Theorem ptrecube 37609

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 13916	. . . . 5 ⊢ (1...𝑁) ∈ Fin
3		retop 24684	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
4		fnconstg 6731	. . . . . 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 2729	. . . . . 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 23492	. . . . 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 692	. . . 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 2752	. . 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 2820	. 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 22887	. . 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 23494	. . . . 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 6854	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ V
14	13	fvconst2 7161	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
15	14	eleq2d 2814	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔‘𝑛) ∈ (topGen‘ran (,))))
16	15	ralbiia 3073	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)))
17		elixp2 8852	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)))
18	17	simp3bi 1147	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛))
19		r19.26 3091	. . . . . . . . . . . . . 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 24685	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ = ∪ (topGen‘ran (,))
21	20	eltopss 22829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝑔‘𝑛) ∈ (topGen‘ran (,))) → (𝑔‘𝑛) ⊆ ℝ)
22	3, 21	mpan 690	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑛) ∈ (topGen‘ran (,)) → (𝑔‘𝑛) ⊆ ℝ)
23	22	sselda 3943	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → (𝑃‘𝑛) ∈ ℝ)
24		ptrecube.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
25	24	rexmet 24714	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ∈ (∞Met‘ℝ)
26		eqid 2729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
27	24, 26	tgioo 24719	. . . . . . . . . . . . . . . . . . . 20 ⊢ (topGen‘ran (,)) = (MetOpen‘𝐷)
28	27	mopni2 24416	. . . . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
32	31	ralimi 3066	. . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (ℎ‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)𝑦) = ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
34	33	sseq1d 3975	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (ℎ‘𝑛) → (((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)))
35	34	anbi2d 630	. . . . . . . . . . . . . . . . 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 9208	. . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . 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 12934	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ+
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40		frn 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ+)
41	40	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ+)
42		ffn 6671	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ Fn (1...𝑁))
43		fnfi 9120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ℎ ∈ Fin)
44	42, 2, 43	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ ∈ Fin)
45		rnfi 9268	. . . . . . . . . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (dom ℎ = ∅ ↔ ran ℎ = ∅)
49		fdm 6680	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → dom ℎ = (1...𝑁))
50	49	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
53	52	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ≠ ∅)
54		rpssre 12938	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ⊆ ℝ
55	40, 54	sstrdi 3956	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ)
56	55	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ)
57		ltso 11233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ < Or ℝ
58		fiinfcl 9431	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( < Or ℝ ∧ (ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ)) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
59	57, 58	mpan 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
60	47, 53, 56, 59	syl3anc 1373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
61	41, 60	sseldd 3944	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ℝ+)
62	39, 61	ifclda 4520	. . . . . . . . . . . . . . . . . 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 12975	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ*)
66		ffvelcdm 7036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ+)
67	66	rpxrd 12975	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ*)
68		ne0i 4300	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69		ifnefalse 4496	. . . . . . . . . . . . . . . . . . . . . . . . . 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 11155	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
74		rpge0 12944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
75	74	rgen 3046	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∀𝑦 ∈ ℝ+ 0 ≤ 𝑦
76		ssralv 4012	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
79	78	ralbidv 3156	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
80	79	rspcev 3585	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
81	73, 77, 80	sylancr 587	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7035	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℎ Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
84	42, 83	sylan 580	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
85		infrelb 12147	. . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → inf(ran ℎ, ℝ, < ) ≤ (ℎ‘𝑛))
87	71, 86	eqbrtrd 5124	. . . . . . . . . . . . . . . . . . . . . . 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 24346	. . . . . . . . . . . . . . . . . . . . . . . . 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 1120	. . . . . . . . . . . . . . . . . . . . . . . 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 700	. . . . . . . . . . . . . . . . . . . . . . 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 580	. . . . . . . . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
100	99	oveqi 7383	. . . . . . . . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . . . . . . 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 3075	. . . . . . . . . . . . . . . . . . 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 1914	. . . . . . . . . . . . . . . . . . 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 1853	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)
105		oveq2 7378	. . . . . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 1930	. . . . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 581	. . . . . . . . . . 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 3950	. . . . . . . . . . . . 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 8861	. . . . . . . . . . . . 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 3148	. . . . . . . . . . 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 2817	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
122		sseq1 3969	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑧 ⊆ 𝑆 ↔ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆))
123	121, 122	anbi12d 632	. . . . . . . . . 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 1134	. . . . . . 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 1930	. . . . 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 3127	. . 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 581	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 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ∖ cdif 3908   ⊆ wss 3911  ∅c0 4292  ifcif 4484  {csn 4585  ∪ cuni 4867   class class class wbr 5102   Or wor 5538   × cxp 5629  dom cdm 5631  ran crn 5632   ↾ cres 5633   ∘ ccom 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7370  Xcixp 8848  Fincfn 8896  infcinf 9369  ℝcr 11046  0cc0 11047  1c1 11048  ℝ^*cxr 11186   < clt 11187   ≤ cle 11188   − cmin 11384  ℝ⁺crp 12930  (,)cioo 13285  ...cfz 13447  abscabs 15178  topGenctg 17378  ∏_tcpt 17379  ∞Metcxmet 21283  ballcbl 21285  MetOpencmopn 21288  Topctop 22815

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7692  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6453  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7824  df-1st 7948  df-2nd 7949  df-frecs 8238  df-wrecs 8269  df-recs 8318  df-rdg 8356  df-1o 8412  df-er 8649  df-map 8779  df-ixp 8849  df-en 8897  df-dom 8898  df-sdom 8899  df-fin 8900  df-sup 9370  df-inf 9371  df-pnf 11189  df-mnf 11190  df-xr 11191  df-ltxr 11192  df-le 11193  df-sub 11386  df-neg 11387  df-div 11815  df-nn 12166  df-2 12228  df-3 12229  df-n0 12422  df-z 12509  df-uz 12773  df-q 12887  df-rp 12931  df-xneg 13051  df-xadd 13052  df-xmul 13053  df-ioo 13289  df-fz 13448  df-seq 13946  df-exp 14006  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-topgen 17384  df-pt 17385  df-psmet 21290  df-xmet 21291  df-met 21292  df-bl 21293  df-mopn 21294  df-top 22816  df-topon 22833  df-bases 22868

This theorem is referenced by:  poimirlem29  37638