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Theorem ptrecube 33539
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 12811 . . . . 5 (1...𝑁) ∈ Fin
3 retop 22612 . . . . . 6 (topGen‘ran (,)) ∈ Top
4 fnconstg 6131 . . . . . 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 2651 . . . . . 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 21421 . . . . 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 708 . . . 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 2673 . . 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 2722 . 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 20817 . . 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 21423 . . . . 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 6239 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ V
1413fvconst2 6510 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
1514eleq2d 2716 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔𝑛) ∈ (topGen‘ran (,))))
1615ralbiia 3008 . . . . . . . . . . . 12 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)))
17 elixp2 7954 . . . . . . . . . . . . . 14 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
1817simp3bi 1098 . . . . . . . . . . . . 13 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛))
19 r19.26 3093 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
20 uniretop 22613 . . . . . . . . . . . . . . . . . . . . 21 ℝ = (topGen‘ran (,))
2120eltopss 20760 . . . . . . . . . . . . . . . . . . . 20 (((topGen‘ran (,)) ∈ Top ∧ (𝑔𝑛) ∈ (topGen‘ran (,))) → (𝑔𝑛) ⊆ ℝ)
223, 21mpan 706 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑛) ∈ (topGen‘ran (,)) → (𝑔𝑛) ⊆ ℝ)
2322sselda 3636 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → (𝑃𝑛) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
2524rexmet 22641 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Met‘ℝ)
26 eqid 2651 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐷) = (MetOpen‘𝐷)
2724, 26tgioo 22646 . . . . . . . . . . . . . . . . . . . 20 (topGen‘ran (,)) = (MetOpen‘𝐷)
2827mopni2 22345 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
2925, 28mp3an1 1451 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
30 r19.42v 3121 . . . . . . . . . . . . . . . . . 18 (∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3123, 29, 30sylanbrc 699 . . . . . . . . . . . . . . . . 17 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3231ralimi 2981 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
33 oveq2 6698 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑛) → ((𝑃𝑛)(ball‘𝐷)𝑦) = ((𝑃𝑛)(ball‘𝐷)(𝑛)))
3433sseq1d 3665 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛) → (((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)))
3534anbi2d 740 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑛) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
3635ac6sfi 8245 . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
38 1rp 11874 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40 frn 6091 . . . . . . . . . . . . . . . . . . . . 21 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ+)
4140adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ+)
42 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ Fn (1...𝑁))
43 fnfi 8279 . . . . . . . . . . . . . . . . . . . . . . . 24 (( Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ∈ Fin)
4442, 2, 43sylancl 695 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ ∈ Fin)
45 rnfi 8290 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ Fin → ran ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ∈ Fin)
4746adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ∈ Fin)
48 dm0rn0 5374 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom = ∅ ↔ ran = ∅)
49 fdm 6089 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:(1...𝑁)⟶ℝ+ → dom = (1...𝑁))
5049eqeq1d 2653 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ → (dom = ∅ ↔ (1...𝑁) = ∅))
5148, 50syl5bbr 274 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ → (ran = ∅ ↔ (1...𝑁) = ∅))
5251necon3abid 2859 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → (ran ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
5352biimpar 501 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ≠ ∅)
54 rpssre 11881 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
5540, 54syl6ss 3648 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ)
5655adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ)
57 ltso 10156 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 8448 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ)) → inf(ran , ℝ, < ) ∈ ran )
5957, 58mpan 706 . . . . . . . . . . . . . . . . . . . . 21 ((ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ) → inf(ran , ℝ, < ) ∈ ran )
6047, 53, 56, 59syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ran )
6141, 60sseldd 3637 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4153 . . . . . . . . . . . . . . . . . 18 (:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6362adantr 480 . . . . . . . . . . . . . . . . 17 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6462adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6564rpxrd 11911 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ*)
66 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ+)
6766rpxrd 11911 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ*)
68 ne0i 3954 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69 ifnefalse 4131 . . . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7255adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ran ⊆ ℝ)
73 0re 10078 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 11883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
7574rgen 2951 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑦 ∈ ℝ+ 0 ≤ 𝑦
76 ssralv 3699 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4688 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
7978ralbidv 3015 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 0 → (∀𝑦 ∈ ran 𝑥𝑦 ↔ ∀𝑦 ∈ ran 0 ≤ 𝑦))
8079rspcev 3340 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8173, 77, 80sylancr 696 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8281adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
83 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
8442, 83sylan 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
85 infrelb 11046 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦 ∧ (𝑛) ∈ ran ) → inf(ran , ℝ, < ) ≤ (𝑛))
8672, 82, 84, 85syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → inf(ran , ℝ, < ) ≤ (𝑛))
8771, 86eqbrtrd 4707 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))
8865, 67, 87jca31 556 . . . . . . . . . . . . . . . . . . . . . 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 22275 . . . . . . . . . . . . . . . . . . . . . . . . 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 1285 . . . . . . . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . . 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 3643 . . . . . . . . . . . . . . . . . . . . 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 628 . . . . . . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . 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 6232 . . . . . . . . . . . . . . . . . . . . . 22 (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
10099oveqi 6703 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) = ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )))
101100sseq1i 3662 . . . . . . . . . . . . . . . . . . . 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 3009 . . . . . . . . . . . . . . . . . . 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 1883 . . . . . . . . . . . . . . . . . . 19 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
104102, 103nfxfr 1819 . . . . . . . . . . . . . . . . . 18 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
105 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → ((𝑃𝑛)(ball‘𝐷)𝑑) = ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))))
106105sseq1d 3665 . . . . . . . . . . . . . . . . . . 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 3015 . . . . . . . . . . . . . . . . . 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 3335 . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . 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 1898 . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . 13 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11318, 112sylan2 490 . . . . . . . . . . . 12 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11416, 113sylanb 488 . . . . . . . . . . 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 3643 . . . . . . . . . . . . 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 7963 . . . . . . . . . . . . 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 3045 . . . . . . . . . . 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 628 . . . . . . . . 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 2719 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑃𝑧𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
122 sseq1 3659 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑧𝑆X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆))
123121, 122anbi12d 747 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) ↔ (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆)))
124123imbi1d 330 . . . . . . . . 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 237 . . . . . . . 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 1103 . . . . . . 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 444 . . . . . 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 1898 . . . . 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 207 . . . 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 3056 . . 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 488 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  wss 3607  c0 3948  ifcif 4119  {csn 4210   cuni 4468   class class class wbr 4685   Or wor 5063   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Xcixp 7950  Fincfn 7997  infcinf 8388  cr 9973  0cc0 9974  1c1 9975  *cxr 10111   < clt 10112  cle 10113  cmin 10304  +crp 11870  (,)cioo 12213  ...cfz 12364  abscabs 14018  topGenctg 16145  tcpt 16146  ∞Metcxmt 19779  ballcbl 19781  MetOpencmopn 19784  Topctop 20746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-topgen 16151  df-pt 16152  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798
This theorem is referenced by:  poimirlem29  33568
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