opnvonmbllem2 - Mathbox for Glauco Siliprandi

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Theorem opnvonmbllem2 47082

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 2737	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
3	2	rrxmetfi 25392	. . . . . . . . . . 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 24312	. . . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
8		opnvonmbllem2.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))
9		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
10	9	rrxval 25367	. . . . . . . . . . . . 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 6839	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
13		ovex 7394	. . . . . . . . . . . . 13 ⊢ (ℝ_fld freeLMod 𝑋) ∈ V
14		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋))
15		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
16		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
17	14, 15, 16	tcphtopn 25206	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (ℝ^‘𝑋))
21	20	fveq2d 6839	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(ℝ^‘𝑋)))
22	21	fveq2d 6839	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
23	12, 19, 22	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
24	8, 23	eleqtrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
25	24	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
26		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
27		eqid 2737	. . . . . . . . 9 ⊢ (MetOpen‘(dist‘(ℝ^‘𝑋))) = (MetOpen‘(dist‘(ℝ^‘𝑋)))
28	27	mopni2 24471	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
30	1	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
31		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
32	31	rrxtoponfi 46740	. . . . . . . . . . . . . . . . 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 22905	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)) ∧ 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
35	33, 8, 34	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ ↑_m 𝑋))
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
37	36, 26	sseldd 3923	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
38	37	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
39		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
40	30, 38, 39	hoiqssbl 47074	. . . . . . . . . . 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 1133	. . . . . . . . . 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 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
43		nfv 1916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
44		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑥
45		nfixp1 8860	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
46	44, 45	nfel 2914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
47		nfcv 2899	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
48	45, 47	nfss 3915	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
49	46, 48	nfan 1901	. . . . . . . . . . . . . . . 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 1903	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → 𝑋 ∈ Fin)
52	51	3ad2ant1 1134	. . . . . . . . . . . . . . 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 8790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
54	53	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℚ)
55	54	3ad2ant2 1135	. . . . . . . . . . . . . . 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 8790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
57	56	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑑:𝑋⟶ℚ)
58	57	3ad2ant2 1135	. . . . . . . . . . . . . . 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 1204	. . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . 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 1203	. . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩) = (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩)
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 47081	. . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . 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 716	. . . . . . . . . . . 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 1132	. . . . . . . . . . 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 3194	. . . . . . . . . 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 1120	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝑒 ∈ ℝ⁺ → ((𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
71	70	rexlimdv 3137	. . . . . . 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 4938	. . . . . 6 ⊢ (𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
74	72, 73	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
75	74	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
76		dfss3 3911	. . . 4 ⊢ (𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
77	75, 76	sylibr 234	. . 3 ⊢ (𝜑 → 𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
78	62	eleq2i 2829	. . . . . . . . 9 ⊢ (ℎ ∈ 𝐾 ↔ ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
79	78	biimpi 216	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
80	79	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
81		rabid 3411	. . . . . . 7 ⊢ (ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ↔ (ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺))
82	80, 81	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺))
83	82	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
84	83	ralrimiva 3130	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
85		iunss 4988	. . . 4 ⊢ (∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺 ↔ ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
86	84, 85	sylibr 234	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
87	77, 86	eqssd 3940	. 2 ⊢ (𝜑 → 𝐺 = ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
88		opnvonmbllem2.n	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
89	1, 88	dmovnsal 47061	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
90		ssrab2 4021	. . . . . 6 ⊢ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ⊆ ((ℚ × ℚ) ↑_m 𝑋)
91	62, 90	eqsstri 3969	. . . . 5 ⊢ 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋)
92	91	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋))
93		qct 45813	. . . . . . 7 ⊢ ℚ ≼ ω
94	93	a1i 11	. . . . . 6 ⊢ (𝜑 → ℚ ≼ ω)
95		xpct 9932	. . . . . 6 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
96	94, 94, 95	syl2anc 585	. . . . 5 ⊢ (𝜑 → (ℚ × ℚ) ≼ ω)
97	96, 1	mpct 45651	. . . 4 ⊢ (𝜑 → ((ℚ × ℚ) ↑_m 𝑋) ≼ ω)
98		ssct 8990	. . . 4 ⊢ ((𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋) ∧ ((ℚ × ℚ) ↑_m 𝑋) ≼ ω) → 𝐾 ≼ ω)
99	92, 97, 98	syl2anc 585	. . 3 ⊢ (𝜑 → 𝐾 ≼ ω)
100		reex 11123	. . . . . . . . . 10 ⊢ ℝ ∈ V
101	100, 100	xpex 7701	. . . . . . . . 9 ⊢ (ℝ × ℝ) ∈ V
102		qssre 12903	. . . . . . . . . 10 ⊢ ℚ ⊆ ℝ
103		xpss12 5640	. . . . . . . . . 10 ⊢ ((ℚ ⊆ ℝ ∧ ℚ ⊆ ℝ) → (ℚ × ℚ) ⊆ (ℝ × ℝ))
104	102, 102, 103	mp2an 693	. . . . . . . . 9 ⊢ (ℚ × ℚ) ⊆ (ℝ × ℝ)
105		mapss 8831	. . . . . . . . 9 ⊢ (((ℝ × ℝ) ∈ V ∧ (ℚ × ℚ) ⊆ (ℝ × ℝ)) → ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋))
106	101, 104, 105	mp2an 693	. . . . . . . 8 ⊢ ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋)
107	91	sseli 3918	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋))
108	106, 107	sselid 3920	. . . . . . 7 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋))
109		elmapi 8790	. . . . . . 7 ⊢ (ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋) → ℎ:𝑋⟶(ℝ × ℝ))
110	108, 109	syl 17	. . . . . 6 ⊢ (ℎ ∈ 𝐾 → ℎ:𝑋⟶(ℝ × ℝ))
111	110	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ:𝑋⟶(ℝ × ℝ))
112		2fveq3 6840	. . . . . 6 ⊢ (𝑘 = 𝑖 → (1^st ‘(ℎ‘𝑘)) = (1^st ‘(ℎ‘𝑖)))
113	112	cbvmptv 5190	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑖)))
114		2fveq3 6840	. . . . . 6 ⊢ (𝑘 = 𝑖 → (2^nd ‘(ℎ‘𝑘)) = (2^nd ‘(ℎ‘𝑖)))
115	114	cbvmptv 5190	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑖)))
116	111, 113, 115	hoicoto2 47054	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) = X𝑖 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘)))‘𝑖)[,)((𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘)))‘𝑖)))
117	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → 𝑋 ∈ Fin)
118	111	ffvelcdmda 7031	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑘) ∈ (ℝ × ℝ))
119		xp1st 7968	. . . . . . 7 ⊢ ((ℎ‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(ℎ‘𝑘)) ∈ ℝ)
120	118, 119	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (1^st ‘(ℎ‘𝑘)) ∈ ℝ)
121	120	fmpttd 7062	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))):𝑋⟶ℝ)
122		xp2nd 7969	. . . . . . 7 ⊢ ((ℎ‘𝑘) ∈ (ℝ × ℝ) → (2^nd ‘(ℎ‘𝑘)) ∈ ℝ)
123	118, 122	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(ℎ‘𝑘)) ∈ ℝ)
124	123	fmpttd 7062	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))):𝑋⟶ℝ)
125	117, 88, 121, 124	hoimbl 47080	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (((𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘)))‘𝑖)[,)((𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘)))‘𝑖)) ∈ 𝑆)
126	116, 125	eqeltrd 2837	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
127	89, 99, 126	saliuncl 46772	. 2 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
128	87, 127	eqeltrd 2837	1 ⊢ (𝜑 → 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ⊆ wss 3890 ⟨cop 4574 ∪ ciun 4934 class class class wbr 5086 ↦ cmpt 5167 × cxp 5623 dom cdm 5625 ∘ ccom 5629 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ωcom 7811 1^st c1st 7934 2^nd c2nd 7935 ↑_m cmap 8767 Xcixp 8839 ≼ cdom 8885 Fincfn 8887 ℝcr 11031 ℚcq 12892 ℝ⁺crp 12936 [,)cico 13294 distcds 17223 TopOpenctopn 17378 ∞Metcxmet 21332 Metcmet 21333 ballcbl 21334 MetOpencmopn 21337 ℝ_fldcrefld 21597 freeLMod cfrlm 21739 TopOnctopon 22888 toℂPreHilctcph 25147 ℝ^crrx 25363 volncvoln 46987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cc 10351 ax-ac2 10379 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9819 df-card 9857 df-acn 9860 df-ac 10032 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-fl 13745 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-rlim 15445 df-sum 15643 df-prod 15863 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-sca 17230 df-vsca 17231 df-ip 17232 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-hom 17238 df-cco 17239 df-rest 17379 df-topn 17380 df-0g 17398 df-gsum 17399 df-topgen 17400 df-prds 17404 df-pws 17406 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-ghm 19182 df-cntz 19286 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20517 df-subrg 20541 df-drng 20702 df-field 20703 df-abv 20780 df-staf 20810 df-srng 20811 df-lmod 20851 df-lss 20921 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 21598 df-phl 21619 df-dsmm 21725 df-frlm 21740 df-top 22872 df-topon 22889 df-topsp 22911 df-bases 22924 df-cmp 23365 df-xms 24298 df-ms 24299 df-nm 24560 df-ngp 24561 df-tng 24562 df-nrg 24563 df-nlm 24564 df-clm 25043 df-cph 25148 df-tcph 25149 df-rrx 25365 df-ovol 25444 df-vol 25445 df-salg 46758 df-sumge0 46812 df-mea 46899 df-ome 46939 df-caragen 46941 df-ovoln 46986 df-voln 46988

This theorem is referenced by: opnvonmbl 47083

W3C validator