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Theorem ptrecube 36488
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 13937 . . . . 5 (1...𝑁) ∈ Fin
3 retop 24278 . . . . . 6 (topGenβ€˜ran (,)) ∈ Top
4 fnconstg 6780 . . . . . 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 2733 . . . . . 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 23074 . . . . 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 691 . . . 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 2761 . . 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 2826 . 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 22468 . . 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 23076 . . . . 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 6905 . . . . . . . . . . . . . . 15 (topGenβ€˜ran (,)) ∈ V
1413fvconst2 7205 . . . . . . . . . . . . . 14 (𝑛 ∈ (1...𝑁) β†’ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) = (topGenβ€˜ran (,)))
1514eleq2d 2820 . . . . . . . . . . . . 13 (𝑛 ∈ (1...𝑁) β†’ ((π‘”β€˜π‘›) ∈ (((1...𝑁) Γ— {(topGenβ€˜ran (,))})β€˜π‘›) ↔ (π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,))))
1615ralbiia 3092 . . . . . . . . . . . 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 8895 . . . . . . . . . . . . . 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 1148 . . . . . . . . . . . . 13 (𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ βˆ€π‘› ∈ (1...𝑁)(π‘ƒβ€˜π‘›) ∈ (π‘”β€˜π‘›))
19 r19.26 3112 . . . . . . . . . . . . . 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 24279 . . . . . . . . . . . . . . . . . . . . 21 ℝ = βˆͺ (topGenβ€˜ran (,))
2120eltopss 22409 . . . . . . . . . . . . . . . . . . . 20 (((topGenβ€˜ran (,)) ∈ Top ∧ (π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,))) β†’ (π‘”β€˜π‘›) βŠ† ℝ)
223, 21mpan 689 . . . . . . . . . . . . . . . . . . 19 ((π‘”β€˜π‘›) ∈ (topGenβ€˜ran (,)) β†’ (π‘”β€˜π‘›) βŠ† ℝ)
2322sselda 3983 . . . . . . . . . . . . . . . . . 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 24307 . . . . . . . . . . . . . . . . . . 19 𝐷 ∈ (∞Metβ€˜β„)
26 eqid 2733 . . . . . . . . . . . . . . . . . . . . 21 (MetOpenβ€˜π·) = (MetOpenβ€˜π·)
2724, 26tgioo 24312 . . . . . . . . . . . . . . . . . . . 20 (topGenβ€˜ran (,)) = (MetOpenβ€˜π·)
2827mopni2 24002 . . . . . . . . . . . . . . . . . . 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 1449 . . . . . . . . . . . . . . . . . 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 3191 . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . 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 3084 . . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (β„Žβ€˜π‘›) β†’ ((π‘ƒβ€˜π‘›)(ballβ€˜π·)𝑦) = ((π‘ƒβ€˜π‘›)(ballβ€˜π·)(β„Žβ€˜π‘›)))
3433sseq1d 4014 . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . 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 9287 . . . . . . . . . . . . . . . 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 588 . . . . . . . . . . . . . . 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 12978 . . . . . . . . . . . . . . . . . . . 20 1 ∈ ℝ+
3938a1i 11 . . . . . . . . . . . . . . . . . . 19 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ (1...𝑁) = βˆ…) β†’ 1 ∈ ℝ+)
40 frn 6725 . . . . . . . . . . . . . . . . . . . . 21 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž βŠ† ℝ+)
4140adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž βŠ† ℝ+)
42 ffn 6718 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ β„Ž Fn (1...𝑁))
43 fnfi 9181 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž Fn (1...𝑁) ∧ (1...𝑁) ∈ Fin) β†’ β„Ž ∈ Fin)
4442, 2, 43sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ β„Ž ∈ Fin)
45 rnfi 9335 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž ∈ Fin β†’ ran β„Ž ∈ Fin)
4644, 45syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž ∈ Fin)
4746adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž ∈ Fin)
48 dm0rn0 5925 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom β„Ž = βˆ… ↔ ran β„Ž = βˆ…)
49 fdm 6727 . . . . . . . . . . . . . . . . . . . . . . . . 25 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ dom β„Ž = (1...𝑁))
5049eqeq1d 2735 . . . . . . . . . . . . . . . . . . . . . . . 24 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (dom β„Ž = βˆ… ↔ (1...𝑁) = βˆ…))
5148, 50bitr3id 285 . . . . . . . . . . . . . . . . . . . . . . 23 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (ran β„Ž = βˆ… ↔ (1...𝑁) = βˆ…))
5251necon3abid 2978 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ (ran β„Ž β‰  βˆ… ↔ Β¬ (1...𝑁) = βˆ…))
5352biimpar 479 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž β‰  βˆ…)
54 rpssre 12981 . . . . . . . . . . . . . . . . . . . . . . 23 ℝ+ βŠ† ℝ
5540, 54sstrdi 3995 . . . . . . . . . . . . . . . . . . . . . 22 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ ran β„Ž βŠ† ℝ)
5655adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ ran β„Ž βŠ† ℝ)
57 ltso 11294 . . . . . . . . . . . . . . . . . . . . . 22 < Or ℝ
58 fiinfcl 9496 . . . . . . . . . . . . . . . . . . . . . 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 689 . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . 20 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ inf(ran β„Ž, ℝ, < ) ∈ ran β„Ž)
6141, 60sseldd 3984 . . . . . . . . . . . . . . . . . . 19 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ Β¬ (1...𝑁) = βˆ…) β†’ inf(ran β„Ž, ℝ, < ) ∈ ℝ+)
6239, 61ifclda 4564 . . . . . . . . . . . . . . . . . 18 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+)
6362adantr 482 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ+)
6564rpxrd 13017 . . . . . . . . . . . . . . . . . . . . . . 23 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ if((1...𝑁) = βˆ…, 1, inf(ran β„Ž, ℝ, < )) ∈ ℝ*)
66 ffvelcdm 7084 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ℝ+)
6766rpxrd 13017 . . . . . . . . . . . . . . . . . . . . . . 23 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ℝ*)
68 ne0i 4335 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (1...𝑁) β†’ (1...𝑁) β‰  βˆ…)
69 ifnefalse 4541 . . . . . . . . . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ ran β„Ž βŠ† ℝ)
73 0re 11216 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 0 ∈ ℝ
74 rpge0 12987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ℝ+ β†’ 0 ≀ 𝑦)
7574rgen 3064 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 βˆ€π‘¦ ∈ ℝ+ 0 ≀ 𝑦
76 ssralv 4051 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (π‘₯ = 0 β†’ (π‘₯ ≀ 𝑦 ↔ 0 ≀ 𝑦))
7978ralbidv 3178 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘₯ = 0 β†’ (βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦 ↔ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦))
8079rspcev 3613 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ βˆ€π‘¦ ∈ ran β„Ž0 ≀ 𝑦) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
8173, 77, 80sylancr 588 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (β„Ž:(1...𝑁)βŸΆβ„+ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
8281adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran β„Ž π‘₯ ≀ 𝑦)
83 fnfvelrn 7083 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((β„Ž Fn (1...𝑁) ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ran β„Ž)
8442, 83sylan 581 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ (β„Žβ€˜π‘›) ∈ ran β„Ž)
85 infrelb 12199 . . . . . . . . . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . . . . . . . . . 24 ((β„Ž:(1...𝑁)βŸΆβ„+ ∧ 𝑛 ∈ (1...𝑁)) β†’ inf(ran β„Ž, ℝ, < ) ≀ (β„Žβ€˜π‘›))
8771, 86eqbrtrd 5171 . . . . . . . . . . . . . . . . . . . . . . 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 516 . . . . . . . . . . . . . . . . . . . . . 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 23929 . . . . . . . . . . . . . . . . . . . . . . . . 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 1121 . . . . . . . . . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . . . . . . . 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 3990 . . . . . . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . . . . . . 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 3168 . . . . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . . . . . . . 22 (ballβ€˜π·) = (ballβ€˜((abs ∘ βˆ’ ) β†Ύ (ℝ Γ— ℝ)))
10099oveqi 7422 . . . . . . . . . . . . . . . . . . . . 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 4011 . . . . . . . . . . . . . . . . . . . 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 3094 . . . . . . . . . . . . . . . . . . 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 1918 . . . . . . . . . . . . . . . . . . 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 1856 . . . . . . . . . . . . . . . . . 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 7417 . . . . . . . . . . . . . . . . . . . 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 4014 . . . . . . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . . . . . . 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 3602 . . . . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . . . . 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 1934 . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . 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 582 . . . . . . . . . . 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 3990 . . . . . . . . . . . . 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 8904 . . . . . . . . . . . . 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 3171 . . . . . . . . . . 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 455 . . . . . . . . 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 2823 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›)))
122 sseq1 4008 . . . . . . . . . . 11 (𝑧 = X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) β†’ (𝑧 βŠ† 𝑆 ↔ X𝑛 ∈ (1...𝑁)(π‘”β€˜π‘›) βŠ† 𝑆))
123121, 122anbi12d 632 . . . . . . . . . 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 342 . . . . . . . . 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 1135 . . . . . . 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 408 . . . . . 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 1934 . . . . 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 3149 . . 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 582 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   βˆ– cdif 3946   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {csn 4629  βˆͺ cuni 4909   class class class wbr 5149   Or wor 5588   Γ— cxp 5675  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   ∘ ccom 5681   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Xcixp 8891  Fincfn 8939  infcinf 9436  β„cr 11109  0cc0 11110  1c1 11111  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„+crp 12974  (,)cioo 13324  ...cfz 13484  abscabs 15181  topGenctg 17383  βˆtcpt 17384  βˆžMetcxmet 20929  ballcbl 20931  MetOpencmopn 20934  Topctop 22395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-fz 13485  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-topgen 17389  df-pt 17390  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449
This theorem is referenced by:  poimirlem29  36517
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