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Theorem ptrecube 35397
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 13432 . . . . 5 (1...𝑁) ∈ Fin
3 retop 23515 . . . . . 6 (topGen‘ran (,)) ∈ Top
4 fnconstg 6567 . . . . . 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 2738 . . . . . 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 22322 . . . . 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 692 . . . 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 2824 . 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 21717 . . 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 22324 . . . . 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 6688 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ V
1413fvconst2 6977 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
1514eleq2d 2818 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔𝑛) ∈ (topGen‘ran (,))))
1615ralbiia 3079 . . . . . . . . . . . 12 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)))
17 elixp2 8512 . . . . . . . . . . . . . 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 3084 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
20 uniretop 23516 . . . . . . . . . . . . . . . . . . . . 21 ℝ = (topGen‘ran (,))
2120eltopss 21659 . . . . . . . . . . . . . . . . . . . 20 (((topGen‘ran (,)) ∈ Top ∧ (𝑔𝑛) ∈ (topGen‘ran (,))) → (𝑔𝑛) ⊆ ℝ)
223, 21mpan 690 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑛) ∈ (topGen‘ran (,)) → (𝑔𝑛) ⊆ ℝ)
2322sselda 3878 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → (𝑃𝑛) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
2524rexmet 23544 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Met‘ℝ)
26 eqid 2738 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐷) = (MetOpen‘𝐷)
2724, 26tgioo 23549 . . . . . . . . . . . . . . . . . . . 20 (topGen‘ran (,)) = (MetOpen‘𝐷)
2827mopni2 23247 . . . . . . . . . . . . . . . . . . 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 3254 . . . . . . . . . . . . . . . . . 18 (∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3123, 29, 30sylanbrc 586 . . . . . . . . . . . . . . . . 17 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3231ralimi 3075 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
33 oveq2 7179 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑛) → ((𝑃𝑛)(ball‘𝐷)𝑦) = ((𝑃𝑛)(ball‘𝐷)(𝑛)))
3433sseq1d 3909 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛) → (((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)))
3534anbi2d 632 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑛) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
3635ac6sfi 8837 . . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
38 1rp 12477 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40 frn 6512 . . . . . . . . . . . . . . . . . . . . 21 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ+)
4140adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ+)
42 ffn 6505 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ Fn (1...𝑁))
43 fnfi 8779 . . . . . . . . . . . . . . . . . . . . . . . 24 (( Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ∈ Fin)
4442, 2, 43sylancl 589 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ ∈ Fin)
45 rnfi 8881 . . . . . . . . . . . . . . . . . . . . . . 23 ( ∈ Fin → ran ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ∈ Fin)
4746adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ∈ Fin)
48 dm0rn0 5769 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom = ∅ ↔ ran = ∅)
49 fdm 6514 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:(1...𝑁)⟶ℝ+ → dom = (1...𝑁))
5049eqeq1d 2740 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ → (dom = ∅ ↔ (1...𝑁) = ∅))
5148, 50bitr3id 288 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ → (ran = ∅ ↔ (1...𝑁) = ∅))
5251necon3abid 2970 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → (ran ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
5352biimpar 481 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ≠ ∅)
54 rpssre 12480 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
5540, 54sstrdi 3890 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ)
5655adantr 484 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ)
57 ltso 10800 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 9039 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ)) → inf(ran , ℝ, < ) ∈ ran )
5957, 58mpan 690 . . . . . . . . . . . . . . . . . . . . 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 3879 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4450 . . . . . . . . . . . . . . . . . 18 (:(1...𝑁)⟶ℝ+ → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6362adantr 484 . . . . . . . . . . . . . . . . 17 ((:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6462adantr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ+)
6564rpxrd 12516 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ*)
66 ffvelrn 6860 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ+)
6766rpxrd 12516 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ*)
68 ne0i 4224 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69 ifnefalse 4427 . . . . . . . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) = inf(ran , ℝ, < ))
7255adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ran ⊆ ℝ)
73 0re 10722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 12486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
7574rgen 3063 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑦 ∈ ℝ+ 0 ≤ 𝑦
76 ssralv 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5034 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
7978ralbidv 3109 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 0 → (∀𝑦 ∈ ran 𝑥𝑦 ↔ ∀𝑦 ∈ ran 0 ≤ 𝑦))
8079rspcev 3527 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8173, 77, 80sylancr 590 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (:(1...𝑁)⟶ℝ+ → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8281adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
83 fnfvelrn 6859 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
8442, 83sylan 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
85 infrelb 11704 . . . . . . . . . . . . . . . . . . . . . . . . 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 5053 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ≤ (𝑛))
8865, 67, 87jca31 518 . . . . . . . . . . . . . . . . . . . . . 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 23177 . . . . . . . . . . . . . . . . . . . . . . . . 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 700 . . . . . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . 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 3885 . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . 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 3091 . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . 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 6678 . . . . . . . . . . . . . . . . . . . . . 22 (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
10099oveqi 7184 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) = ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )))
101100sseq1i 3906 . . . . . . . . . . . . . . . . . . . 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 3080 . . . . . . . . . . . . . . . . . . 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 1920 . . . . . . . . . . . . . . . . . . 19 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
104102, 103nfxfr 1859 . . . . . . . . . . . . . . . . . 18 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
105 oveq2 7179 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → ((𝑃𝑛)(ball‘𝐷)𝑑) = ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))))
106105sseq1d 3909 . . . . . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . . . . . . 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 3516 . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . 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 1936 . . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . 13 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11318, 112sylan2 596 . . . . . . . . . . . 12 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11416, 113sylanb 584 . . . . . . . . . . 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 3885 . . . . . . . . . . . . 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 8521 . . . . . . . . . . . . 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 3183 . . . . . . . . . . 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 457 . . . . . . . . 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 2821 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑃𝑧𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
122 sseq1 3903 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑧𝑆X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆))
123121, 122anbi12d 634 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) ↔ (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆)))
124123imbi1d 345 . . . . . . . . 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 250 . . . . . . . 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 410 . . . . . 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 1936 . . . . 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 220 . . . 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 3190 . . 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 584 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 399  w3a 1088   = wceq 1542  wex 1786  wcel 2113  {cab 2716  wne 2934  wral 3053  wrex 3054  Vcvv 3398  cdif 3841  wss 3844  c0 4212  ifcif 4415  {csn 4517   cuni 4797   class class class wbr 5031   Or wor 5442   × cxp 5524  dom cdm 5526  ran crn 5527  cres 5528  ccom 5530   Fn wfn 6335  wf 6336  cfv 6340  (class class class)co 7171  Xcixp 8508  Fincfn 8556  infcinf 8979  cr 10615  0cc0 10616  1c1 10617  *cxr 10753   < clt 10754  cle 10755  cmin 10949  +crp 12473  (,)cioo 12822  ...cfz 12982  abscabs 14684  topGenctg 16815  tcpt 16816  ∞Metcxmet 20203  ballcbl 20205  MetOpencmopn 20208  Topctop 21645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480  ax-cnex 10672  ax-resscn 10673  ax-1cn 10674  ax-icn 10675  ax-addcl 10676  ax-addrcl 10677  ax-mulcl 10678  ax-mulrcl 10679  ax-mulcom 10680  ax-addass 10681  ax-mulass 10682  ax-distr 10683  ax-i2m1 10684  ax-1ne0 10685  ax-1rid 10686  ax-rnegex 10687  ax-rrecex 10688  ax-cnre 10689  ax-pre-lttri 10690  ax-pre-lttrn 10691  ax-pre-ltadd 10692  ax-pre-mulgt0 10693  ax-pre-sup 10694
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7128  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-1st 7715  df-2nd 7716  df-wrecs 7977  df-recs 8038  df-rdg 8076  df-1o 8132  df-er 8321  df-map 8440  df-ixp 8509  df-en 8557  df-dom 8558  df-sdom 8559  df-fin 8560  df-sup 8980  df-inf 8981  df-pnf 10756  df-mnf 10757  df-xr 10758  df-ltxr 10759  df-le 10760  df-sub 10951  df-neg 10952  df-div 11377  df-nn 11718  df-2 11780  df-3 11781  df-n0 11978  df-z 12064  df-uz 12326  df-q 12432  df-rp 12474  df-xneg 12591  df-xadd 12592  df-xmul 12593  df-ioo 12826  df-fz 12983  df-seq 13462  df-exp 13523  df-cj 14549  df-re 14550  df-im 14551  df-sqrt 14685  df-abs 14686  df-topgen 16821  df-pt 16822  df-psmet 20210  df-xmet 20211  df-met 20212  df-bl 20213  df-mopn 20214  df-top 21646  df-topon 21663  df-bases 21698
This theorem is referenced by:  poimirlem29  35426
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