opnvonmbllem2 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > opnvonmbllem2		Structured version Visualization version GIF version

Theorem opnvonmbllem2 46588

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 2734	. . . . . . . . . . . 12 ⊢ (dist‘(ℝ^‘𝑋)) = (dist‘(ℝ^‘𝑋))
3	2	rrxmetfi 25459	. . . . . . . . . . 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 24359	. . . . . . . . . 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 2734	. . . . . . . . . . . . . 14 ⊢ (ℝ^‘𝑋) = (ℝ^‘𝑋)
10	9	rrxval 25434	. . . . . . . . . . . . 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 6910	. . . . . . . . . . 11 ⊢ (𝜑 → (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))))
13		ovex 7463	. . . . . . . . . . . . 13 ⊢ (ℝ_fld freeLMod 𝑋) ∈ V
14		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (toℂPreHil‘(ℝ_fld freeLMod 𝑋))
15		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
16		eqid 2734	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (TopOpen‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋)))
17	14, 15, 16	tcphtopn 25273	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (𝜑 → (toℂPreHil‘(ℝ_fld freeLMod 𝑋)) = (ℝ^‘𝑋))
21	20	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘(toℂPreHil‘(ℝ_fld freeLMod 𝑋))) = (dist‘(ℝ^‘𝑋)))
22	21	fveq2d 6910	. . . . . . . . . . 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 2840	. . . . . . . . 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 2734	. . . . . . . . 9 ⊢ (MetOpen‘(dist‘(ℝ^‘𝑋))) = (MetOpen‘(dist‘(ℝ^‘𝑋)))
28	27	mopni2 24521	. . . . . . . 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 1370	. . . . . . 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 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘(ℝ^‘𝑋)) = (TopOpen‘(ℝ^‘𝑋))
32	31	rrxtoponfi 46246	. . . . . . . . . . . . . . . . 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 22948	. . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . 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 46580	. . . . . . . . . . 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 1131	. . . . . . . . . 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 1911	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺)
43		nfv 1911	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑖(𝑐 ∈ (ℚ ↑_m 𝑋) ∧ 𝑑 ∈ (ℚ ↑_m 𝑋))
44		nfcv 2902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖𝑥
45		nfixp1 8956	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
46	44, 45	nfel 2917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖 𝑥 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖))
47		nfcv 2902	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑖(𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
48	45, 47	nfss 3987	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑖X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)) ⊆ (𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒)
49	46, 48	nfan 1896	. . . . . . . . . . . . . . . 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 1898	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . 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 8887	. . . . . . . . . . . . . . . . 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 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))) → 𝑐:𝑋⟶ℚ)
56		elmapi 8887	. . . . . . . . . . . . . . . . 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 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))) → 𝑑:𝑋⟶ℚ)
59		simp3r 1201	. . . . . . . . . . . . . . 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 1197	. . . . . . . . . . . . . . 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 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 ∈ X𝑖 ∈ 𝑋 ((𝑐‘𝑖)[,)(𝑑‘𝑖)))
62		opnvonmbl.k	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺}
63		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩) = (𝑖 ∈ 𝑋 ↦ ⟨(𝑐‘𝑖), (𝑑‘𝑖)⟩)
64	50, 52, 55, 58, 59, 60, 61, 62, 63	opnvonmbllem1 46587	. . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . 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 1130	. . . . . . . . . . 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 3209	. . . . . . . . . 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 1118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐺) → (𝑒 ∈ ℝ⁺ → ((𝑥(ball‘(dist‘(ℝ^‘𝑋)))𝑒) ⊆ 𝐺 → ∃ℎ ∈ 𝐾 𝑥 ∈ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))))
71	70	rexlimdv 3150	. . . . . . 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 4999	. . . . . 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 3143	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 𝑥 ∈ ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
76		dfss3 3983	. . . 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 2830	. . . . . . . . 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 3454	. . . . . . 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 3143	. . . 4 ⊢ (𝜑 → ∀ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺)
85		iunss 5049	. . . 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 4012	. 2 ⊢ (𝜑 → 𝐺 = ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖))
88		opnvonmbllem2.n	. . . 4 ⊢ 𝑆 = dom (voln‘𝑋)
89	1, 88	dmovnsal 46567	. . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg)
90		ssrab2 4089	. . . . . 6 ⊢ {ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋) ∣ X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ⊆ 𝐺} ⊆ ((ℚ × ℚ) ↑_m 𝑋)
91	62, 90	eqsstri 4029	. . . . 5 ⊢ 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋)
92	91	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ⊆ ((ℚ × ℚ) ↑_m 𝑋))
93		qct 45311	. . . . . . 7 ⊢ ℚ ≼ ω
94	93	a1i 11	. . . . . 6 ⊢ (𝜑 → ℚ ≼ ω)
95		xpct 10053	. . . . . 6 ⊢ ((ℚ ≼ ω ∧ ℚ ≼ ω) → (ℚ × ℚ) ≼ ω)
96	94, 94, 95	syl2anc 584	. . . . 5 ⊢ (𝜑 → (ℚ × ℚ) ≼ ω)
97	96, 1	mpct 45143	. . . 4 ⊢ (𝜑 → ((ℚ × ℚ) ↑_m 𝑋) ≼ ω)
98		ssct 9089	. . . 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 11243	. . . . . . . . . 10 ⊢ ℝ ∈ V
101	100, 100	xpex 7771	. . . . . . . . 9 ⊢ (ℝ × ℝ) ∈ V
102		qssre 12998	. . . . . . . . . 10 ⊢ ℚ ⊆ ℝ
103		xpss12 5703	. . . . . . . . . 10 ⊢ ((ℚ ⊆ ℝ ∧ ℚ ⊆ ℝ) → (ℚ × ℚ) ⊆ (ℝ × ℝ))
104	102, 102, 103	mp2an 692	. . . . . . . . 9 ⊢ (ℚ × ℚ) ⊆ (ℝ × ℝ)
105		mapss 8927	. . . . . . . . 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 3990	. . . . . . . 8 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℚ × ℚ) ↑_m 𝑋))
108	106, 107	sselid 3992	. . . . . . 7 ⊢ (ℎ ∈ 𝐾 → ℎ ∈ ((ℝ × ℝ) ↑_m 𝑋))
109		elmapi 8887	. . . . . . 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 6911	. . . . . 6 ⊢ (𝑘 = 𝑖 → (1^st ‘(ℎ‘𝑘)) = (1^st ‘(ℎ‘𝑖)))
113	112	cbvmptv 5260	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑖)))
114		2fveq3 6911	. . . . . 6 ⊢ (𝑘 = 𝑖 → (2^nd ‘(ℎ‘𝑘)) = (2^nd ‘(ℎ‘𝑖)))
115	114	cbvmptv 5260	. . . . 5 ⊢ (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))) = (𝑖 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑖)))
116	111, 113, 115	hoicoto2 46560	. . . 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 7103	. . . . . . 7 ⊢ (((𝜑 ∧ ℎ ∈ 𝐾) ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑘) ∈ (ℝ × ℝ))
119		xp1st 8044	. . . . . . 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 7134	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (1^st ‘(ℎ‘𝑘))):𝑋⟶ℝ)
122		xp2nd 8045	. . . . . . 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 7134	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → (𝑘 ∈ 𝑋 ↦ (2^nd ‘(ℎ‘𝑘))):𝑋⟶ℝ)
125	117, 88, 121, 124	hoimbl 46586	. . . 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 2838	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ 𝐾) → X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
127	89, 99, 126	saliuncl 46278	. 2 ⊢ (𝜑 → ∪ ℎ ∈ 𝐾 X𝑖 ∈ 𝑋 (([,) ∘ ℎ)‘𝑖) ∈ 𝑆)
128	87, 127	eqeltrd 2838	1 ⊢ (𝜑 → 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ⊆ wss 3962 ⟨cop 4636 ∪ ciun 4995 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 dom cdm 5688 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ωcom 7886 1^st c1st 8010 2^nd c2nd 8011 ↑_m cmap 8864 Xcixp 8935 ≼ cdom 8981 Fincfn 8983 ℝcr 11151 ℚcq 12987 ℝ⁺crp 13031 [,)cico 13385 distcds 17306 TopOpenctopn 17467 ∞Metcxmet 21366 Metcmet 21367 ballcbl 21368 MetOpencmopn 21371 ℝ_fldcrefld 21639 freeLMod cfrlm 21783 TopOnctopon 22931 toℂPreHilctcph 25214 ℝ^crrx 25430 volncvoln 46493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cc 10472 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-prod 15936 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-prds 17493 df-pws 17495 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-drng 20747 df-field 20748 df-abv 20826 df-staf 20856 df-srng 20857 df-lmod 20876 df-lss 20947 df-lmhm 21038 df-lvec 21119 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-refld 21640 df-phl 21661 df-dsmm 21769 df-frlm 21784 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cmp 23410 df-xms 24345 df-ms 24346 df-nm 24610 df-ngp 24611 df-tng 24612 df-nrg 24613 df-nlm 24614 df-clm 25109 df-cph 25215 df-tcph 25216 df-rrx 25432 df-ovol 25512 df-vol 25513 df-salg 46264 df-sumge0 46318 df-mea 46405 df-ome 46445 df-caragen 46447 df-ovoln 46492 df-voln 46494

This theorem is referenced by: opnvonmbl 46589

W3C validator