq 14149 df-exp 14209 df-hash 14479 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-clim 15661 df-sum 15860 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-prds 17624 df-pws 17626 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-mhm 18984 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-ghm 19434 df-cntz 19537 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-cring 20468 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-rhm 20708 df-subrng 20804 df-subrg 20828 df-drng 20988 df-field 20989 df-abv 21072 df-staf 21102 df-srng 21103 df-lmod 21143 df-lss 21213 df-lmhm 21303 df-lvec 21384 df-sra 21454 df-rgmod 21455 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-cnfld 21685 df-refld 21917 df-phl 21938 df-dsmm 22044 df-frlm 22059 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-xms 24645 df-ms 24646 df-nm 24907 df-ngp 24908 df-tng 24909 df-nrg 24910 df-nlm 24911 df-clm 25390 df-cph 25495 df-tcph 25496 df-rrx 25712

This theorem is used by: ioorrnopnxr 47411

W3C validator