ptrecube - Mathbox for Brendan Leahy

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Theorem ptrecube 37790

Description: Any point in an open set of N-space is surrounded by an open cube within that set. (Contributed by Brendan Leahy, 21-Aug-2020.) (Proof shortened by AV, 28-Sep-2020.)

**Hypotheses**
Ref	Expression
ptrecube.r	⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
ptrecube.d	⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))

**Assertion**
Ref	Expression
ptrecube	⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)

Distinct variable groups: 𝑛,𝑑,𝑁 𝑃,𝑑,𝑛 𝑆,𝑑,𝑛 Allowed substitution hints: 𝐷(𝑛,𝑑) 𝑅(𝑛,𝑑)

Proof of Theorem ptrecube Dummy variables 𝑔 ℎ 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptrecube.r	. . . 4 ⊢ 𝑅 = (∏_t‘((1...𝑁) × {(topGen‘ran (,))}))
2		fzfi 13897	. . . . 5 ⊢ (1...𝑁) ∈ Fin
3		retop 24707	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
4		fnconstg 6721	. . . . . 6 ⊢ ((topGen‘ran (,)) ∈ Top → ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁))
5	3, 4	ax-mp 5	. . . . 5 ⊢ ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁)
6		eqid 2735	. . . . . 6 ⊢ {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))} = {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))}
7	6	ptval 23516	. . . . 5 ⊢ (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁)) → (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) = (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))}))
8	2, 5, 7	mp2an 693	. . . 4 ⊢ (∏_t‘((1...𝑁) × {(topGen‘ran (,))})) = (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))})
9	1, 8	eqtri 2758	. . 3 ⊢ 𝑅 = (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))})
10	9	eleq2i 2827	. 2 ⊢ (𝑆 ∈ 𝑅 ↔ 𝑆 ∈ (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))}))
11		tg2 22911	. . 3 ⊢ ((𝑆 ∈ (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))}) ∧ 𝑃 ∈ 𝑆) → ∃𝑧 ∈ {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))} (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆))
12	6	elpt 23518	. . . . 5 ⊢ (𝑧 ∈ {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))} ↔ ∃𝑔((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
13		fvex 6846	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ V
14	13	fvconst2 7150	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
15	14	eleq2d 2821	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔‘𝑛) ∈ (topGen‘ran (,))))
16	15	ralbiia 3079	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)))
17		elixp2 8841	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)))
18	17	simp3bi 1148	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛))
19		r19.26 3095	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)))
20		uniretop 24708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ = ∪ (topGen‘ran (,))
21	20	eltopss 22853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝑔‘𝑛) ∈ (topGen‘ran (,))) → (𝑔‘𝑛) ⊆ ℝ)
22	3, 21	mpan 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑛) ∈ (topGen‘ran (,)) → (𝑔‘𝑛) ⊆ ℝ)
23	22	sselda 3932	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → (𝑃‘𝑛) ∈ ℝ)
24		ptrecube.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
25	24	rexmet 24737	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ∈ (∞Met‘ℝ)
26		eqid 2735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
27	24, 26	tgioo 24742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (topGen‘ran (,)) = (MetOpen‘𝐷)
28	27	mopni2 24439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛))
29	25, 28	mp3an1 1451	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛))
30		r19.42v 3167	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)) ↔ ((𝑃‘𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
31	23, 29, 30	sylanbrc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
32	31	ralimi 3072	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
33		oveq2 7366	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (ℎ‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)𝑦) = ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
34	33	sseq1d 3964	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (ℎ‘𝑛) → (((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)))
35	34	anbi2d 631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = (ℎ‘𝑛) → (((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)) ↔ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))))
36	35	ac6sfi 9186	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ⁺ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛))) → ∃ℎ(ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))))
37	2, 32, 36	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃ℎ(ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))))
38		1rp 12911	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ⁺
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ⁺)
40		frn 6668	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ran ℎ ⊆ ℝ⁺)
41	40	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ⁺)
42		ffn 6661	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ℎ Fn (1...𝑁))
43		fnfi 9104	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ℎ ∈ Fin)
44	42, 2, 43	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ℎ ∈ Fin)
45		rnfi 9242	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ Fin → ran ℎ ∈ Fin)
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ran ℎ ∈ Fin)
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ∈ Fin)
48		dm0rn0 5872	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (dom ℎ = ∅ ↔ ran ℎ = ∅)
49		fdm 6670	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → dom ℎ = (1...𝑁))
50	49	eqeq1d 2737	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → (dom ℎ = ∅ ↔ (1...𝑁) = ∅))
51	48, 50	bitr3id 285	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → (ran ℎ = ∅ ↔ (1...𝑁) = ∅))
52	51	necon3abid 2967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → (ran ℎ ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
53	52	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ≠ ∅)
54		rpssre 12915	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ⁺ ⊆ ℝ
55	40, 54	sstrdi 3945	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ran ℎ ⊆ ℝ)
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ)
57		ltso 11215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ < Or ℝ
58		fiinfcl 9408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( < Or ℝ ∧ (ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ)) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
59	57, 58	mpan 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
60	47, 53, 56, 59	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
61	41, 60	sseldd 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ℝ⁺)
62	39, 61	ifclda 4514	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ⁺)
63	62	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ⁺)
64	62	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ⁺)
65	64	rpxrd 12952	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^*)
66		ffvelcdm 7026	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ⁺)
67	66	rpxrd 12952	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ^*)
68		ne0i 4292	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69		ifnefalse 4490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1...𝑁) ≠ ∅ → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ (1...𝑁) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
71	70	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
72	55	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → ran ℎ ⊆ ℝ)
73		0re 11136	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
74		rpge0 12921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℝ⁺ → 0 ≤ 𝑦)
75	74	rgen 3052	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∀𝑦 ∈ ℝ⁺ 0 ≤ 𝑦
76		ssralv 4001	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ran ℎ ⊆ ℝ⁺ → (∀𝑦 ∈ ℝ⁺ 0 ≤ 𝑦 → ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
77	40, 75, 76	mpisyl 21	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ∀𝑦 ∈ ran ℎ0 ≤ 𝑦)
78		breq1 5100	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
79	78	ralbidv 3158	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
80	79	rspcev 3575	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
81	73, 77, 80	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
82	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
83		fnfvelrn 7025	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℎ Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
84	42, 83	sylan 581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
85		infrelb 12129	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ran ℎ ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ∧ (ℎ‘𝑛) ∈ ran ℎ) → inf(ran ℎ, ℝ, < ) ≤ (ℎ‘𝑛))
86	72, 82, 84, 85	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → inf(ran ℎ, ℝ, < ) ≤ (ℎ‘𝑛))
87	71, 86	eqbrtrd 5119	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛))
88	65, 67, 87	jca31 514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → ((if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^* ∧ (ℎ‘𝑛) ∈ ℝ^*) ∧ if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛)))
89		ssbl 24369	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑃‘𝑛) ∈ ℝ) ∧ (if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^* ∧ (ℎ‘𝑛) ∈ ℝ^*) ∧ if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛)) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
90	89	3expb 1121	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑃‘𝑛) ∈ ℝ) ∧ ((if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^* ∧ (ℎ‘𝑛) ∈ ℝ^*) ∧ if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛))) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
91	25, 90	mpanl1 701	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃‘𝑛) ∈ ℝ ∧ ((if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^* ∧ (ℎ‘𝑛) ∈ ℝ^*) ∧ if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛))) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
92	91	ancoms 458	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ^* ∧ (ℎ‘𝑛) ∈ ℝ^*) ∧ if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛)) ∧ (𝑃‘𝑛) ∈ ℝ) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
93	88, 92	sylan 581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑃‘𝑛) ∈ ℝ) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
94		sstr2 3939	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) → (((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) ∧ (𝑃‘𝑛) ∈ ℝ) → (((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
96	95	expimpd 453	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ 𝑛 ∈ (1...𝑁)) → (((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)) → ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
97	96	ralimdva 3147	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ:(1...𝑁)⟶ℝ⁺ → (∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)) → ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
98	97	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))) → ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛))
99	24	fveq2i 6836	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
100	99	oveqi 7371	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) = ((𝑃‘𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )))
101	100	sseq1i 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛))
102	101	ralbii 3081	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛))
103		nfv 1916	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)
104	102, 103	nfxfr 1855	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)
105		oveq2 7366	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) → ((𝑃‘𝑛)(ball‘𝐷)𝑑) = ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))))
106	105	sseq1d 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) → (((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
107	106	ralbidv 3158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) → (∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)))
108	104, 107	rspce 3564	. . . . . . . . . . . . . . . . 17 ⊢ ((if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
109	63, 98, 108	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
110	109	exlimiv 1932	. . . . . . . . . . . . . . 15 ⊢ (∃ℎ(ℎ:(1...𝑁)⟶ℝ⁺ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛))) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
111	37, 110	syl 17	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
112	19, 111	sylbir 235	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
113	18, 112	sylan2 594	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
114	16, 113	sylanb 582	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)) → ∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛))
115		sstr2 3939	. . . . . . . . . . . . 13 ⊢ (X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆 → X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
116		ss2ixp 8850	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛) → X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛))
117	115, 116	syl11 33	. . . . . . . . . . . 12 ⊢ (X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆 → (∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛) → X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
118	117	reximdv 3150	. . . . . . . . . . 11 ⊢ (X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆 → (∃𝑑 ∈ ℝ⁺ ∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔‘𝑛) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
119	114, 118	syl5com 31	. . . . . . . . . 10 ⊢ ((∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)) → (X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆 → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
120	119	expimpd 453	. . . . . . . . 9 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) → ((𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
121		eleq2 2824	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
122		sseq1 3958	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑧 ⊆ 𝑆 ↔ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆))
123	121, 122	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆)))
124	123	imbi1d 341	. . . . . . . . 9 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆) ↔ ((𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
125	120, 124	syl5ibrcom 247	. . . . . . . 8 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) → (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
126	125	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) → (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
127	126	imp 406	. . . . . 6 ⊢ (((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)) → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
128	127	exlimiv 1932	. . . . 5 ⊢ (∃𝑔((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)) → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
129	12, 128	sylbi 217	. . . 4 ⊢ (𝑧 ∈ {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))} → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
130	129	rexlimiv 3129	. . 3 ⊢ (∃𝑧 ∈ {𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))} (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
131	11, 130	syl 17	. 2 ⊢ ((𝑆 ∈ (topGen‘{𝑥 ∣ ∃ℎ((ℎ Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(ℎ‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(ℎ‘𝑛) = ∪ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(ℎ‘𝑛))}) ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
132	10, 131	sylanb 582	1 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ⁺ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2713 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 Vcvv 3439 ∖ cdif 3897 ⊆ wss 3900 ∅c0 4284 ifcif 4478 {csn 4579 ∪ cuni 4862 class class class wbr 5097 Or wor 5530 × cxp 5621 dom cdm 5623 ran crn 5624 ↾ cres 5625 ∘ ccom 5627 Fn wfn 6486 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 Xcixp 8837 Fincfn 8885 infcinf 9346 ℝcr 11027 0cc0 11028 1c1 11029 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11366 ℝ⁺crp 12907 (,)cioo 13263 ...cfz 13425 abscabs 15159 topGenctg 17359 ∏_tcpt 17360 ∞Metcxmet 21296 ballcbl 21298 MetOpencmopn 21301 Topctop 22839

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-fz 13426 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-topgen 17365 df-pt 17366 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-top 22840 df-topon 22857 df-bases 22892

This theorem is referenced by: poimirlem29 37819

W3C validator