Theorem ptrecube 33409

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 12771	. . . . 5 ⊢ (1...𝑁) ∈ Fin
3		retop 22565	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
4		fnconstg 6093	. . . . . 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 2622	. . . . . 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 21373	. . . . 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 708	. . . 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 2644	. . 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 2693	. 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 20769	. . 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 21375	. . . . 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 6201	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ V
14	13	fvconst2 6469	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
15	14	eleq2d 2687	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔‘𝑛) ∈ (topGen‘ran (,))))
16	15	ralbiia 2979	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ∈ (topGen‘ran (,)))
17		elixp2 7912	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛)))
18	17	simp3bi 1078	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃‘𝑛) ∈ (𝑔‘𝑛))
19		r19.26 3064	. . . . . . . . . . . . . 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 22566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ = ∪ (topGen‘ran (,))
21	20	eltopss 20712	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝑔‘𝑛) ∈ (topGen‘ran (,))) → (𝑔‘𝑛) ⊆ ℝ)
22	3, 21	mpan 706	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑛) ∈ (topGen‘ran (,)) → (𝑔‘𝑛) ⊆ ℝ)
23	22	sselda 3603	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → (𝑃‘𝑛) ∈ ℝ)
24		ptrecube.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
25	24	rexmet 22594	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 ∈ (∞Met‘ℝ)
26		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
27	24, 26	tgioo 22599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (topGen‘ran (,)) = (MetOpen‘𝐷)
28	27	mopni2 22298	. . . . . . . . . . . . . . . . . . 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 1411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛))
30		r19.42v 3092	. . . . . . . . . . . . . . . . . 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 698	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔‘𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃‘𝑛) ∈ (𝑔‘𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃‘𝑛) ∈ ℝ ∧ ((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛)))
32	31	ralimi 2952	. . . . . . . . . . . . . . . 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 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (ℎ‘𝑛) → ((𝑃‘𝑛)(ball‘𝐷)𝑦) = ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)))
34	33	sseq1d 3632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (ℎ‘𝑛) → (((𝑃‘𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔‘𝑛) ↔ ((𝑃‘𝑛)(ball‘𝐷)(ℎ‘𝑛)) ⊆ (𝑔‘𝑛)))
35	34	anbi2d 740	. . . . . . . . . . . . . . . . 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 8204	. . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . 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 11836	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℝ+
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40		frn 6053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ+)
41	40	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ+)
42		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ Fn (1...𝑁))
43		fnfi 8238	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ℎ ∈ Fin)
44	42, 2, 43	sylancl 694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ℎ ∈ Fin)
45		rnfi 8249	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ ∈ Fin → ran ℎ ∈ Fin)
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ∈ Fin)
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ∈ Fin)
48		dm0rn0 5342	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (dom ℎ = ∅ ↔ ran ℎ = ∅)
49		fdm 6051	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → dom ℎ = (1...𝑁))
50	49	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (dom ℎ = ∅ ↔ (1...𝑁) = ∅))
51	48, 50	syl5bbr 274	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ = ∅ ↔ (1...𝑁) = ∅))
52	51	necon3abid 2830	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → (ran ℎ ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
53	52	biimpar 502	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ≠ ∅)
54		rpssre 11843	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℝ+ ⊆ ℝ
55	40, 54	syl6ss 3615	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ran ℎ ⊆ ℝ)
56	55	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ℎ ⊆ ℝ)
57		ltso 10118	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ < Or ℝ
58		fiinfcl 8407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (( < Or ℝ ∧ (ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ)) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
59	57, 58	mpan 706	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ran ℎ ∈ Fin ∧ ran ℎ ≠ ∅ ∧ ran ℎ ⊆ ℝ) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
60	47, 53, 56, 59	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ran ℎ)
61	41, 60	sseldd 3604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran ℎ, ℝ, < ) ∈ ℝ+)
62	39, 61	ifclda 4120	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ+)
63	62	adantr 481	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ+)
65	64	rpxrd 11873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ∈ ℝ*)
66		ffvelrn 6357	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ+)
67	66	rpxrd 11873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ℝ*)
68		ne0i 3921	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69		ifnefalse 4098	. . . . . . . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) = inf(ran ℎ, ℝ, < ))
72	55	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → ran ℎ ⊆ ℝ)
73		0re 10040	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 0 ∈ ℝ
74		rpge0 11845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
75	74	rgen 2922	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∀𝑦 ∈ ℝ+ 0 ≤ 𝑦
76		ssralv 3666	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦))
79	78	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦))
80	79	rspcev 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ℎ0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
81	73, 77, 80	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (ℎ:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
82	81	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ℎ 𝑥 ≤ 𝑦)
83		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℎ Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
84	42, 83	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → (ℎ‘𝑛) ∈ ran ℎ)
85		infrelb 11008	. . . . . . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → inf(ran ℎ, ℝ, < ) ≤ (ℎ‘𝑛))
87	71, 86	eqbrtrd 4675	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℎ:(1...𝑁)⟶ℝ+ ∧ 𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < )) ≤ (ℎ‘𝑛))
88	65, 67, 87	jca31 557	. . . . . . . . . . . . . . . . . . . . . 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 22228	. . . . . . . . . . . . . . . . . . . . . . . . 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 1266	. . . . . . . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . 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 3610	. . . . . . . . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . 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 6194	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
100	99	oveqi 6663	. . . . . . . . . . . . . . . . . . . . 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 3629	. . . . . . . . . . . . . . . . . . . 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 2980	. . . . . . . . . . . . . . . . . . 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 1843	. . . . . . . . . . . . . . . . . . 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 1779	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑∀𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran ℎ, ℝ, < ))) ⊆ (𝑔‘𝑛)
105		oveq2 6658	. . . . . . . . . . . . . . . . . . . 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 3632	. . . . . . . . . . . . . . . . . . 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 2986	. . . . . . . . . . . . . . . . . 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 3304	. . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . 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 1858	. . . . . . . . . . . . . . 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 225	. . . . . . . . . . . . 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 491	. . . . . . . . . . . 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 489	. . . . . . . . . . 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 3610	. . . . . . . . . . . . 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 7921	. . . . . . . . . . . . 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 3016	. . . . . . . . . . 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 629	. . . . . . . . 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 2690	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛)))
122		sseq1 3626	. . . . . . . . . . 11 ⊢ (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) → (𝑧 ⊆ 𝑆 ↔ X𝑛 ∈ (1...𝑁)(𝑔‘𝑛) ⊆ 𝑆))
123	121, 122	anbi12d 747	. . . . . . . . . 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 331	. . . . . . . . 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 237	. . . . . . . 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 1083	. . . . . . 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 445	. . . . . 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 1858	. . . . 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 207	. . . 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 3027	. . 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 489	1 ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ∪ cuni 4436   class class class wbr 4653   Or wor 5034   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Xcixp 7908  Fincfn 7955  infcinf 8347  ℝcr 9935  0cc0 9936  1c1 9937  ℝ^*cxr 10073   < clt 10074   ≤ cle 10075   − cmin 10266  ℝ⁺crp 11832  (,)cioo 12175  ...cfz 12326  abscabs 13974  topGenctg 16098  ∏_tcpt 16099  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-pt 16105  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  poimirlem29  33438