Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ptrecube Structured version   Visualization version   GIF version

Theorem ptrecube 36426
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 ((𝑆𝑅𝑃𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(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 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptrecube.r . . . 4 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))
2 fzfi 13933 . . . . 5 (1...𝑁) ∈ Fin
3 retop 24260 . . . . . 6 (topGen‘ran (,)) ∈ Top
4 fnconstg 6776 . . . . . 6 ((topGen‘ran (,)) ∈ Top → ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁))
53, 4ax-mp 5 . . . . 5 ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁)
6 eqid 2733 . . . . . 6 {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))} = {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))}
76ptval 23056 . . . . 5 (((1...𝑁) ∈ Fin ∧ ((1...𝑁) × {(topGen‘ran (,))}) Fn (1...𝑁)) → (∏t‘((1...𝑁) × {(topGen‘ran (,))})) = (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))}))
82, 5, 7mp2an 691 . . . 4 (∏t‘((1...𝑁) × {(topGen‘ran (,))})) = (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))})
91, 8eqtri 2761 . . 3 𝑅 = (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))})
109eleq2i 2826 . 2 (𝑆𝑅𝑆 ∈ (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))}))
11 tg2 22450 . . 3 ((𝑆 ∈ (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))}) ∧ 𝑃𝑆) → ∃𝑧 ∈ {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))} (𝑃𝑧𝑧𝑆))
126elpt 23058 . . . . 5 (𝑧 ∈ {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))} ↔ ∃𝑔((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
13 fvex 6901 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ V
1413fvconst2 7200 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
1514eleq2d 2820 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔𝑛) ∈ (topGen‘ran (,))))
1615ralbiia 3092 . . . . . . . . . . . 12 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)))
17 elixp2 8891 . . . . . . . . . . . . . 14 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
1817simp3bi 1148 . . . . . . . . . . . . 13 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛))
19 r19.26 3112 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
20 uniretop 24261 . . . . . . . . . . . . . . . . . . . . 21 ℝ = (topGen‘ran (,))
2120eltopss 22391 . . . . . . . . . . . . . . . . . . . 20 (((topGen‘ran (,)) ∈ Top ∧ (𝑔𝑛) ∈ (topGen‘ran (,))) → (𝑔𝑛) ⊆ ℝ)
223, 21mpan 689 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑛) ∈ (topGen‘ran (,)) → (𝑔𝑛) ⊆ ℝ)
2322sselda 3981 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → (𝑃𝑛) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
2524rexmet 24289 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Met‘ℝ)
26 eqid 2733 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐷) = (MetOpen‘𝐷)
2724, 26tgioo 24294 . . . . . . . . . . . . . . . . . . . 20 (topGen‘ran (,)) = (MetOpen‘𝐷)
2827mopni2 23984 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
2925, 28mp3an1 1449 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
30 r19.42v 3191 . . . . . . . . . . . . . . . . . 18 (∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3123, 29, 30sylanbrc 584 . . . . . . . . . . . . . . . . 17 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3231ralimi 3084 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
33 oveq2 7412 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑛) → ((𝑃𝑛)(ball‘𝐷)𝑦) = ((𝑃𝑛)(ball‘𝐷)(𝑛)))
3433sseq1d 4012 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛) → (((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)))
3534anbi2d 630 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑛) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
3635ac6sfi 9283 . . . . . . . . . . . . . . . 16 (((1...𝑁) ∈ Fin ∧ ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
372, 32, 36sylancr 588 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
38 1rp 12974 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40 frn 6721 . . . . . . . . . . . . . . . . . . . . 21 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ+)
4140adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ+)
42 ffn 6714 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ Fn (1...𝑁))
43 fnfi 9177 . . . . . . . . . . . . . . . . . . . . . . . 24 (( Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ∈ Fin)
4442, 2, 43sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ ∈ Fin)
45 rnfi 9331 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ Fin → ran ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ∈ Fin)
4746adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ∈ Fin)
48 dm0rn0 5922 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom = ∅ ↔ ran = ∅)
49 fdm 6723 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:(1...𝑁)⟶ℝ+ → dom = (1...𝑁))
5049eqeq1d 2735 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ → (dom = ∅ ↔ (1...𝑁) = ∅))
5148, 50bitr3id 285 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ → (ran = ∅ ↔ (1...𝑁) = ∅))
5251necon3abid 2978 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → (ran ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
5352biimpar 479 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ≠ ∅)
54 rpssre 12977 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
5540, 54sstrdi 3993 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ)
5655adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ)
57 ltso 11290 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 9492 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ)) → inf(ran , ℝ, < ) ∈ ran )
5957, 58mpan 689 . . . . . . . . . . . . . . . . . . . . 21 ((ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ) → inf(ran , ℝ, < ) ∈ ran )
6047, 53, 56, 59syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ran )
6141, 60sseldd 3982 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4562 . . . . . . . . . . . . . . . . . 18 (:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6362adantr 482 . . . . . . . . . . . . . . . . 17 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6462adantr 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6564rpxrd 13013 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ*)
66 ffvelcdm 7079 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ+)
6766rpxrd 13013 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ*)
68 ne0i 4333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69 ifnefalse 4539 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1...𝑁) ≠ ∅ → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑁) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7170adantl 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7255adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ran ⊆ ℝ)
73 0re 11212 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 12983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
7574rgen 3064 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑦 ∈ ℝ+ 0 ≤ 𝑦
76 ssralv 4049 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ran ⊆ ℝ+ → (∀𝑦 ∈ ℝ+ 0 ≤ 𝑦 → ∀𝑦 ∈ ran 0 ≤ 𝑦))
7740, 75, 76mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (:(1...𝑁)⟶ℝ+ → ∀𝑦 ∈ ran 0 ≤ 𝑦)
78 breq1 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
7978ralbidv 3178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 0 → (∀𝑦 ∈ ran 𝑥𝑦 ↔ ∀𝑦 ∈ ran 0 ≤ 𝑦))
8079rspcev 3612 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8173, 77, 80sylancr 588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8281adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
83 fnfvelrn 7078 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
8442, 83sylan 581 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
85 infrelb 12195 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦 ∧ (𝑛) ∈ ran ) → inf(ran , ℝ, < ) ≤ (𝑛))
8672, 82, 84, 85syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → inf(ran , ℝ, < ) ≤ (𝑛))
8771, 86eqbrtrd 5169 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))
8865, 67, 87jca31 516 . . . . . . . . . . . . . . . . . . . . . 22 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ((if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ* ∧ (𝑛) ∈ ℝ*) ∧ if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛)))
89 ssbl 23911 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑃𝑛) ∈ ℝ) ∧ (if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ* ∧ (𝑛) ∈ ℝ*) ∧ if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛)) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)))
90893expb 1121 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑃𝑛) ∈ ℝ) ∧ ((if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ* ∧ (𝑛) ∈ ℝ*) ∧ if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)))
9125, 90mpanl1 699 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃𝑛) ∈ ℝ ∧ ((if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ* ∧ (𝑛) ∈ ℝ*) ∧ if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)))
9291ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 ((((if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ* ∧ (𝑛) ∈ ℝ*) ∧ if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛)) ∧ (𝑃𝑛) ∈ ℝ) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)))
9388, 92sylan 581 . . . . . . . . . . . . . . . . . . . . 21 (((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) ∧ (𝑃𝑛) ∈ ℝ) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)))
94 sstr2 3988 . . . . . . . . . . . . . . . . . . . . 21 (((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ ((𝑃𝑛)(ball‘𝐷)(𝑛)) → (((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
9593, 94syl 17 . . . . . . . . . . . . . . . . . . . 20 (((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) ∧ (𝑃𝑛) ∈ ℝ) → (((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
9695expimpd 455 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)) → ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
9796ralimdva 3168 . . . . . . . . . . . . . . . . . 18 (:(1...𝑁)⟶ℝ+ → (∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
9897imp 408 . . . . . . . . . . . . . . . . 17 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → ∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛))
9924fveq2i 6891 . . . . . . . . . . . . . . . . . . . . . 22 (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
10099oveqi 7417 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) = ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )))
101100sseq1i 4009 . . . . . . . . . . . . . . . . . . . 20 (((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛))
102101ralbii 3094 . . . . . . . . . . . . . . . . . . 19 (∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛) ↔ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛))
103 nfv 1918 . . . . . . . . . . . . . . . . . . 19 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
104102, 103nfxfr 1856 . . . . . . . . . . . . . . . . . 18 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
105 oveq2 7412 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → ((𝑃𝑛)(ball‘𝐷)𝑑) = ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))))
106105sseq1d 4012 . . . . . . . . . . . . . . . . . . 19 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → (((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
107106ralbidv 3178 . . . . . . . . . . . . . . . . . 18 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → (∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛) ↔ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)))
108104, 107rspce 3601 . . . . . . . . . . . . . . . . 17 ((if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
10963, 98, 108syl2anc 585 . . . . . . . . . . . . . . . 16 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
110109exlimiv 1934 . . . . . . . . . . . . . . 15 (∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11137, 110syl 17 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11219, 111sylbir 234 . . . . . . . . . . . . 13 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11318, 112sylan2 594 . . . . . . . . . . . 12 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11416, 113sylanb 582 . . . . . . . . . . 11 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
115 sstr2 3988 . . . . . . . . . . . . 13 (X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
116 ss2ixp 8900 . . . . . . . . . . . . 13 (∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛) → X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ X𝑛 ∈ (1...𝑁)(𝑔𝑛))
117115, 116syl11 33 . . . . . . . . . . . 12 (X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆 → (∀𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛) → X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
118117reximdv 3171 . . . . . . . . . . 11 (X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆 → (∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
119114, 118syl5com 31 . . . . . . . . . 10 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → (X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆 → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
120119expimpd 455 . . . . . . . . 9 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) → ((𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
121 eleq2 2823 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑃𝑧𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
122 sseq1 4006 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑧𝑆X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆))
123121, 122anbi12d 632 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) ↔ (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆)))
124123imbi1d 342 . . . . . . . . 9 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆) ↔ ((𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
125120, 124syl5ibrcom 246 . . . . . . . 8 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) → (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
1261253ad2ant2 1135 . . . . . . 7 ((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) → (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)))
127126imp 408 . . . . . 6 (((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
128127exlimiv 1934 . . . . 5 (∃𝑔((𝑔 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑧 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑧)(𝑔𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
12912, 128sylbi 216 . . . 4 (𝑧 ∈ {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))} → ((𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆))
130129rexlimiv 3149 . . 3 (∃𝑧 ∈ {𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))} (𝑃𝑧𝑧𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
13111, 130syl 17 . 2 ((𝑆 ∈ (topGen‘{𝑥 ∣ ∃(( Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ ((1...𝑁) ∖ 𝑤)(𝑛) = (((1...𝑁) × {(topGen‘ran (,))})‘𝑛)) ∧ 𝑥 = X𝑛 ∈ (1...𝑁)(𝑛))}) ∧ 𝑃𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
13210, 131sylanb 582 1 ((𝑆𝑅𝑃𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wral 3062  wrex 3071  Vcvv 3475  cdif 3944  wss 3947  c0 4321  ifcif 4527  {csn 4627   cuni 4907   class class class wbr 5147   Or wor 5586   × cxp 5673  dom cdm 5675  ran crn 5676  cres 5677  ccom 5679   Fn wfn 6535  wf 6536  cfv 6540  (class class class)co 7404  Xcixp 8887  Fincfn 8935  infcinf 9432  cr 11105  0cc0 11106  1c1 11107  *cxr 11243   < clt 11244  cle 11245  cmin 11440  +crp 12970  (,)cioo 13320  ...cfz 13480  abscabs 15177  topGenctg 17379  tcpt 17380  ∞Metcxmet 20914  ballcbl 20916  MetOpencmopn 20919  Topctop 22377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-fz 13481  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-topgen 17385  df-pt 17386  df-psmet 20921  df-xmet 20922  df-met 20923  df-bl 20924  df-mopn 20925  df-top 22378  df-topon 22395  df-bases 22431
This theorem is referenced by:  poimirlem29  36455
  Copyright terms: Public domain W3C validator