Theorem ptrecube 38458

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 14083	. . . . 5 ⊢ (1...𝑁) ∈ Fin
3		retop 25041	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
4		fnconstg 6758	. . . . . 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 2760	. . . . . 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 23850	. . . . 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 705	. . . 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 2783	. . 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 2852	. 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 23244	. . 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 23852	. . . . 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 6886	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ V
14	13	fvconst2 7198	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
15	14	eleq2d 2846	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔‘𝑛) ∈ (topGen‘ran (,))))
16	15	ralbiia 3106	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)))
17		elixp2 8907	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)))
18	17	simp3bi 1165	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛))
19		r19.26 3122	. . . . . . . . . . . . . 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 25042	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ = ∪ (topGen‘ran (,))
21	20	eltopss 23186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝑔‘𝑛) ∈ (topGen‘ran (,))) → (𝑔‘𝑛) ⊆ ℝ)
22	3, 21	mpan 703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑛) ∈ (topGen‘ran (,)) → (𝑔‘𝑛) ⊆ ℝ)
23	22	sselda 3930	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → (𝑃‘𝑛) ∈ ℝ)
24		ptrecube.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
25	24	rexmet 25071	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ∈ (∞Met‘ℝ)
26		eqid 2760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
27	24, 26	tgioo 25076	. . . . . . . . . . . . . . . . . . . 20 ⊢ (topGen‘ran (,)) = (MetOpen‘𝐷)
28	27	mopni2 24773	. . . . . . . . . . . . . . . . . . 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 1477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛))
30		r19.42v 3194	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
32	31	ralimi 3099	. . . . . . . . . . . . . . . 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 7416	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (ℎ‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)𝑦) = ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
34	33	sseq1d 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (ℎ‘𝑛) → (((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)))
35	34	anbi2d 642	. . . . . . . . . . . . . . . . 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 9253	. . . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . . 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 13093	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ+
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40		frn 6705	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ+)
41	40	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ+)
42		ffn 6697	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ Fn (1...𝑁))
43		fnfi 9171	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ℎ ∈ Fin)
44	42, 2, 43	sylancl 598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ ∈ Fin)
45		rnfi 9307	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ Fin → ran ℎ ∈ Fin)
46	44, 45	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ∈ Fin)
47	46	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ∈ Fin)
48		dm0rn0 5902	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (dom ℎ = ∅ ↔ ran ℎ = ∅)
49		fdm 6707	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → dom ℎ = (1...𝑁))
50	49	eqeq1d 2762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (dom ℎ = ∅ ↔ (1...𝑁) = ∅))
51	48, 50	bitr3id 288	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ = ∅ ↔ (1...𝑁) = ∅))
52	51	necon3abid 2991	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
53	52	biimpar 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ≠ ∅)
54		rpssre 13097	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ⊆ ℝ
55	40, 54	sstrdi 3942	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ)
56	55	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ)
57		ltso 11361	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ < Or ℝ
58		fiinfcl 9473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( < Or ℝ ∧ (ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ)) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
59	57, 58	mpan 703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
60	47, 53, 56, 59	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
61	41, 60	sseldd 3931	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ℝ+)
62	39, 61	ifclda 4517	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ+)
63	62	adantr 486	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ+)
65	64	rpxrd 13134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ*)
66		ffvelcdm 7069	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ+)
67	66	rpxrd 13134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ*)
68		ne0i 4286	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69		ifnefalse 4493	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1...𝑁) ≠ ∅ → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
70	68, 69	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ (1...𝑁) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
71	70	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
72	55	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → ran ℎ ⊆ ℝ)
73		0re 11281	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
74		rpge0 13103	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
75	74	rgen 3078	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∀𝑦 ∈ ℝ+ 0 ≤ 𝑦
76		ssralv 3999	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ran ℎ ⊆ ℝ+ → (∀𝑦 ∈ ℝ+ 0 ≤ 𝑦 → ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
77	40, 75, 76	mpisyl 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ∀𝑦 ∈ ran ℎ0 ≤ 𝑦)
78		breq1 5105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
79	78	ralbidv 3185	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
80	79	rspcev 3576	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
81	73, 77, 80	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
82	81	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
83		fnfvelrn 7068	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℎ Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
84	42, 83	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
85		infrelb 12271	. . . . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → inf(ran ℎ, ℝ, < ) ≤ (ℎ‘𝑛))
87	71, 86	eqbrtrd 5126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛))
88	65, 67, 87	jca31 524	. . . . . . . . . . . . . . . . . . . . . 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 24703	. . . . . . . . . . . . . . . . . . . . . . . . 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 1138	. . . . . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . 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 3174	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . 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 6876	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
100	99	oveqi 7421	. . . . . . . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . . . . . 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 3108	. . . . . . . . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . . 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 1886	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)
105		oveq2 7416	. . . . . . . . . . . . . . . . . . . 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 3961	. . . . . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . 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 1963	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 238	. . . . . . . . . . . . 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 605	. . . . . . . . . . . 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 593	. . . . . . . . . . 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 3937	. . . . . . . . . . . . 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 8916	. . . . . . . . . . . . 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 34	. . . . . . . . . . . 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 3177	. . . . . . . . . . 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 32	. . . . . . . . . 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 459	. . . . . . . . 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 2849	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
122		sseq1 3955	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑧 ⊆ 𝑆 ↔ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆))
123	121, 122	anbi12d 644	. . . . . . . . . 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 344	. . . . . . . . 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 250	. . . . . . . 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 1152	. . . . . . 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 412	. . . . . 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 1963	. . . . 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 220	. . . 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 3156	. . 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 18	. 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 593	1 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∖ cdif 3895   ⊆ wss 3898  ∅c0 4278  ifcif 4481  {csn 4583  ∪ cuni 4866   class class class wbr 5102   Or wor 5554   × cxp 5645  dom cdm 5647  ran crn 5648   ↾ cres 5649   ∘ ccom 5651   Fn wfn 6522  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408  Xcixp 8903  Fincfn 8951  infcinf 9411  ℝcr 11170  0cc0 11171  1c1 11172  ℝ^*cxr 11313   < clt 11314   ≤ cle 11315   − cmin 11512  ℝ⁺crp 13089  (,)cioo 13445  ...cfz 13608  abscabs 15368  topGenctg 17569  ∏_tcpt 17570  ∞Metcxmet 21624  ballcbl 21626  MetOpencmopn 21629  Topctop 23172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  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 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-sup 9412  df-inf 9413  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-nn 12305  df-2 12374  df-3 12375  df-n0 12576  df-z 12663  df-uz 12935  df-q 13045  df-rp 13090  df-xneg 13210  df-xadd 13211  df-xmul 13212  df-ioo 13449  df-fz 13609  df-seq 14113  df-exp 14173  df-cj 15233  df-re 15234  df-im 15235  df-sqrt 15369  df-abs 15370  df-topgen 17575  df-pt 17576  df-psmet 21631  df-xmet 21632  df-met 21633  df-bl 21634  df-mopn 21635  df-top 23173  df-topon 23190  df-bases 23225

This theorem is used by:  poimirlem29  38487