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Theorem prdsxmslem2 22704
Description: Lemma for prdsxms 22705. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)
Hypotheses
Ref Expression
prdsxms.y 𝑌 = (𝑆Xs𝑅)
prdsxms.s (𝜑𝑆𝑊)
prdsxms.i (𝜑𝐼 ∈ Fin)
prdsxms.d 𝐷 = (dist‘𝑌)
prdsxms.b 𝐵 = (Base‘𝑌)
prdsxms.r (𝜑𝑅:𝐼⟶∞MetSp)
prdsxms.j 𝐽 = (TopOpen‘𝑌)
prdsxms.v 𝑉 = (Base‘(𝑅𝑘))
prdsxms.e 𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))
prdsxms.k 𝐾 = (TopOpen‘(𝑅𝑘))
prdsxms.c 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))}
Assertion
Ref Expression
prdsxmslem2 (𝜑𝐽 = (MetOpen‘𝐷))
Distinct variable groups:   𝑔,𝑘,𝐵   𝑥,𝑔,𝐷,𝑘   𝑧,𝑔,𝐼,𝑘,𝑥   𝑔,𝐸   𝑆,𝑔,𝑘,𝑥   𝑔,𝑊,𝑘,𝑥   𝑔,𝑌,𝑘,𝑥   𝜑,𝑔,𝑘,𝑥   𝑅,𝑔,𝑘,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑥,𝑧)   𝐶(𝑥,𝑧,𝑔,𝑘)   𝐷(𝑧)   𝑆(𝑧)   𝐸(𝑥,𝑧,𝑘)   𝐽(𝑥,𝑧,𝑔,𝑘)   𝐾(𝑥,𝑧,𝑔,𝑘)   𝑉(𝑥,𝑧,𝑔,𝑘)   𝑊(𝑧)   𝑌(𝑧)

Proof of Theorem prdsxmslem2
Dummy variables 𝑝 𝑟 𝑤 𝑦 𝑚 𝑢 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsxms.i . . . 4 (𝜑𝐼 ∈ Fin)
2 topnfn 16439 . . . . 5 TopOpen Fn V
3 prdsxms.r . . . . . . 7 (𝜑𝑅:𝐼⟶∞MetSp)
43ffnd 6279 . . . . . 6 (𝜑𝑅 Fn 𝐼)
5 dffn2 6280 . . . . . 6 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
64, 5sylib 210 . . . . 5 (𝜑𝑅:𝐼⟶V)
7 fnfco 6306 . . . . 5 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
82, 6, 7sylancr 581 . . . 4 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
9 prdsxms.c . . . . 5 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))}
109ptval 21744 . . . 4 ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
111, 8, 10syl2anc 579 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
12 eldifsn 4536 . . . . . . . 8 (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) ↔ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅))
13 prdsxms.y . . . . . . . . . . . 12 𝑌 = (𝑆Xs𝑅)
14 prdsxms.s . . . . . . . . . . . 12 (𝜑𝑆𝑊)
15 prdsxms.d . . . . . . . . . . . 12 𝐷 = (dist‘𝑌)
16 prdsxms.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑌)
1713, 14, 1, 15, 16, 3prdsxmslem1 22703 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝐵))
18 blrn 22584 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝𝐵𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
1917, 18syl 17 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝𝐵𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
2017adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
21 simprl 787 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑝𝐵)
22 simprr 789 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
23 xbln0 22589 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2420, 21, 22, 23syl3anc 1494 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2513ad2ant1 1167 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
26 mptexg 6740 . . . . . . . . . . . . . . . . . 18 (𝐼 ∈ Fin → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
2725, 26syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
28 ovex 6937 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
2928rgenw 3133 . . . . . . . . . . . . . . . . . 18 𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
30 eqid 2825 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))
3130fnmpt 6253 . . . . . . . . . . . . . . . . . 18 (∀𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3229, 31mp1i 13 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3333ad2ant1 1167 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
3433ffvelrnda 6608 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
35 prdsxms.v . . . . . . . . . . . . . . . . . . . . . 22 𝑉 = (Base‘(𝑅𝑘))
36 prdsxms.e . . . . . . . . . . . . . . . . . . . . . 22 𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))
3735, 36xmsxmet 22631 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
3834, 37syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
39 eqid 2825 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
40 eqid 2825 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
41143ad2ant1 1167 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆𝑊)
4234ralrimiva 3175 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑅𝑘) ∈ ∞MetSp)
43 simp2l 1260 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝𝐵)
4433feqmptd 6496 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘𝐼 ↦ (𝑅𝑘)))
4544oveq2d 6921 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4613, 45syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4746fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4816, 47syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4943, 48eleqtrd 2908 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
5039, 40, 41, 25, 42, 35, 49prdsbascl 16496 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑝𝑘) ∈ 𝑉)
5150r19.21bi 3141 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑝𝑘) ∈ 𝑉)
52 simp2r 1261 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
5352adantr 474 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝑟 ∈ ℝ*)
54 eqid 2825 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐸) = (MetOpen‘𝐸)
5554blopn 22675 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝𝑘) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
5638, 51, 53, 55syl3anc 1494 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
57 2fveq3 6438 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → (dist‘(𝑅𝑛)) = (dist‘(𝑅𝑘)))
58 2fveq3 6438 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = (Base‘(𝑅𝑘)))
5958, 35syl6eqr 2879 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = 𝑉)
6059sqxpeqd 5374 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))) = (𝑉 × 𝑉))
6157, 60reseq12d 5630 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉)))
6261, 36syl6eqr 2879 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = 𝐸)
6362fveq2d 6437 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))))) = (ball‘𝐸))
64 fveq2 6433 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝑝𝑛) = (𝑝𝑘))
65 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘𝑟 = 𝑟)
6663, 64, 65oveq123d 6926 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) = ((𝑝𝑘)(ball‘𝐸)𝑟))
67 ovex 6937 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ V
6866, 30, 67fvmpt 6529 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐼 → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
6968adantl 475 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
70 fvco3 6522 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
71 prdsxms.k . . . . . . . . . . . . . . . . . . . . . 22 𝐾 = (TopOpen‘(𝑅𝑘))
7270, 71syl6eqr 2879 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7333, 72sylan 575 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7471, 35, 36xmstopn 22626 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
7534, 74syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
7673, 75eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
7756, 69, 763eltr4d 2921 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7877ralrimiva 3175 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7933feqmptd 6496 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛𝐼 ↦ (𝑅𝑛)))
8079oveq2d 6921 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8113, 80syl5eq 2873 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8281fveq2d 6437 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8315, 82syl5eq 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8483fveq2d 6437 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))))
8584oveqd 6922 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟))
86 fveq2 6433 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑅𝑛) = (𝑅𝑘))
8786cbvmptv 4973 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐼 ↦ (𝑅𝑛)) = (𝑘𝐼 ↦ (𝑅𝑘))
8887oveq2i 6916 . . . . . . . . . . . . . . . . . . 19 (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
89 eqid 2825 . . . . . . . . . . . . . . . . . . 19 (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
90 eqid 2825 . . . . . . . . . . . . . . . . . . 19 (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
9181fveq2d 6437 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9216, 91syl5eq 2873 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9343, 92eleqtrd 2908 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
94 simp3 1172 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
9588, 89, 35, 36, 90, 41, 25, 34, 38, 93, 52, 94prdsbl 22666 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
9685, 95eqtrd 2861 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
97 fneq1 6212 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (𝑔 Fn 𝐼 ↔ (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼))
98 fveq1 6432 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (𝑔𝑘) = ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘))
9998eleq1d 2891 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → ((𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))
10099ralbidv 3195 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))
10197, 100anbi12d 624 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))))
10298, 68sylan9eq 2881 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∧ 𝑘𝐼) → (𝑔𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
103102ixpeq2dva 8190 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → X𝑘𝐼 (𝑔𝑘) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
104103eqeq2d 2835 . . . . . . . . . . . . . . . . . . . 20 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → ((𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)))
105101, 104anbi12d 624 . . . . . . . . . . . . . . . . . . 19 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)) ↔ (((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))))
106105spcegv 3511 . . . . . . . . . . . . . . . . . 18 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V → ((((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
1071063impib 1148 . . . . . . . . . . . . . . . . 17 (((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V ∧ ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
10827, 32, 78, 96, 107syl121anc 1498 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
1091083expia 1154 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → (0 < 𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
11024, 109sylbid 232 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
111110adantr 474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
112 simpr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
113112neeq1d 3058 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
114 ral0 4298 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)
115 difeq2 3949 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝐼 → (𝐼𝑧) = (𝐼𝐼))
116 difid 4178 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝐼) = ∅
117115, 116syl6eq 2877 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝐼 → (𝐼𝑧) = ∅)
118117raleqdv 3356 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)))
119118rspcev 3526 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
1201, 114, 119sylancl 580 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
121120adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
122121biantrud 527 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))))
123 df-3an 1113 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)))
124122, 123syl6rbbr 282 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))))
125 eqeq1 2829 . . . . . . . . . . . . . . 15 (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘𝐼 (𝑔𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
126124, 125bi2anan9 629 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
127126exbidv 2020 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
128111, 113, 1273imtr4d 286 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))))
129128ex 403 . . . . . . . . . . 11 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘)))))
130129rexlimdvva 3248 . . . . . . . . . 10 (𝜑 → (∃𝑝𝐵𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘)))))
13119, 130sylbid 232 . . . . . . . . 9 (𝜑 → (𝑥 ∈ ran (ball‘𝐷) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘)))))
132131impd 400 . . . . . . . 8 (𝜑 → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))))
13312, 132syl5bi 234 . . . . . . 7 (𝜑 → (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))))
134133alrimiv 2026 . . . . . 6 (𝜑 → ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))))
135 ssab 3897 . . . . . 6 ((ran (ball‘𝐷) ∖ {∅}) ⊆ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))} ↔ ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))))
136134, 135sylibr 226 . . . . 5 (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘𝐼 (𝑔𝑘))})
137136, 9syl6sseqr 3877 . . . 4 (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
138 ssv 3850 . . . . . . . . . 10 ∞MetSp ⊆ V
139 fnssres 6237 . . . . . . . . . 10 ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
1402, 138, 139mp2an 683 . . . . . . . . 9 (TopOpen ↾ ∞MetSp) Fn ∞MetSp
141 fvres 6452 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
142 xmstps 22628 . . . . . . . . . . . 12 (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
143 eqid 2825 . . . . . . . . . . . . 13 (TopOpen‘𝑥) = (TopOpen‘𝑥)
144143tpstop 21112 . . . . . . . . . . . 12 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
145142, 144syl 17 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
146141, 145eqeltrd 2906 . . . . . . . . . 10 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
147146rgen 3131 . . . . . . . . 9 𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
148 ffnfv 6637 . . . . . . . . 9 ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
149140, 147, 148mpbir2an 702 . . . . . . . 8 (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
150 fco2 6296 . . . . . . . 8 (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
151149, 3, 150sylancr 581 . . . . . . 7 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
152 eqid 2825 . . . . . . . 8 X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
1539, 152ptbasfi 21755 . . . . . . 7 ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅):𝐼⟶Top) → 𝐶 = (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))))
1541, 151, 153syl2anc 579 . . . . . 6 (𝜑𝐶 = (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))))
155 eqid 2825 . . . . . . . . 9 (MetOpen‘𝐷) = (MetOpen‘𝐷)
156155mopntop 22615 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
15717, 156syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ∈ Top)
15813, 16, 14, 1, 4prdsbas2 16482 . . . . . . . . . . . 12 (𝜑𝐵 = X𝑘𝐼 (Base‘(𝑅𝑘)))
1593, 72sylan 575 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
1603ffvelrnda 6608 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
161 xmstps 22628 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑘) ∈ ∞MetSp → (𝑅𝑘) ∈ TopSp)
162160, 161syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopSp)
16335, 71istps 21109 . . . . . . . . . . . . . . . . 17 ((𝑅𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
164162, 163sylib 210 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → 𝐾 ∈ (TopOn‘𝑉))
165159, 164eqeltrd 2906 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
166 toponuni 21089 . . . . . . . . . . . . . . 15 (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
167165, 166syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
16835, 167syl5eqr 2875 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (Base‘(𝑅𝑘)) = ((TopOpen ∘ 𝑅)‘𝑘))
169168ixpeq2dva 8190 . . . . . . . . . . . 12 (𝜑X𝑘𝐼 (Base‘(𝑅𝑘)) = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
170158, 169eqtrd 2861 . . . . . . . . . . 11 (𝜑𝐵 = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
171 fveq2 6433 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
172171unieqd 4668 . . . . . . . . . . . 12 (𝑘 = 𝑛 ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
173172cbvixpv 8193 . . . . . . . . . . 11 X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
174170, 173syl6eq 2877 . . . . . . . . . 10 (𝜑𝐵 = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛))
175155mopntopon 22614 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
17617, 175syl 17 . . . . . . . . . . 11 (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
177 toponmax 21101 . . . . . . . . . . 11 ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
178176, 177syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ (MetOpen‘𝐷))
179174, 178eqeltrrd 2907 . . . . . . . . 9 (𝜑X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
180179snssd 4558 . . . . . . . 8 (𝜑 → {X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
181174mpteq1d 4961 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
182181ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
183182cnveqd 5530 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
184183imaeq1d 5706 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢))
185 fveq1 6432 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑝 → (𝑤𝑘) = (𝑝𝑘))
186185eleq1d 2891 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑝 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑝𝑘) ∈ 𝑢))
187 eqid 2825 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤𝐵 ↦ (𝑤𝑘))
188187mptpreima 5869 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢}
189186, 188elrab2 3589 . . . . . . . . . . . . . . . . . 18 (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))
190160, 37syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
191190adantr 474 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
192 simprl 787 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢𝐾)
193160, 74syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
194193adantr 474 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
195192, 194eleqtrd 2908 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
196 simprrr 800 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → (𝑝𝑘) ∈ 𝑢)
19754mopni2 22668 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
198191, 195, 196, 197syl3anc 1494 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
19917ad3antrrr 721 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
200 simprrl 799 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑝𝐵)
201200adantr 474 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝𝐵)
202 rpxr 12123 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
203202ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
204155blopn 22675 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
205199, 201, 203, 204syl3anc 1494 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
206 simprl 787 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
207 blcntr 22588 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
208199, 201, 206, 207syl3anc 1494 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
209 blssm 22593 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
210199, 201, 203, 209syl3anc 1494 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
211 simplrr 796 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
212 simplll 791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
213 rpgt0 12126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑟 ∈ ℝ+ → 0 < 𝑟)
214213ad2antrl 719 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
215212, 201, 203, 214, 96syl121anc 1498 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
216215eleq2d 2892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)))
217216biimpa 470 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
218 vex 3417 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑤 ∈ V
219218elixp 8182 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
220219simprbi 492 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) → ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
221217, 220syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
222 simp-4r 803 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘𝐼)
223 rsp 3138 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟) → (𝑘𝐼 → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
224221, 222, 223sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
225211, 224sseldd 3828 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ 𝑢)
226210, 225ssrabdv 3906 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢})
227226, 188syl6sseqr 3877 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))
228 eleq2 2895 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝𝑦𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
229 sseq1 3851 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
230228, 229anbi12d 624 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑝(ball‘𝐷)𝑟) → ((𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)) ↔ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
231230rspcev 3526 . . . . . . . . . . . . . . . . . . . . 21 (((𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷) ∧ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
232205, 208, 227, 231syl12anc 870 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
233198, 232rexlimddv 3245 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
234233expr 450 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
235189, 234syl5bi 234 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
236235ralrimiv 3174 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
237157ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (MetOpen‘𝐷) ∈ Top)
238 eltop2 21150 . . . . . . . . . . . . . . . . 17 ((MetOpen‘𝐷) ∈ Top → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
239237, 238syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
240236, 239mpbird 249 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
241184, 240eqeltrrd 2907 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242241ralrimiva 3175 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
243159raleqdv 3356 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)))
244242, 243mpbird 249 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
245244ralrimiva 3175 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
246 fveq2 6433 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
247 fveq2 6433 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑤𝑘) = (𝑤𝑚))
248247mpteq2dv 4968 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
249248cnveqd 5530 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚(𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
250249imaeq1d 5706 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
251250eleq1d 2891 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
252246, 251raleqbidv 3364 . . . . . . . . . . . 12 (𝑘 = 𝑚 → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
253252cbvralv 3383 . . . . . . . . . . 11 (∀𝑘𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
254245, 253sylib 210 . . . . . . . . . 10 (𝜑 → ∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
255 eqid 2825 . . . . . . . . . . 11 (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) = (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
256255fmpt2x 7499 . . . . . . . . . 10 (∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
257254, 256sylib 210 . . . . . . . . 9 (𝜑 → (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
258257frnd 6285 . . . . . . . 8 (𝜑 → ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
259180, 258unssd 4016 . . . . . . 7 (𝜑 → ({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷))
260 fiss 8599 . . . . . . 7 (((MetOpen‘𝐷) ∈ Top ∧ ({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷)) → (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
261157, 259, 260syl2anc 579 . . . . . 6 (𝜑 → (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
262154, 261eqsstrd 3864 . . . . 5 (𝜑𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
263 fitop 21075 . . . . . . 7 ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
264157, 263syl 17 . . . . . 6 (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
265155mopnval 22613 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
26617, 265syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
267 tgdif0 21167 . . . . . . 7 (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
268266, 267syl6eqr 2879 . . . . . 6 (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
269264, 268eqtrd 2861 . . . . 5 (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
270262, 269sseqtrd 3866 . . . 4 (𝜑𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
271 2basgen 21165 . . . 4 (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
272137, 270, 271syl2anc 579 . . 3 (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
27311, 272eqtr4d 2864 . 2 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
274 prdsxms.j . . 3 𝐽 = (TopOpen‘𝑌)
27513, 14, 1, 4, 274prdstopn 21802 . 2 (𝜑𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
276273, 275, 2683eqtr4d 2871 1 (𝜑𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111  wal 1654   = wceq 1656  wex 1878  wcel 2164  {cab 2811  wne 2999  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  cdif 3795  cun 3796  wss 3798  c0 4144  {csn 4397   cuni 4658   ciun 4740   class class class wbr 4873  cmpt 4952   × cxp 5340  ccnv 5341  ran crn 5343  cres 5344  cima 5345  ccom 5346   Fn wfn 6118  wf 6119  cfv 6123  (class class class)co 6905  cmpt2 6907  Xcixp 8175  Fincfn 8222  ficfi 8585  0cc0 10252  *cxr 10390   < clt 10391  +crp 12112  Basecbs 16222  distcds 16314  TopOpenctopn 16435  topGenctg 16451  tcpt 16452  Xscprds 16459  ∞Metcxmet 20091  ballcbl 20093  MetOpencmopn 20096  Topctop 21068  TopOnctopon 21085  TopSpctps 21107  ∞MetSpcxms 22492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-sup 8617  df-inf 8618  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-q 12072  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-icc 12470  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-hom 16329  df-cco 16330  df-rest 16436  df-topn 16437  df-topgen 16457  df-pt 16458  df-prds 16461  df-psmet 20098  df-xmet 20099  df-bl 20101  df-mopn 20102  df-top 21069  df-topon 21086  df-topsp 21108  df-bases 21121  df-xms 22495
This theorem is referenced by:  prdsxms  22705
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