opnvonmbllem2 - Mathbox for Glauco Siliprandi

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Theorem opnvonmbllem2 46662

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 2735	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
3	2	rrxmetfi 25364	. . . . . . . . . . 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 24273	. . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
10	9	rrxval 25339	. . . . . . . . . . . . 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 6880	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
13		ovex 7438	. . . . . . . . . . . . 13 ⊢ (ℝ_fld freeLMod 𝑋) ∈ V
14		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋))
15		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
16		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
17	14, 15, 16	tcphtopn 25178	. . . . . . . . . . . . 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 2741	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (ℝ^‘𝑋))
21	20	fveq2d 6880	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(ℝ^‘𝑋)))
22	21	fveq2d 6880	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
23	12, 19, 22	3eqtrd 2774	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
24	8, 23	eleqtrd 2836	. . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (MetOpen‘(dist‘(ℝ^‘𝑋))) = (MetOpen‘(dist‘(ℝ^‘𝑋)))
28	27	mopni2 24432	. . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
30	1	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
31		eqid 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
32	31	rrxtoponfi 46320	. . . . . . . . . . . . . . . . 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 22865	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)) ∧ 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
35	33, 8, 34	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ ↑_m 𝑋))
36	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
37	36, 26	sseldd 3959	. . . . . . . . . . . . 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 46654	. . . . . . . . . . 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 1132	. . . . . . . . . 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 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
43		nfv 1914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
44		nfcv 2898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑥
45		nfixp1 8932	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
46	44, 45	nfel 2913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
47		nfcv 2898	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
48	45, 47	nfss 3951	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
49	46, 48	nfan 1899	. . . . . . . . . . . . . . . 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 1901	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . . 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 8863	. . . . . . . . . . . . . . . . 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 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))) → 𝑐:𝑋⟶ℚ)
56		elmapi 8863	. . . . . . . . . . . . . . . . 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 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))) → 𝑑:𝑋⟶ℚ)
59		simp3r 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))) → X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒))
60		simp1r 1199	. . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩) = (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩)
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 46661	. . . . . . . . . . . . . 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 1119	. . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 1131	. . . . . . . . . . 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 3197	. . . . . . . . . 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 1119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝑒 ∈ ℝ⁺ → ((𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
71	70	rexlimdv 3139	. . . . . . 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 4971	. . . . . 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 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
76		dfss3 3947	. . . 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 2826	. . . . . . . . 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 3437	. . . . . . 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 3132	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
85		iunss 5021	. . . 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 3976	. 2 ⊢ (𝜑 → 𝐺 = ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
88		opnvonmbllem2.n	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
89	1, 88	dmovnsal 46641	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
90		ssrab2 4055	. . . . . 6 ⊢ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ⊆ ((ℚ × ℚ) ↑_m 𝑋)
91	62, 90	eqsstri 4005	. . . . 5 ⊢ 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋)
92	91	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋))
93		qct 45389	. . . . . . 7 ⊢ ℚ ≼ ω
94	93	a1i 11	. . . . . 6 ⊢ (𝜑 → ℚ ≼ ω)
95		xpct 10030	. . . . . 6 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
96	94, 94, 95	syl2anc 584	. . . . 5 ⊢ (𝜑 → (ℚ × ℚ) ≼ ω)
97	96, 1	mpct 45225	. . . 4 ⊢ (𝜑 → ((ℚ × ℚ) ↑_m 𝑋) ≼ ω)
98		ssct 9065	. . . 4 ⊢ ((𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋) ∧ ((ℚ × ℚ) ↑_m 𝑋) ≼ ω) → 𝐾 ≼ ω)
99	92, 97, 98	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐾 ≼ ω)
100		reex 11220	. . . . . . . . . 10 ⊢ ℝ ∈ V
101	100, 100	xpex 7747	. . . . . . . . 9 ⊢ (ℝ × ℝ) ∈ V
102		qssre 12975	. . . . . . . . . 10 ⊢ ℚ ⊆ ℝ
103		xpss12 5669	. . . . . . . . . 10 ⊢ ((ℚ ⊆ ℝ ∧ ℚ ⊆ ℝ) → (ℚ × ℚ) ⊆ (ℝ × ℝ))
104	102, 102, 103	mp2an 692	. . . . . . . . 9 ⊢ (ℚ × ℚ) ⊆ (ℝ × ℝ)
105		mapss 8903	. . . . . . . . 9 ⊢ (((ℝ × ℝ) ∈ V ∧ (ℚ × ℚ) ⊆ (ℝ × ℝ)) → ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋))
106	101, 104, 105	mp2an 692	. . . . . . . 8 ⊢ ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋)
107	91	sseli 3954	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋))
108	106, 107	sselid 3956	. . . . . . 7 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋))
109		elmapi 8863	. . . . . . 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 6881	. . . . . 6 ⊢ (𝑘 = 𝑖 → (1^st ‘(ℎ‘𝑘)) = (1^st ‘(ℎ‘𝑖)))
113	112	cbvmptv 5225	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑖)))
114		2fveq3 6881	. . . . . 6 ⊢ (𝑘 = 𝑖 → (2^nd ‘(ℎ‘𝑘)) = (2^nd ‘(ℎ‘𝑖)))
115	114	cbvmptv 5225	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑖)))
116	111, 113, 115	hoicoto2 46634	. . . 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 7074	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑘) ∈ (ℝ × ℝ))
119		xp1st 8020	. . . . . . 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 7105	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))):𝑋⟶ℝ)
122		xp2nd 8021	. . . . . . 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 7105	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))):𝑋⟶ℝ)
125	117, 88, 121, 124	hoimbl 46660	. . . 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 2834	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
127	89, 99, 126	saliuncl 46352	. 2 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
128	87, 127	eqeltrd 2834	1 ⊢ (𝜑 → 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 {crab 3415 Vcvv 3459 ⊆ wss 3926 ⟨cop 4607 ∪ ciun 4967 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 dom cdm 5654 ∘ ccom 5658 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ωcom 7861 1^st c1st 7986 2^nd c2nd 7987 ↑_m cmap 8840 Xcixp 8911 ≼ cdom 8957 Fincfn 8959 ℝcr 11128 ℚcq 12964 ℝ⁺crp 13008 [,)cico 13364 distcds 17280 TopOpenctopn 17435 ∞Metcxmet 21300 Metcmet 21301 ballcbl 21302 MetOpencmopn 21305 ℝ_fldcrefld 21564 freeLMod cfrlm 21706 TopOnctopon 22848 toℂPreHilctcph 25119 ℝ^crrx 25335 volncvoln 46567

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cc 10449 ax-ac2 10477 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-disj 5087 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-dju 9915 df-card 9953 df-acn 9956 df-ac 10130 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-rlim 15505 df-sum 15703 df-prod 15920 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-prds 17461 df-pws 17463 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-ghm 19196 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-drng 20691 df-field 20692 df-abv 20769 df-staf 20799 df-srng 20800 df-lmod 20819 df-lss 20889 df-lmhm 20980 df-lvec 21061 df-sra 21131 df-rgmod 21132 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-cnfld 21316 df-refld 21565 df-phl 21586 df-dsmm 21692 df-frlm 21707 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cmp 23325 df-xms 24259 df-ms 24260 df-nm 24521 df-ngp 24522 df-tng 24523 df-nrg 24524 df-nlm 24525 df-clm 25014 df-cph 25120 df-tcph 25121 df-rrx 25337 df-ovol 25417 df-vol 25418 df-salg 46338 df-sumge0 46392 df-mea 46479 df-ome 46519 df-caragen 46521 df-ovoln 46566 df-voln 46568

This theorem is referenced by: opnvonmbl 46663

W3C validator