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Theorem ptrecube 33080
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 12727 . . . . 5 (1...𝑁) ∈ Fin
3 retop 22505 . . . . . 6 (topGen‘ran (,)) ∈ Top
4 fnconstg 6060 . . . . . 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 2621 . . . . . 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 21313 . . . . 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 707 . . . 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 2643 . . 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 2690 . 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 20709 . . 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 21315 . . . . 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 6168 . . . . . . . . . . . . . . 15 (topGen‘ran (,)) ∈ V
1413fvconst2 6434 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) → (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) = (topGen‘ran (,)))
1514eleq2d 2684 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) → ((𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ (𝑔𝑛) ∈ (topGen‘ran (,))))
1615ralbiia 2975 . . . . . . . . . . . 12 (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (((1...𝑁) × {(topGen‘ran (,))})‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)))
17 elixp2 7872 . . . . . . . . . . . . . 14 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
1817simp3bi 1076 . . . . . . . . . . . . 13 (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛))
19 r19.26 3059 . . . . . . . . . . . . . 14 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) ↔ (∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ ∀𝑛 ∈ (1...𝑁)(𝑃𝑛) ∈ (𝑔𝑛)))
20 uniretop 22506 . . . . . . . . . . . . . . . . . . . . 21 ℝ = (topGen‘ran (,))
2120eltopss 20652 . . . . . . . . . . . . . . . . . . . 20 (((topGen‘ran (,)) ∈ Top ∧ (𝑔𝑛) ∈ (topGen‘ran (,))) → (𝑔𝑛) ⊆ ℝ)
223, 21mpan 705 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑛) ∈ (topGen‘ran (,)) → (𝑔𝑛) ⊆ ℝ)
2322sselda 3588 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → (𝑃𝑛) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))
2524rexmet 22534 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Met‘ℝ)
26 eqid 2621 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐷) = (MetOpen‘𝐷)
2724, 26tgioo 22539 . . . . . . . . . . . . . . . . . . . 20 (topGen‘ran (,)) = (MetOpen‘𝐷)
2827mopni2 22238 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (∞Met‘ℝ) ∧ (𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
2925, 28mp3an1 1408 . . . . . . . . . . . . . . . . . 18 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛))
30 r19.42v 3086 . . . . . . . . . . . . . . . . . 18 (∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ∃𝑦 ∈ ℝ+ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3123, 29, 30sylanbrc 697 . . . . . . . . . . . . . . . . 17 (((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
3231ralimi 2948 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∀𝑛 ∈ (1...𝑁)∃𝑦 ∈ ℝ+ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)))
33 oveq2 6623 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑛) → ((𝑃𝑛)(ball‘𝐷)𝑦) = ((𝑃𝑛)(ball‘𝐷)(𝑛)))
3433sseq1d 3617 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑛) → (((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛) ↔ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛)))
3534anbi2d 739 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑛) → (((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)𝑦) ⊆ (𝑔𝑛)) ↔ ((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
3635ac6sfi 8164 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ (1...𝑁)((𝑔𝑛) ∈ (topGen‘ran (,)) ∧ (𝑃𝑛) ∈ (𝑔𝑛)) → ∃(:(1...𝑁)⟶ℝ+ ∧ ∀𝑛 ∈ (1...𝑁)((𝑃𝑛) ∈ ℝ ∧ ((𝑃𝑛)(ball‘𝐷)(𝑛)) ⊆ (𝑔𝑛))))
38 1rp 11796 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ (1...𝑁) = ∅) → 1 ∈ ℝ+)
40 frn 6020 . . . . . . . . . . . . . . . . . . . . 21 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ+)
4140adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ+)
42 ffn 6012 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ Fn (1...𝑁))
43 fnfi 8198 . . . . . . . . . . . . . . . . . . . . . . . 24 (( Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) → ∈ Fin)
4442, 2, 43sylancl 693 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ ∈ Fin)
45 rnfi 8209 . . . . . . . . . . . . . . . . . . . . . . 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 5312 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom = ∅ ↔ ran = ∅)
49 fdm 6018 . . . . . . . . . . . . . . . . . . . . . . . . 25 (:(1...𝑁)⟶ℝ+ → dom = (1...𝑁))
5049eqeq1d 2623 . . . . . . . . . . . . . . . . . . . . . . . 24 (:(1...𝑁)⟶ℝ+ → (dom = ∅ ↔ (1...𝑁) = ∅))
5148, 50syl5bbr 274 . . . . . . . . . . . . . . . . . . . . . . 23 (:(1...𝑁)⟶ℝ+ → (ran = ∅ ↔ (1...𝑁) = ∅))
5251necon3abid 2826 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → (ran ≠ ∅ ↔ ¬ (1...𝑁) = ∅))
5352biimpar 502 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ≠ ∅)
54 rpssre 11803 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
5540, 54syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . 22 (:(1...𝑁)⟶ℝ+ → ran ⊆ ℝ)
5655adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → ran ⊆ ℝ)
57 ltso 10078 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 8367 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ)) → inf(ran , ℝ, < ) ∈ ran )
5957, 58mpan 705 . . . . . . . . . . . . . . . . . . . . 21 ((ran ∈ Fin ∧ ran ≠ ∅ ∧ ran ⊆ ℝ) → inf(ran , ℝ, < ) ∈ ran )
6047, 53, 56, 59syl3anc 1323 . . . . . . . . . . . . . . . . . . . 20 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ran )
6141, 60sseldd 3589 . . . . . . . . . . . . . . . . . . 19 ((:(1...𝑁)⟶ℝ+ ∧ ¬ (1...𝑁) = ∅) → inf(ran , ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4098 . . . . . . . . . . . . . . . . . 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 11833 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) ∈ ℝ*)
66 ffvelrn 6323 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ+)
6766rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . 23 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ℝ*)
68 ne0i 3903 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) → (1...𝑁) ≠ ∅)
69 ifnefalse 4076 . . . . . . . . . . . . . . . . . . . . . . . . . 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 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 11805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
7574rgen 2918 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑦 ∈ ℝ+ 0 ≤ 𝑦
76 ssralv 3651 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
7978ralbidv 2982 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 0 → (∀𝑦 ∈ ran 𝑥𝑦 ↔ ∀𝑦 ∈ ran 0 ≤ 𝑦))
8079rspcev 3299 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran 0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦)
8173, 77, 80sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6322 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (( Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
8442, 83sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → (𝑛) ∈ ran )
85 infrelb 10968 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑥𝑦 ∧ (𝑛) ∈ ran ) → inf(ran , ℝ, < ) ≤ (𝑛))
8672, 82, 84, 85syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . 24 ((:(1...𝑁)⟶ℝ+𝑛 ∈ (1...𝑁)) → inf(ran , ℝ, < ) ≤ (𝑛))
8771, 86eqbrtrd 4645 . . . . . . . . . . . . . . . . . . . . . . 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 22168 . . . . . . . . . . . . . . . . . . . . . . . . 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 1263 . . . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . . . . . . . . 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 2958 . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . 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 6161 . . . . . . . . . . . . . . . . . . . . . 22 (ball‘𝐷) = (ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))
10099oveqi 6628 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) = ((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )))
101100sseq1i 3614 . . . . . . . . . . . . . . . . . . . 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 2976 . . . . . . . . . . . . . . . . . . 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 1840 . . . . . . . . . . . . . . . . . . 19 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
104102, 103nfxfr 1776 . . . . . . . . . . . . . . . . . 18 𝑑𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))) ⊆ (𝑔𝑛)
105 oveq2 6623 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = ∅, 1, inf(ran , ℝ, < )) → ((𝑃𝑛)(ball‘𝐷)𝑑) = ((𝑃𝑛)(ball‘𝐷)if((1...𝑁) = ∅, 1, inf(ran , ℝ, < ))))
106105sseq1d 3617 . . . . . . . . . . . . . . . . . . 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 2982 . . . . . . . . . . . . . . . . . 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 3294 . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . 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 1855 . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . 12 ((∀𝑛 ∈ (1...𝑁)(𝑔𝑛) ∈ (topGen‘ran (,)) ∧ 𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)) → ∃𝑑 ∈ ℝ+𝑛 ∈ (1...𝑁)((𝑃𝑛)(ball‘𝐷)𝑑) ⊆ (𝑔𝑛))
11416, 113sylanb 489 . . . . . . . . . . 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 3595 . . . . . . . . . . . . 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 7881 . . . . . . . . . . . . 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 3012 . . . . . . . . . . 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 2687 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑃𝑧𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛)))
122 sseq1 3611 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → (𝑧𝑆X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆))
123121, 122anbi12d 746 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(𝑔𝑛) → ((𝑃𝑧𝑧𝑆) ↔ (𝑃X𝑛 ∈ (1...𝑁)(𝑔𝑛) ∧ X𝑛 ∈ (1...𝑁)(𝑔𝑛) ⊆ 𝑆)))
124123imbi1d 331 . . . . . . . . 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 1081 . . . . . . 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 445 . . . . . 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 1855 . . . . 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 3022 . . 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 489 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 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2908  wrex 2909  Vcvv 3190  cdif 3557  wss 3560  c0 3897  ifcif 4064  {csn 4155   cuni 4409   class class class wbr 4623   Or wor 5004   × cxp 5082  dom cdm 5084  ran crn 5085  cres 5086  ccom 5088   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  Xcixp 7868  Fincfn 7915  infcinf 8307  cr 9895  0cc0 9896  1c1 9897  *cxr 10033   < clt 10034  cle 10035  cmin 10226  +crp 11792  (,)cioo 12133  ...cfz 12284  abscabs 13924  topGenctg 16038  tcpt 16039  ∞Metcxmt 19671  ballcbl 19673  MetOpencmopn 19676  Topctop 20638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-fz 12285  df-seq 12758  df-exp 12817  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-topgen 16044  df-pt 16045  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690
This theorem is referenced by:  poimirlem29  33109
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