opnvonmbllem2 - Mathbox for Glauco Siliprandi

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Theorem opnvonmbllem2 47204

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 2762	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
3	2	rrxmetfi 25471	. . . . . . . . . . 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 24391	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (dist‘(ℝ^‘𝑋)) ∈ (∞Met‘(ℝ ↑_m 𝑋)))
8		opnvonmbllem2.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋)))
9		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
10	9	rrxval 25446	. . . . . . . . . . . . 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 6871	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
13		ovex 7429	. . . . . . . . . . . . 13 ⊢ (ℝ_fld freeLMod 𝑋) ∈ V
14		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋))
15		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
16		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
17	14, 15, 16	tcphtopn 25285	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (ℝ^‘𝑋))
21	20	fveq2d 6871	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(ℝ^‘𝑋)))
22	21	fveq2d 6871	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘(dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
23	12, 19, 22	3eqtrd 2801	. . . . . . . . . 10 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (MetOpen‘(dist‘(ℝ^‘𝑋))))
24	8, 23	eleqtrd 2864	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
25	24	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ∈ (MetOpen‘(dist‘(ℝ^‘𝑋))))
26		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ 𝐺)
27		eqid 2762	. . . . . . . . 9 ⊢ (MetOpen‘(dist‘(ℝ^‘𝑋))) = (MetOpen‘(dist‘(ℝ^‘𝑋)))
28	27	mopni2 24550	. . . . . . . 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 1390	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → ∃𝑒 ∈ ℝ⁺ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
30	1	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑋 ∈ Fin)
31		eqid 2762	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
32	31	rrxtoponfi 46862	. . . . . . . . . . . . . . . . 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 22984	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘(ℝ^‘𝑋)) ∈ (TopOn‘(ℝ ↑_m 𝑋)) ∧ 𝐺 ∈ (TopOpen‘(ℝ^‘𝑋))) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
35	33, 8, 34	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ⊆ (ℝ ↑_m 𝑋))
36	35	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝐺 ⊆ (ℝ ↑_m 𝑋))
37	36, 26	sseldd 3937	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
38	37	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑥 ∈ (ℝ ↑_m 𝑋))
39		simpr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐺) ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
40	30, 38, 39	hoiqssbl 47196	. . . . . . . . . . 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 1145	. . . . . . . . . 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 1934	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
43		nfv 1934	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
44		nfcv 2924	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑥
45		nfixp1 8900	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
46	44, 45	nfel 2938	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
47		nfcv 2924	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
48	45, 47	nfss 3929	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
49	46, 48	nfan 1919	. . . . . . . . . . . . . . . 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 1921	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺) → 𝑋 ∈ Fin)
52	51	3ad2ant1 1146	. . . . . . . . . . . . . . 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 8830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ (ℚ ↑_m 𝑋) → 𝑐:𝑋⟶ℚ)
54	53	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑐:𝑋⟶ℚ)
55	54	3ad2ant2 1147	. . . . . . . . . . . . . . 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 8830	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (ℚ ↑_m 𝑋) → 𝑑:𝑋⟶ℚ)
57	56	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋)) → 𝑑:𝑋⟶ℚ)
58	57	3ad2ant2 1147	. . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩) = (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩)
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 47203	. . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 1144	. . . . . . . . . . 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 3218	. . . . . . . . . 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 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝑒 ∈ ℝ⁺ → ((𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
71	70	rexlimdv 3161	. . . . . . 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 4953	. . . . . 6 ⊢ (𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
74	72, 73	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
75	74	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
76		dfss3 3925	. . . 4 ⊢ (𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ↔ ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
77	75, 76	sylibr 236	. . 3 ⊢ (𝜑 → 𝐺 ⊆ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
78	62	eleq2i 2854	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 ↔ ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
79	78	bilani 508	. . . . . . 7 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ ∈ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺})
80		rabid 3435	. . . . . . 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 220	. . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∧ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺))
82	81	simprd 499	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
83	82	ralrimiva 3154	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
84		iunss 5002	. . . 4 ⊢ (∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺 ↔ ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
85	83, 84	sylibr 236	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
86	77, 85	eqssd 3953	. 2 ⊢ (𝜑 → 𝐺 = ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
87		opnvonmbllem2.n	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
88	1, 87	dmovnsal 47183	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
89		ssrab2 4033	. . . . . 6 ⊢ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ⊆ ((ℚ × ℚ) ↑_m 𝑋)
90	62, 89	eqsstri 3982	. . . . 5 ⊢ 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋)
91	90	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋))
92		qct 45935	. . . . . . 7 ⊢ ℚ ≼ ω
93	92	a1i 11	. . . . . 6 ⊢ (𝜑 → ℚ ≼ ω)
94		xpct 9972	. . . . . 6 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
95	93, 93, 94	syl2anc 593	. . . . 5 ⊢ (𝜑 → (ℚ × ℚ) ≼ ω)
96	95, 1	mpct 45775	. . . 4 ⊢ (𝜑 → ((ℚ × ℚ) ↑_m 𝑋) ≼ ω)
97		ssct 9030	. . . 4 ⊢ ((𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋) ∧ ((ℚ × ℚ) ↑_m 𝑋) ≼ ω) → 𝐾 ≼ ω)
98	91, 96, 97	syl2anc 593	. . 3 ⊢ (𝜑 → 𝐾 ≼ ω)
99		reex 11164	. . . . . . . . . 10 ⊢ ℝ ∈ V
100	99, 99	xpex 7736	. . . . . . . . 9 ⊢ (ℝ × ℝ) ∈ V
101		qssre 12960	. . . . . . . . . 10 ⊢ ℚ ⊆ ℝ
102		xpss12 5662	. . . . . . . . . 10 ⊢ ((ℚ ⊆ ℝ ∧ ℚ ⊆ ℝ) → (ℚ × ℚ) ⊆ (ℝ × ℝ))
103	101, 101, 102	mp2an 702	. . . . . . . . 9 ⊢ (ℚ × ℚ) ⊆ (ℝ × ℝ)
104		mapss 8871	. . . . . . . . 9 ⊢ (((ℝ × ℝ) ∈ V ∧ (ℚ × ℚ) ⊆ (ℝ × ℝ)) → ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋))
105	100, 103, 104	mp2an 702	. . . . . . . 8 ⊢ ((ℚ × ℚ) ↑_m 𝑋) ⊆ ((ℝ × ℝ) ↑_m 𝑋)
106	90	sseli 3932	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋))
107	105, 106	sselid 3934	. . . . . . 7 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋))
108		elmapi 8830	. . . . . . 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 485	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → ℎ:𝑋⟶(ℝ × ℝ))
111		2fveq3 6872	. . . . . 6 ⊢ (𝑘 = 𝑖 → (1^st ‘(ℎ‘𝑘)) = (1^st ‘(ℎ‘𝑖)))
112	111	cbvmptv 5204	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑖)))
113		2fveq3 6872	. . . . . 6 ⊢ (𝑘 = 𝑖 → (2^nd ‘(ℎ‘𝑘)) = (2^nd ‘(ℎ‘𝑖)))
114	113	cbvmptv 5204	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑖)))
115	110, 112, 114	hoicoto2 47176	. . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → 𝑋 ∈ Fin)
117	110	ffvelcdmda 7065	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑘) ∈ (ℝ × ℝ))
118		xp1st 8002	. . . . . . 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 7096	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))):𝑋⟶ℝ)
121		xp2nd 8003	. . . . . . 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 7096	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))):𝑋⟶ℝ)
124	116, 87, 120, 123	hoimbl 47202	. . . 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 2862	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
126	88, 98, 125	saliuncl 46894	. 2 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
127	86, 126	eqeltrd 2862	1 ⊢ (𝜑 → 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∃wrex 3086 {crab 3414 Vcvv 3454 ⊆ wss 3904 ⟨cop 4588 ∪ ciun 4949 class class class wbr 5100 ↦ cmpt 5181 × cxp 5645 dom cdm 5647 ∘ ccom 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ωcom 7846 1^st c1st 7968 2^nd c2nd 7969 ↑_m cmap 8808 Xcixp 8879 ≼ cdom 8925 Fincfn 8927 ℝcr 11072 ℚcq 12949 ℝ⁺crp 12993 [,)cico 13351 distcds 17295 TopOpenctopn 17450 ∞Metcxmet 21406 Metcmet 21407 ballcbl 21408 MetOpencmopn 21411 ℝ_fldcrefld 21653 freeLMod cfrlm 21795 TopOnctopon 22967 toℂPreHilctcph 25226 ℝ^crrx 25442 volncvoln 47109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cc 10392 ax-ac2 10420 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 ax-mulf 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-acn 9900 df-ac 10072 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-rlim 15516 df-sum 15714 df-prod 15934 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-prds 17476 df-pws 17478 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-mhm 18817 df-submnd 18818 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-ghm 19254 df-cntz 19357 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20228 df-ring 20281 df-cring 20282 df-oppr 20382 df-dvdsr 20402 df-unit 20403 df-invr 20433 df-dvr 20446 df-rhm 20517 df-subrng 20592 df-subrg 20616 df-drng 20777 df-field 20778 df-abv 20855 df-staf 20885 df-srng 20886 df-lmod 20926 df-lss 20996 df-lmhm 21086 df-lvec 21167 df-sra 21237 df-rgmod 21238 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-cnfld 21422 df-refld 21654 df-phl 21675 df-dsmm 21781 df-frlm 21796 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cmp 23444 df-xms 24377 df-ms 24378 df-nm 24639 df-ngp 24640 df-tng 24641 df-nrg 24642 df-nlm 24643 df-clm 25122 df-cph 25227 df-tcph 25228 df-rrx 25444 df-ovol 25523 df-vol 25524 df-salg 46880 df-sumge0 46934 df-mea 47021 df-ome 47061 df-caragen 47063 df-ovoln 47108 df-voln 47110

This theorem is referenced by: opnvonmbl 47205

W3C validator