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Theorem ptrecube 36791
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 = (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))}))
ptrecube.d 𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))
Assertion
Ref Expression
ptrecube ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 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 β„Ž 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptrecube.r . . . 4 𝑅 = (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))}))
2 fzfi 13941 . . . . 5 (1...𝑁) ∈ Fin
3 retop 24498 . . . . . 6 (topGenβ€˜ran (,)) ∈ Top
4 fnconstg 6778 . . . . . 6 ((topGenβ€˜ran (,)) ∈ Top β†’ ((1...𝑁) Γ— {(topGenβ€˜ran (,))}) Fn (1...𝑁))
53, 4ax-mp 5 . . . . 5 ((1...𝑁) Γ— {(topGenβ€˜ran (,))}) Fn (1...𝑁)
6 eqid 2730 . . . . . 6 {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))} = {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))}
76ptval 23294 . . . . 5 (((1...𝑁) ∈ Fin ∧ ((1...𝑁) Γ— {(topGenβ€˜ran (,))}) Fn (1...𝑁)) β†’ (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))})) = (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))}))
82, 5, 7mp2an 688 . . . 4 (∏tβ€˜((1...𝑁) Γ— {(topGenβ€˜ran (,))})) = (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))})
91, 8eqtri 2758 . . 3 𝑅 = (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))})
109eleq2i 2823 . 2 (𝑆 ∈ 𝑅 ↔ 𝑆 ∈ (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))}))
11 tg2 22688 . . 3 ((𝑆 ∈ (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))}) ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘§ ∈ {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))} (𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆))
126elpt 23296 . . . . 5 (𝑧 ∈ {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))} ↔ βˆƒπ‘”((𝑔 Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑧)(π‘”β€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)))
13 fvex 6903 . . . . . . . . . . . . . . 15 (topGenβ€˜ran (,)) ∈ V
1413fvconst2 7206 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) = (topGenβ€˜ran (,)))
1514eleq2d 2817 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) β†’ ((π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ↔ (π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,))))
1615ralbiia 3089 . . . . . . . . . . . 12 (βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ↔ βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)))
17 elixp2 8897 . . . . . . . . . . . . . 14 (𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) ↔ (𝑃 ∈ V ∧ 𝑃 Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)))
1817simp3bi 1145 . . . . . . . . . . . . 13 (𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ βˆ€π‘› ∈ (1...𝑁)(π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›))
19 r19.26 3109 . . . . . . . . . . . . . 14 (βˆ€π‘› ∈ (1...𝑁)((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) ↔ (βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)))
20 uniretop 24499 . . . . . . . . . . . . . . . . . . . . 21 ℝ = βˆͺ (topGenβ€˜ran (,))
2120eltopss 22629 . . . . . . . . . . . . . . . . . . . 20 (((topGenβ€˜ran (,)) ∈ Top ∧ (π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,))) β†’ (π‘”β€˜π‘›) βŠ† ℝ)
223, 21mpan 686 . . . . . . . . . . . . . . . . . . 19 ((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) β†’ (π‘”β€˜π‘›) βŠ† ℝ)
2322sselda 3981 . . . . . . . . . . . . . . . . . 18 (((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ (π‘ƒβ€˜π‘›) ∈ ℝ)
24 ptrecube.d . . . . . . . . . . . . . . . . . . . 20 𝐷 = ((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ))
2524rexmet 24527 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Metβ€˜β„)
26 eqid 2730 . . . . . . . . . . . . . . . . . . . . 21 (MetOpenβ€˜π·) = (MetOpenβ€˜π·)
2724, 26tgioo 24532 . . . . . . . . . . . . . . . . . . . 20 (topGenβ€˜ran (,)) = (MetOpenβ€˜π·)
2827mopni2 24222 . . . . . . . . . . . . . . . . . . 19 ((𝐷 ∈ (∞Metβ€˜β„) ∧ (π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›))
2925, 28mp3an1 1446 . . . . . . . . . . . . . . . . . 18 (((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›))
30 r19.42v 3188 . . . . . . . . . . . . . . . . . 18 (βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›)) ↔ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›)))
3123, 29, 30sylanbrc 581 . . . . . . . . . . . . . . . . 17 (((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›)))
3231ralimi 3081 . . . . . . . . . . . . . . . 16 (βˆ€π‘› ∈ (1...𝑁)((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆ€π‘› ∈ (1...𝑁)βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›)))
33 oveq2 7419 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (β„Žβ€˜π‘›) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) = ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
3433sseq1d 4012 . . . . . . . . . . . . . . . . . 18 (𝑦 = (β„Žβ€˜π‘›) β†’ (((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›) ↔ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›)))
3534anbi2d 627 . . . . . . . . . . . . . . . . 17 (𝑦 = (β„Žβ€˜π‘›) β†’ (((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›)) ↔ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))))
3635ac6sfi 9289 . . . . . . . . . . . . . . . 16 (((1...𝑁) ∈ Fin ∧ βˆ€π‘› ∈ (1...𝑁)βˆƒπ‘¦ ∈ ℝ+ ((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) βŠ† (π‘”β€˜π‘›))) β†’ βˆƒβ„Ž(β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))))
372, 32, 36sylancr 585 . . . . . . . . . . . . . . 15 (βˆ€π‘› ∈ (1...𝑁)((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒβ„Ž(β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))))
38 1rp 12982 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ (1...𝑁) = βˆ…) β†’ 1 ∈ ℝ+)
40 frn 6723 . . . . . . . . . . . . . . . . . . . . 21 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž βŠ† ℝ+)
4140adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž βŠ† ℝ+)
42 ffn 6716 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ β„Ž Fn (1...𝑁))
43 fnfi 9183 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) β†’ β„Ž ∈ Fin)
4442, 2, 43sylancl 584 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ β„Ž ∈ Fin)
45 rnfi 9337 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž ∈ Fin β†’ ran β„Ž ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž ∈ Fin)
4746adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž ∈ Fin)
48 dm0rn0 5923 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom β„Ž = βˆ… ↔ ran β„Ž = βˆ…)
49 fdm 6725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ dom β„Ž = (1...𝑁))
5049eqeq1d 2732 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (dom β„Ž = βˆ… ↔ (1...𝑁) = βˆ…))
5148, 50bitr3id 284 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (ran β„Ž = βˆ… ↔ (1...𝑁) = βˆ…))
5251necon3abid 2975 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (ran β„Ž β‰  βˆ… ↔ Β¬ (1...𝑁) = βˆ…))
5352biimpar 476 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž β‰  βˆ…)
54 rpssre 12985 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ+ βŠ† ℝ
5540, 54sstrdi 3993 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž βŠ† ℝ)
5655adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž βŠ† ℝ)
57 ltso 11298 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 9498 . . . . . . . . . . . . . . . . . . . . . 22 (( < Or ℝ ∧ (ran β„Ž ∈ Fin ∧ ran β„Ž β‰  βˆ… ∧ ran β„Ž βŠ† ℝ)) β†’ inf(ran β„Ž, ℝ, < ) ∈ ran β„Ž)
5957, 58mpan 686 . . . . . . . . . . . . . . . . . . . . 21 ((ran β„Ž ∈ Fin ∧ ran β„Ž β‰  βˆ… ∧ ran β„Ž βŠ† ℝ) β†’ inf(ran β„Ž, ℝ, < ) ∈ ran β„Ž)
6047, 53, 56, 59syl3anc 1369 . . . . . . . . . . . . . . . . . . . 20 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ inf(ran β„Ž, ℝ, < ) ∈ ran β„Ž)
6141, 60sseldd 3982 . . . . . . . . . . . . . . . . . . 19 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ inf(ran β„Ž, ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4562 . . . . . . . . . . . . . . . . . 18 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+)
6362adantr 479 . . . . . . . . . . . . . . . . 17 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+)
6462adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+)
6564rpxrd 13021 . . . . . . . . . . . . . . . . . . . . . . 23 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ*)
66 ffvelcdm 7082 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ℝ+)
6766rpxrd 13021 . . . . . . . . . . . . . . . . . . . . . . 23 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ℝ*)
68 ne0i 4333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) β†’ (1...𝑁) β‰  βˆ…)
69 ifnefalse 4539 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1...𝑁) β‰  βˆ… β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) = inf(ran β„Ž, ℝ, < ))
7068, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑁) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) = inf(ran β„Ž, ℝ, < ))
7170adantl 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) = inf(ran β„Ž, ℝ, < ))
7255adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ ran β„Ž βŠ† ℝ)
73 0re 11220 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 12991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ β†’ 0 ≀ 𝑦)
7574rgen 3061 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 βˆ€π‘¦ ∈ ℝ+ 0 ≀ 𝑦
76 ssralv 4049 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (ran β„Ž βŠ† ℝ+ β†’ (βˆ€π‘¦ ∈ ℝ+ 0 ≀ 𝑦 β†’ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦))
7740, 75, 76mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦)
78 breq1 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘₯ = 0 β†’ (π‘₯ ≀ 𝑦 ↔ 0 ≀ 𝑦))
7978ralbidv 3175 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘₯ = 0 β†’ (βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦 ↔ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦))
8079rspcev 3611 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
8173, 77, 80sylancr 585 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
8281adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
83 fnfvelrn 7081 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β„Ž Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ran β„Ž)
8442, 83sylan 578 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ran β„Ž)
85 infrelb 12203 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran β„Ž βŠ† ℝ ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦 ∧ (β„Žβ€˜π‘›) ∈ ran β„Ž) β†’ inf(ran β„Ž, ℝ, < ) ≀ (β„Žβ€˜π‘›))
8672, 82, 84, 85syl3anc 1369 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ inf(ran β„Ž, ℝ, < ) ≀ (β„Žβ€˜π‘›))
8771, 86eqbrtrd 5169 . . . . . . . . . . . . . . . . . . . . . . 23 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›))
8865, 67, 87jca31 513 . . . . . . . . . . . . . . . . . . . . . 22 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ ((if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ* ∧ (β„Žβ€˜π‘›) ∈ ℝ*) ∧ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›)))
89 ssbl 24149 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐷 ∈ (∞Metβ€˜β„) ∧ (π‘ƒβ€˜π‘›) ∈ ℝ) ∧ (if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ* ∧ (β„Žβ€˜π‘›) ∈ ℝ*) ∧ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›)) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
90893expb 1118 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐷 ∈ (∞Metβ€˜β„) ∧ (π‘ƒβ€˜π‘›) ∈ ℝ) ∧ ((if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ* ∧ (β„Žβ€˜π‘›) ∈ ℝ*) ∧ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›))) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
9125, 90mpanl1 696 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ* ∧ (β„Žβ€˜π‘›) ∈ ℝ*) ∧ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›))) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
9291ancoms 457 . . . . . . . . . . . . . . . . . . . . . 22 ((((if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ* ∧ (β„Žβ€˜π‘›) ∈ ℝ*) ∧ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ≀ (β„Žβ€˜π‘›)) ∧ (π‘ƒβ€˜π‘›) ∈ ℝ) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
9388, 92sylan 578 . . . . . . . . . . . . . . . . . . . . 21 (((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) ∧ (π‘ƒβ€˜π‘›) ∈ ℝ) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
94 sstr2 3988 . . . . . . . . . . . . . . . . . . . . 21 (((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) β†’ (((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
9593, 94syl 17 . . . . . . . . . . . . . . . . . . . 20 (((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) ∧ (π‘ƒβ€˜π‘›) ∈ ℝ) β†’ (((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
9695expimpd 452 . . . . . . . . . . . . . . . . . . 19 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›)) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
9796ralimdva 3165 . . . . . . . . . . . . . . . . . 18 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›)) β†’ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
9897imp 405 . . . . . . . . . . . . . . . . 17 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))) β†’ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›))
9924fveq2i 6893 . . . . . . . . . . . . . . . . . . . . . 22 (ballβ€˜π·) = (ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))
10099oveqi 7424 . . . . . . . . . . . . . . . . . . . . 21 ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) = ((π‘ƒβ€˜π‘›)(ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )))
101100sseq1i 4009 . . . . . . . . . . . . . . . . . . . 20 (((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›) ↔ ((π‘ƒβ€˜π‘›)(ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›))
102101ralbii 3091 . . . . . . . . . . . . . . . . . . 19 (βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›) ↔ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›))
103 nfv 1915 . . . . . . . . . . . . . . . . . . 19 β„²π‘‘βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)
104102, 103nfxfr 1853 . . . . . . . . . . . . . . . . . 18 β„²π‘‘βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)
105 oveq2 7419 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) = ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))))
106105sseq1d 4012 . . . . . . . . . . . . . . . . . . 19 (𝑑 = if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) β†’ (((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›) ↔ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
107106ralbidv 3175 . . . . . . . . . . . . . . . . . 18 (𝑑 = if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) β†’ (βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›) ↔ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)))
108104, 107rspce 3600 . . . . . . . . . . . . . . . . 17 ((if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < ))) βŠ† (π‘”β€˜π‘›)) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
10963, 98, 108syl2anc 582 . . . . . . . . . . . . . . . 16 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
110109exlimiv 1931 . . . . . . . . . . . . . . 15 (βˆƒβ„Ž(β„Ž:(1...𝑁)βŸΆβ„+ ∧ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›) ∈ ℝ ∧ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)) βŠ† (π‘”β€˜π‘›))) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
11137, 110syl 17 . . . . . . . . . . . . . 14 (βˆ€π‘› ∈ (1...𝑁)((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ (π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
11219, 111sylbir 234 . . . . . . . . . . . . 13 ((βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›)) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
11318, 112sylan2 591 . . . . . . . . . . . 12 ((βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
11416, 113sylanb 579 . . . . . . . . . . 11 ((βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)) β†’ βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›))
115 sstr2 3988 . . . . . . . . . . . . 13 (X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆 β†’ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
116 ss2ixp 8906 . . . . . . . . . . . . 13 (βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›) β†’ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›))
117115, 116syl11 33 . . . . . . . . . . . 12 (X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆 β†’ (βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›) β†’ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
118117reximdv 3168 . . . . . . . . . . 11 (X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆 β†’ (βˆƒπ‘‘ ∈ ℝ+ βˆ€π‘› ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† (π‘”β€˜π‘›) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
119114, 118syl5com 31 . . . . . . . . . 10 ((βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)) β†’ (X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆 β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
120119expimpd 452 . . . . . . . . 9 (βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) β†’ ((𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) ∧ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
121 eleq2 2820 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)))
122 sseq1 4006 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (𝑧 βŠ† 𝑆 ↔ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆))
123121, 122anbi12d 629 . . . . . . . . . 10 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) ↔ (𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) ∧ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆)))
124123imbi1d 340 . . . . . . . . 9 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆) ↔ ((𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) ∧ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)))
125120, 124syl5ibrcom 246 . . . . . . . 8 (βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) β†’ (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)))
1261253ad2ant2 1132 . . . . . . 7 ((𝑔 Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑧)(π‘”β€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) β†’ (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)))
127126imp 405 . . . . . 6 (((𝑔 Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑧)(π‘”β€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)) β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
128127exlimiv 1931 . . . . 5 (βˆƒπ‘”((𝑔 Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑧)(π‘”β€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ 𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)) β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
12912, 128sylbi 216 . . . 4 (𝑧 ∈ {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))} β†’ ((𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆))
130129rexlimiv 3146 . . 3 (βˆƒπ‘§ ∈ {π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))} (𝑃 ∈ 𝑧 ∧ 𝑧 βŠ† 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)
13111, 130syl 17 . 2 ((𝑆 ∈ (topGenβ€˜{π‘₯ ∣ βˆƒβ„Ž((β„Ž Fn (1...𝑁) ∧ βˆ€π‘› ∈ (1...𝑁)(β„Žβ€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ∧ βˆƒπ‘€ ∈ Fin βˆ€π‘› ∈ ((1...𝑁) βˆ– 𝑀)(β„Žβ€˜π‘›) = βˆͺ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›)) ∧ π‘₯ = X𝑛 ∈ (1...𝑁)(β„Žβ€˜π‘›))}) ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)
13210, 131sylanb 579 1 ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘‘ ∈ ℝ+ X𝑛 ∈ (1...𝑁)((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑑) βŠ† 𝑆)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  {cab 2707   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   Or wor 5586   Γ— cxp 5673  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Xcixp 8893  Fincfn 8941  infcinf 9438  β„cr 11111  0cc0 11112  1c1 11113  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„+crp 12978  (,)cioo 13328  ...cfz 13488  abscabs 15185  topGenctg 17387  βˆtcpt 17388  βˆžMetcxmet 21129  ballcbl 21131  MetOpencmopn 21134  Topctop 22615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-fz 13489  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-topgen 17393  df-pt 17394  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-top 22616  df-topon 22633  df-bases 22669
This theorem is referenced by:  poimirlem29  36820
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