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Theorem ptrecube 36078
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 13877 . . . . 5 (1...𝑁) ∈ Fin
3 retop 24125 . . . . . 6 (topGen‘ran (,)) ∈ Top
4 fnconstg 6730 . . . . . 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 2736 . . . . . 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 22921 . . . . 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 690 . . . 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 2764 . . 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 2829 . 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 22315 . . 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 22923 . . . . 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 6855 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ V
1413fvconst2 7153 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
1514eleq2d 2823 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔𝑛) ∈ (topGen‘ran (,))))
1615ralbiia 3094 . . . . . . . . . . . 12 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)))
17 elixp2 8839 . . . . . . . . . . . . . 14 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
1817simp3bi 1147 . . . . . . . . . . . . 13 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛))
19 r19.26 3114 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
20 uniretop 24126 . . . . . . . . . . . . . . . . . . . . 21 ℝ = (topGen‘ran (,))
2120eltopss 22256 . . . . . . . . . . . . . . . . . . . 20 (((topGen‘ran (,)) ∈ Top ∧ (𝑔𝑛) ∈ (topGen‘ran (,))) → (𝑔𝑛) ⊆ ℝ)
223, 21mpan 688 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑛) ∈ (topGen‘ran (,)) → (𝑔𝑛) ⊆ ℝ)
2322sselda 3944 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → (𝑃𝑛) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
2524rexmet 24154 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Met‘ℝ)
26 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐷) = (MetOpen‘𝐷)
2724, 26tgioo 24159 . . . . . . . . . . . . . . . . . . . 20 (topGen‘ran (,)) = (MetOpen‘𝐷)
2827mopni2 23849 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
2925, 28mp3an1 1448 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
30 r19.42v 3187 . . . . . . . . . . . . . . . . . 18 (∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3123, 29, 30sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3231ralimi 3086 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
33 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑛) → ((𝑃𝑛)(ball‘𝐷)𝑦) = ((𝑃𝑛)(ball‘𝐷)(𝑛)))
3433sseq1d 3975 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛) → (((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)))
3534anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑛) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
3635ac6sfi 9231 . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
38 1rp 12919 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40 frn 6675 . . . . . . . . . . . . . . . . . . . . 21 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ+)
4140adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ+)
42 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ Fn (1...𝑁))
43 fnfi 9125 . . . . . . . . . . . . . . . . . . . . . . . 24 (( Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ∈ Fin)
4442, 2, 43sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ ∈ Fin)
45 rnfi 9279 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ Fin → ran ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ∈ Fin)
4746adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ∈ Fin)
48 dm0rn0 5880 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom = ∅ ↔ ran = ∅)
49 fdm 6677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:(1...𝑁)⟶ℝ+ → dom = (1...𝑁))
5049eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ → (dom = ∅ ↔ (1...𝑁) = ∅))
5148, 50bitr3id 284 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ → (ran = ∅ ↔ (1...𝑁) = ∅))
5251necon3abid 2980 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → (ran ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
5352biimpar 478 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ≠ ∅)
54 rpssre 12922 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
5540, 54sstrdi 3956 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ)
5655adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ)
57 ltso 11235 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 9437 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ)) → inf(ran , ℝ, < ) ∈ ran )
5957, 58mpan 688 . . . . . . . . . . . . . . . . . . . . 21 ((ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ) → inf(ran , ℝ, < ) ∈ ran )
6047, 53, 56, 59syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ran )
6141, 60sseldd 3945 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4521 . . . . . . . . . . . . . . . . . 18 (:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6362adantr 481 . . . . . . . . . . . . . . . . 17 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6462adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6564rpxrd 12958 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ*)
66 ffvelcdm 7032 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ+)
6766rpxrd 12958 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ*)
68 ne0i 4294 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69 ifnefalse 4498 . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7255adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ran ⊆ ℝ)
73 0re 11157 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 12928 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
7574rgen 3066 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑦 ∈ ℝ+ 0 ≤ 𝑦
76 ssralv 4010 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
7978ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 0 → (∀𝑦 ∈ ran 𝑥𝑦 ↔ ∀𝑦 ∈ ran 0 ≤ 𝑦))
8079rspcev 3581 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8173, 77, 80sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8281adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
83 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
8442, 83sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
85 infrelb 12140 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦 ∧ (𝑛) ∈ ran ) → inf(ran , ℝ, < ) ≤ (𝑛))
8672, 82, 84, 85syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → inf(ran , ℝ, < ) ≤ (𝑛))
8771, 86eqbrtrd 5127 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))
8865, 67, 87jca31 515 . . . . . . . . . . . . . . . . . . . . . 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 23776 . . . . . . . . . . . . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . . . . 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 3951 . . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . . . . 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 6845 . . . . . . . . . . . . . . . . . . . . . 22 (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
10099oveqi 7370 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) = ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )))
101100sseq1i 3972 . . . . . . . . . . . . . . . . . . . 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 3096 . . . . . . . . . . . . . . . . . . 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 1917 . . . . . . . . . . . . . . . . . . 19 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
104102, 103nfxfr 1855 . . . . . . . . . . . . . . . . . 18 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
105 oveq2 7365 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → ((𝑃𝑛)(ball‘𝐷)𝑑) = ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))))
106105sseq1d 3975 . . . . . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . . . . . . 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 3570 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 1933 . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . 12 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11416, 113sylanb 581 . . . . . . . . . . 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 3951 . . . . . . . . . . . . 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 8848 . . . . . . . . . . . . 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 3167 . . . . . . . . . . 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 454 . . . . . . . . 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 2826 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑃𝑧𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
122 sseq1 3969 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑧𝑆X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆))
123121, 122anbi12d 631 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) ↔ (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆)))
124123imbi1d 341 . . . . . . . . 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 1134 . . . . . . 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 407 . . . . . 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 1933 . . . . 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 3145 . . 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 581 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  c0 4282  ifcif 4486  {csn 4586   cuni 4865   class class class wbr 5105   Or wor 5544   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  Xcixp 8835  Fincfn 8883  infcinf 9377  cr 11050  0cc0 11051  1c1 11052  *cxr 11188   < clt 11189  cle 11190  cmin 11385  +crp 12915  (,)cioo 13264  ...cfz 13424  abscabs 15119  topGenctg 17319  tcpt 17320  ∞Metcxmet 20781  ballcbl 20783  MetOpencmopn 20786  Topctop 22242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-topgen 17325  df-pt 17326  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296
This theorem is referenced by:  poimirlem29  36107
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