Theorem ptrecube 37834

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 13900	. . . . 5 ⊢ (1...𝑁) ∈ Fin
3		retop 24710	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
4		fnconstg 6723	. . . . . 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 2737	. . . . . 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 23519	. . . . 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 2760	. . 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 2829	. 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 22914	. . 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 23521	. . . . 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 6848	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ V
14	13	fvconst2 7153	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
15	14	eleq2d 2823	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔‘𝑛) ∈ (topGen‘ran (,))))
16	15	ralbiia 3081	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)))
17		elixp2 8844	. . . . . . . . . . . . . 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 3097	. . . . . . . . . . . . . 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 24711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ = ∪ (topGen‘ran (,))
21	20	eltopss 22856	. . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → (𝑃‘𝑛) ∈ ℝ)
24		ptrecube.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
25	24	rexmet 24740	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ∈ (∞Met‘ℝ)
26		eqid 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
27	24, 26	tgioo 24745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (topGen‘ran (,)) = (MetOpen‘𝐷)
28	27	mopni2 24442	. . . . . . . . . . . . . . . . . . 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 3169	. . . . . . . . . . . . . . . . . 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 3074	. . . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (ℎ‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)𝑦) = ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
34	33	sseq1d 3966	. . . . . . . . . . . . . . . . . 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 9189	. . . . . . . . . . . . . . . 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 12914	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ+
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40		frn 6670	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ+)
41	40	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ+)
42		ffn 6663	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ Fn (1...𝑁))
43		fnfi 9107	. . . . . . . . . . . . . . . . . . . . . . . 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 9245	. . . . . . . . . . . . . . . . . . . . . . 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 5874	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (dom ℎ = ∅ ↔ ran ℎ = ∅)
49		fdm 6672	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → dom ℎ = (1...𝑁))
50	49	eqeq1d 2739	. . . . . . . . . . . . . . . . . . . . . . . 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 2969	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
53	52	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ≠ ∅)
54		rpssre 12918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ⊆ ℝ
55	40, 54	sstrdi 3947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ)
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ)
57		ltso 11218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ < Or ℝ
58		fiinfcl 9411	. . . . . . . . . . . . . . . . . . . . . 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 3935	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ℝ+)
62	39, 61	ifclda 4516	. . . . . . . . . . . . . . . . . 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 12955	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ*)
66		ffvelcdm 7028	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ+)
67	66	rpxrd 12955	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ*)
68		ne0i 4294	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69		ifnefalse 4492	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11139	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
74		rpge0 12924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
75	74	rgen 3054	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∀𝑦 ∈ ℝ+ 0 ≤ 𝑦
76		ssralv 4003	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
79	78	ralbidv 3160	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
80	79	rspcev 3577	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12132	. . . . . . . . . . . . . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . . . . . . . . . . . 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 24372	. . . . . . . . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
100	99	oveqi 7374	. . . . . . . . . . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . . . . . . 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 3083	. . . . . . . . . . . . . . . . . . 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 7369	. . . . . . . . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . . . 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 3566	. . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . 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 8853	. . . . . . . . . . . . 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 3152	. . . . . . . . . . 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 2826	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
122		sseq1 3960	. . . . . . . . . . 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 3131	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4286  ifcif 4480  {csn 4581  ∪ cuni 4864   class class class wbr 5099   Or wor 5532   × cxp 5623  dom cdm 5625  ran crn 5626   ↾ cres 5627   ∘ ccom 5629   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Xcixp 8840  Fincfn 8888  infcinf 9349  ℝcr 11030  0cc0 11031  1c1 11032  ℝ^*cxr 11170   < clt 11171   ≤ cle 11172   − cmin 11369  ℝ⁺crp 12910  (,)cioo 13266  ...cfz 13428  abscabs 15162  topGenctg 17362  ∏_tcpt 17363  ∞Metcxmet 21299  ballcbl 21301  MetOpencmopn 21304  Topctop 22842

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108  ax-pre-sup 11109

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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  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-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-er 8638  df-map 8770  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12151  df-2 12213  df-3 12214  df-n0 12407  df-z 12494  df-uz 12757  df-q 12867  df-rp 12911  df-xneg 13031  df-xadd 13032  df-xmul 13033  df-ioo 13270  df-fz 13429  df-seq 13930  df-exp 13990  df-cj 15027  df-re 15028  df-im 15029  df-sqrt 15163  df-abs 15164  df-topgen 17368  df-pt 17369  df-psmet 21306  df-xmet 21307  df-met 21308  df-bl 21309  df-mopn 21310  df-top 22843  df-topon 22860  df-bases 22895

This theorem is referenced by:  poimirlem29  37863