opnvonmbllem2 - Mathbox for Glauco Siliprandi

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Theorem opnvonmbllem2 47076

Description: An open subset of the n-dimensional Real numbers is Lebesgue measurable. This is Proposition 115G (a) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

**Hypotheses**
Ref	Expression
opnvonmbllem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
opnvonmbllem2.n	⊢ 𝑆 = dom (voln‘𝑋)
opnvonmbllem2.g	⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))
opnvonmbl.k	⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}

**Assertion**
Ref	Expression
opnvonmbllem2	⊢ (𝜑 → 𝐺 ∈ 𝑆)

Distinct variable groups: ℎ,𝐺,𝑖 ℎ,𝐾,𝑖 𝑆,ℎ,𝑖 ℎ,𝑋,𝑖 𝜑,ℎ,𝑖

Proof of Theorem opnvonmbllem2 Dummy variables 𝑐 𝑑 𝑒 𝑥 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opnvonmbllem2.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		eqid 2739	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
3	2	rrxmetfi 25397	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_m 𝑋)))
4	1, 3	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_m 𝑋)))
5		metxmet 24317	. . . . . . . . . 10 ⊢ ((dist‘(ℝ^‘𝑋)) ∈ (Met‘(ℝ ↑_m 𝑋)) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
6	4, 5	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
8		opnvonmbllem2.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))
9		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
10	9	rrxval 25372	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → (ℝ^‘𝑋) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
11	1, 10	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ^‘𝑋) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
12	11	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
13		ovex 7389	. . . . . . . . . . . . 13 ⊢ (ℝ_fld freeLMod 𝑋) ∈ V
14		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋))
15		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
16		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
17	14, 15, 16	tcphtopn 25211	. . . . . . . . . . . . 13 ⊢ ((ℝ_fld freeLMod 𝑋) ∈ V → (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))))
18	13, 17	ax-mp 5	. . . . . . . . . . . 12 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))))
20	11	eqcomd 2745	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (ℝ^‘𝑋))
21	20	fveq2d 6831	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(ℝ^‘𝑋)))
22	21	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
23	12, 19, 22	3eqtrd 2778	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
24	8, 23	eleqtrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
25	24	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
26		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
27		eqid 2739	. . . . . . . . 9 ⊢ (MetOpen‘(dist‘(ℝ^‘𝑋))) = (MetOpen‘(dist‘(ℝ^‘𝑋)))
28	27	mopni2 24476	. . . . . . . 8 ⊢ (((dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)) ∧ 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))) ∧ 𝑥 ∈ 𝐺) → ∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
29	7, 25, 26, 28	syl3anc 1379	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
30	1	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
31		eqid 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
32	31	rrxtoponfi 46734	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ Fin → (TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)))
33	1, 32	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)))
34		toponss 22910	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)) ∧ 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
35	33, 8, 34	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ ↑_m 𝑋))
36	35	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
37	36, 26	sseldd 3916	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
38	37	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
39		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
40	30, 38, 39	hoiqssbl 47068	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)))
41	40	3adant3 1138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → ∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)))
42		nfv 1921	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
43		nfv 1921	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
44		nfcv 2901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑥
45		nfixp1 8856	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
46	44, 45	nfel 2915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
47		nfcv 2901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
48	45, 47	nfss 3908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
49	46, 48	nfan 1906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))
50	42, 43, 49	nf3an 1908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)))
51	1	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → 𝑋 ∈ Fin)
52	51	3ad2ant1 1139	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → 𝑋 ∈ Fin)
53		elmapi 8786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
54	53	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℚ)
55	54	3ad2ant2 1140	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → 𝑐:𝑋⟶ℚ)
56		elmapi 8786	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
57	56	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑑:𝑋⟶ℚ)
58	57	3ad2ant2 1140	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → 𝑑:𝑋⟶ℚ)
59		simp3r 1209	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))
60		simp1r 1205	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
61		simp3l 1208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
62		opnvonmbl.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}
63		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩) = (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩)
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 47075	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) ∧ (𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) ∧ (𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
65	64	3exp 1125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → ((𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
66	65	adantlr 721	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → ((𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
67	66	3adant2 1137	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → ((𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
68	67	rexlimdvv 3195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → (∃𝑐 ∈ (ℚ ↑_m 𝑋)∃𝑑 ∈ (ℚ ↑_m 𝑋)(𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ∧ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)))
69	41, 68	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
70	69	3exp 1125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝑒 ∈ ℝ⁺ → ((𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
71	70	rexlimdv 3138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖)))
72	29, 71	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
73		eliun 4925	. . . . . 6 ⊢ (𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
74	72, 73	sylibr 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
75	74	ralrimiva 3131	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
76		dfss3 3904	. . . 4 ⊢ (𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
77	75, 76	sylibr 235	. . 3 ⊢ (𝜑 → 𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
78	62	eleq2i 2831	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 ↔ ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
79	78	bilani 505	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
80		rabid 3412	. . . . . . 7 ⊢ (ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ↔ (ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺))
81	79, 80	sylib 219	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺))
82	81	simprd 496	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
83	82	ralrimiva 3131	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
84		iunss 4974	. . . 4 ⊢ (∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺 ↔ ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
85	83, 84	sylibr 235	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
86	77, 85	eqssd 3932	. 2 ⊢ (𝜑 → 𝐺 = ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
87		opnvonmbllem2.n	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
88	1, 87	dmovnsal 47055	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
89		ssrab2 4011	. . . . . 6 ⊢ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ⊆ ((ℚ × ℚ) ↑_m 𝑋)
90	62, 89	eqsstri 3961	. . . . 5 ⊢ 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋)
91	90	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋))
92		qct 45807	. . . . . . 7 ⊢ ℚ ≼ ω
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → ℚ ≼ ω)
94		xpct 9929	. . . . . 6 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
95	93, 93, 94	syl2anc 590	. . . . 5 ⊢ (𝜑 → (ℚ × ℚ) ≼ ω)
96	95, 1	mpct 45647	. . . 4 ⊢ (𝜑 → ((ℚ × ℚ) ↑_m 𝑋) ≼ ω)
97		ssct 8986	. . . 4 ⊢ ((𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋) ∧ ((ℚ × ℚ) ↑_m 𝑋) ≼ ω) → 𝐾 ≼ ω)
98	91, 96, 97	syl2anc 590	. . 3 ⊢ (𝜑 → 𝐾 ≼ ω)
99		reex 11120	. . . . . . . . . 10 ⊢ ℝ ∈ V
100	99, 99	xpex 7696	. . . . . . . . 9 ⊢ (ℝ × ℝ) ∈ V
101		qssre 12900	. . . . . . . . . 10 ⊢ ℚ ⊆ ℝ
102		xpss12 5633	. . . . . . . . . 10 ⊢ ((ℚ ⊆ ℝ ∧ ℚ ⊆ ℝ) → (ℚ × ℚ) ⊆ (ℝ × ℝ))
103	101, 101, 102	mp2an 698	. . . . . . . . 9 ⊢ (ℚ × ℚ) ⊆ (ℝ × ℝ)
104		mapss 8827	. . . . . . . . 9 ⊢ (((ℝ × ℝ) ∈ V ∧ (ℚ × ℚ) ⊆ (ℝ × ℝ)) → ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋))
105	100, 103, 104	mp2an 698	. . . . . . . 8 ⊢ ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋)
106	90	sseli 3911	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋))
107	105, 106	sselid 3913	. . . . . . 7 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋))
108		elmapi 8786	. . . . . . 7 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) → ℎ:𝑋⟶(ℝ × ℝ))
109	107, 108	syl 17	. . . . . 6 ⊢ (ℎ ∈ 𝐾 → ℎ:𝑋⟶(ℝ × ℝ))
110	109	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ:𝑋⟶(ℝ × ℝ))
111		2fveq3 6832	. . . . . 6 ⊢ (𝑘 = 𝑖 → (1^st ‘(ℎ‘𝑘)) = (1^st ‘(ℎ‘𝑖)))
112	111	cbvmptv 5176	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑖)))
113		2fveq3 6832	. . . . . 6 ⊢ (𝑘 = 𝑖 → (2^nd ‘(ℎ‘𝑘)) = (2^nd ‘(ℎ‘𝑖)))
114	113	cbvmptv 5176	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑖)))
115	110, 112, 114	hoicoto2 47048	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘)))‘𝑖)[,)((𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘)))‘𝑖)))
116	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → 𝑋 ∈ Fin)
117	110	ffvelcdmda 7025	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑘) ∈ (ℝ × ℝ))
118		xp1st 7963	. . . . . . 7 ⊢ ((ℎ‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(ℎ‘𝑘)) ∈ ℝ)
119	117, 118	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(ℎ‘𝑘)) ∈ ℝ)
120	119	fmpttd 7056	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))):𝑋⟶ℝ)
121		xp2nd 7964	. . . . . . 7 ⊢ ((ℎ‘𝑘) ∈ (ℝ × ℝ) → (2^nd ‘(ℎ‘𝑘)) ∈ ℝ)
122	117, 121	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(ℎ‘𝑘)) ∈ ℝ)
123	122	fmpttd 7056	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))):𝑋⟶ℝ)
124	116, 87, 120, 123	hoimbl 47074	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘)))‘𝑖)[,)((𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘)))‘𝑖)) ∈ 𝑆)
125	115, 124	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
126	88, 98, 125	saliuncl 46766	. 2 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
127	86, 126	eqeltrd 2839	1 ⊢ (𝜑 → 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 {crab 3391 Vcvv 3431 ⊆ wss 3883 ⟨cop 4561 ∪ ciun 4921 class class class wbr 5072 ↦ cmpt 5153 × cxp 5616 dom cdm 5618 ∘ ccom 5622 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ωcom 7806 1^st c1st 7929 2^nd c2nd 7930 ↑_m cmap 8763 Xcixp 8835 ≼ cdom 8881 Fincfn 8883 ℝcr 11028 ℚcq 12889 ℝ⁺crp 12933 [,)cico 13291 distcds 17220 TopOpenctopn 17375 ∞Metcxmet 21332 Metcmet 21333 ballcbl 21334 MetOpencmopn 21337 ℝ_fldcrefld 21579 freeLMod cfrlm 21721 TopOnctopon 22893 toℂPreHilctcph 25152 ℝ^crrx 25368 volncvoln 46981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-prds 17401 df-pws 17403 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-abv 20781 df-staf 20811 df-srng 20812 df-lmod 20852 df-lss 20922 df-lmhm 21012 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-refld 21580 df-phl 21601 df-dsmm 21707 df-frlm 21722 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cmp 23370 df-xms 24303 df-ms 24304 df-nm 24565 df-ngp 24566 df-tng 24567 df-nrg 24568 df-nlm 24569 df-clm 25048 df-cph 25153 df-tcph 25154 df-rrx 25370 df-ovol 25449 df-vol 25450 df-salg 46752 df-sumge0 46806 df-mea 46893 df-ome 46933 df-caragen 46935 df-ovoln 46980 df-voln 46982

This theorem is referenced by: opnvonmbl 47077

W3C validator