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Theorem prdsxmslem2 22257
Description: Lemma for prdsxms 22258. 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 16018 . . . . 5 TopOpen Fn V
3 prdsxms.r . . . . . . 7 (𝜑𝑅:𝐼⟶∞MetSp)
4 ffn 6007 . . . . . . 7 (𝑅:𝐼⟶∞MetSp → 𝑅 Fn 𝐼)
53, 4syl 17 . . . . . 6 (𝜑𝑅 Fn 𝐼)
6 dffn2 6009 . . . . . 6 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
75, 6sylib 208 . . . . 5 (𝜑𝑅:𝐼⟶V)
8 fnfco 6031 . . . . 5 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
92, 7, 8sylancr 694 . . . 4 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
10 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))}
1110ptval 21296 . . . 4 ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
121, 9, 11syl2anc 692 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
13 eldifsn 4292 . . . . . . . 8 (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) ↔ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅))
14 prdsxms.y . . . . . . . . . . . 12 𝑌 = (𝑆Xs𝑅)
15 prdsxms.s . . . . . . . . . . . 12 (𝜑𝑆𝑊)
16 prdsxms.d . . . . . . . . . . . 12 𝐷 = (dist‘𝑌)
17 prdsxms.b . . . . . . . . . . . 12 𝐵 = (Base‘𝑌)
1814, 15, 1, 16, 17, 3prdsxmslem1 22256 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝐵))
19 blrn 22137 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝𝐵𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
2018, 19syl 17 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝𝐵𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
2118adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
22 simprl 793 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑝𝐵)
23 simprr 795 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
24 xbln0 22142 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2521, 22, 23, 24syl3anc 1323 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2613ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
27 mptexg 6444 . . . . . . . . . . . . . . . . . 18 (𝐼 ∈ Fin → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
29 ovex 6638 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
3029rgenw 2919 . . . . . . . . . . . . . . . . . 18 𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
31 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))
3231fnmpt 5982 . . . . . . . . . . . . . . . . . 18 (∀𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3330, 32mp1i 13 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3433ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
3534ffvelrnda 6320 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
36 prdsxms.v . . . . . . . . . . . . . . . . . . . . . 22 𝑉 = (Base‘(𝑅𝑘))
37 prdsxms.e . . . . . . . . . . . . . . . . . . . . . 22 𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))
3836, 37xmsxmet 22184 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
3935, 38syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
40 eqid 2621 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
41 eqid 2621 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
42153ad2ant1 1080 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆𝑊)
4335ralrimiva 2961 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑅𝑘) ∈ ∞MetSp)
44 simp2l 1085 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝𝐵)
4534feqmptd 6211 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘𝐼 ↦ (𝑅𝑘)))
4645oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4714, 46syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4847fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4917, 48syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
5044, 49eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
5140, 41, 42, 26, 43, 36, 50prdsbascl 16075 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑝𝑘) ∈ 𝑉)
5251r19.21bi 2927 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑝𝑘) ∈ 𝑉)
53 simp2r 1086 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
5453adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝑟 ∈ ℝ*)
55 eqid 2621 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐸) = (MetOpen‘𝐸)
5655blopn 22228 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝𝑘) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
5739, 52, 54, 56syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
58 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (𝑅𝑛) = (𝑅𝑘))
5958fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → (dist‘(𝑅𝑛)) = (dist‘(𝑅𝑘)))
6058fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = (Base‘(𝑅𝑘)))
6160, 36syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = 𝑉)
6261sqxpeqd 5106 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))) = (𝑉 × 𝑉))
6359, 62reseq12d 5362 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉)))
6463, 37syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = 𝐸)
6564fveq2d 6157 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))))) = (ball‘𝐸))
66 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝑝𝑛) = (𝑝𝑘))
67 eqidd 2622 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘𝑟 = 𝑟)
6865, 66, 67oveq123d 6631 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) = ((𝑝𝑘)(ball‘𝐸)𝑟))
69 ovex 6638 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ V
7068, 31, 69fvmpt 6244 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐼 → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
7170adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
72 fvco3 6237 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
73 prdsxms.k . . . . . . . . . . . . . . . . . . . . . 22 𝐾 = (TopOpen‘(𝑅𝑘))
7472, 73syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7534, 74sylan 488 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7673, 36, 37xmstopn 22179 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
7735, 76syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
7875, 77eqtrd 2655 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
7957, 71, 783eltr4d 2713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
8079ralrimiva 2961 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
8134feqmptd 6211 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛𝐼 ↦ (𝑅𝑛)))
8281oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8314, 82syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8483fveq2d 6157 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8516, 84syl5eq 2667 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8685fveq2d 6157 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))))
8786oveqd 6627 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟))
8858cbvmptv 4715 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐼 ↦ (𝑅𝑛)) = (𝑘𝐼 ↦ (𝑅𝑘))
8988oveq2i 6621 . . . . . . . . . . . . . . . . . . 19 (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
90 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
91 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
9283fveq2d 6157 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9317, 92syl5eq 2667 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9444, 93eleqtrd 2700 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
95 simp3 1061 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
9689, 90, 36, 37, 91, 42, 26, 35, 39, 94, 53, 95prdsbl 22219 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
9787, 96eqtrd 2655 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
98 fneq1 5942 . . . . . . . . . . . . . . . . . . . . 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))
99 fveq1 6152 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (𝑔𝑘) = ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘))
10099eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . 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)))
101100ralbidv 2981 . . . . . . . . . . . . . . . . . . . . 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)))
10298, 101anbi12d 746 . . . . . . . . . . . . . . . . . . . 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))))
10399, 70sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∧ 𝑘𝐼) → (𝑔𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
104103ixpeq2dva 7875 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → X𝑘𝐼 (𝑔𝑘) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
105104eqeq2d 2631 . . . . . . . . . . . . . . . . . . . 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)))
106102, 105anbi12d 746 . . . . . . . . . . . . . . . . . . 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))))
107106spcegv 3283 . . . . . . . . . . . . . . . . . 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))))
1081073impib 1259 . . . . . . . . . . . . . . . . 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)))
10928, 33, 80, 97, 108syl121anc 1328 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
1101093expia 1264 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → (0 < 𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
11125, 110sylbid 230 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
112111adantr 481 . . . . . . . . . . . . 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))))
113 simpr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
114113neeq1d 2849 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
115 ral0 4053 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)
116 difeq2 3705 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝐼 → (𝐼𝑧) = (𝐼𝐼))
117 difid 3927 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝐼) = ∅
118116, 117syl6eq 2671 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝐼 → (𝐼𝑧) = ∅)
119118raleqdv 3136 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)))
120119rspcev 3298 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
1211, 115, 120sylancl 693 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
122121adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
123122biantrud 528 . . . . . . . . . . . . . . . 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))))
124 df-3an 1038 . . . . . . . . . . . . . . . 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)))
125123, 124syl6rbbr 279 . . . . . . . . . . . . . . 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))))
126 eqeq1 2625 . . . . . . . . . . . . . . 15 (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘𝐼 (𝑔𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
127125, 126bi2anan9 916 . . . . . . . . . . . . . 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))))
128127exbidv 1847 . . . . . . . . . . . . 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))))
129112, 114, 1283imtr4d 283 . . . . . . . . . . . 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))))
130129ex 450 . . . . . . . . . . 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)))))
131130rexlimdvva 3032 . . . . . . . . . 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)))))
13220, 131sylbid 230 . . . . . . . . 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)))))
133132impd 447 . . . . . . . 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))))
13413, 133syl5bi 232 . . . . . . 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))))
135134alrimiv 1852 . . . . . 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))))
136 ssab 3656 . . . . . 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))))
137135, 136sylibr 224 . . . . 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))})
138137, 10syl6sseqr 3636 . . . 4 (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
139 ssv 3609 . . . . . . . . . 10 ∞MetSp ⊆ V
140 fnssres 5967 . . . . . . . . . 10 ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
1412, 139, 140mp2an 707 . . . . . . . . 9 (TopOpen ↾ ∞MetSp) Fn ∞MetSp
142 fvres 6169 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
143 xmstps 22181 . . . . . . . . . . . 12 (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
144 eqid 2621 . . . . . . . . . . . . 13 (TopOpen‘𝑥) = (TopOpen‘𝑥)
145144tpstop 20665 . . . . . . . . . . . 12 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
146143, 145syl 17 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
147142, 146eqeltrd 2698 . . . . . . . . . 10 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
148147rgen 2917 . . . . . . . . 9 𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
149 ffnfv 6349 . . . . . . . . 9 ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
150141, 148, 149mpbir2an 954 . . . . . . . 8 (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
151 fco2 6021 . . . . . . . 8 (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
152150, 3, 151sylancr 694 . . . . . . 7 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
153 eqid 2621 . . . . . . . 8 X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
15410, 153ptbasfi 21307 . . . . . . 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)))))
1551, 152, 154syl2anc 692 . . . . . 6 (𝜑𝐶 = (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))))
156 eqid 2621 . . . . . . . . 9 (MetOpen‘𝐷) = (MetOpen‘𝐷)
157156mopntop 22168 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
15818, 157syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ∈ Top)
15914, 17, 15, 1, 5prdsbas2 16061 . . . . . . . . . . . 12 (𝜑𝐵 = X𝑘𝐼 (Base‘(𝑅𝑘)))
1603, 74sylan 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
1613ffvelrnda 6320 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
162 xmstps 22181 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑘) ∈ ∞MetSp → (𝑅𝑘) ∈ TopSp)
163161, 162syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopSp)
16436, 73istps 20662 . . . . . . . . . . . . . . . . 17 ((𝑅𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
165163, 164sylib 208 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → 𝐾 ∈ (TopOn‘𝑉))
166160, 165eqeltrd 2698 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
167 toponuni 20651 . . . . . . . . . . . . . . 15 (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
168166, 167syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
16936, 168syl5eqr 2669 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (Base‘(𝑅𝑘)) = ((TopOpen ∘ 𝑅)‘𝑘))
170169ixpeq2dva 7875 . . . . . . . . . . . 12 (𝜑X𝑘𝐼 (Base‘(𝑅𝑘)) = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
171159, 170eqtrd 2655 . . . . . . . . . . 11 (𝜑𝐵 = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
172 fveq2 6153 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
173172unieqd 4417 . . . . . . . . . . . 12 (𝑘 = 𝑛 ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
174173cbvixpv 7878 . . . . . . . . . . 11 X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
175171, 174syl6eq 2671 . . . . . . . . . 10 (𝜑𝐵 = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛))
176156mopntopon 22167 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
17718, 176syl 17 . . . . . . . . . . 11 (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
178 toponmax 20652 . . . . . . . . . . 11 ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
179177, 178syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ (MetOpen‘𝐷))
180175, 179eqeltrrd 2699 . . . . . . . . 9 (𝜑X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
181180snssd 4314 . . . . . . . 8 (𝜑 → {X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
182175mpteq1d 4703 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
183182ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
184183cnveqd 5263 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
185184imaeq1d 5429 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢))
186 fveq1 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑝 → (𝑤𝑘) = (𝑝𝑘))
187186eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑝 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑝𝑘) ∈ 𝑢))
188 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤𝐵 ↦ (𝑤𝑘))
189188mptpreima 5592 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢}
190187, 189elrab2 3352 . . . . . . . . . . . . . . . . . 18 (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))
191161, 38syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
192191adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
193 simprl 793 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢𝐾)
194161, 76syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
195194adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
196193, 195eleqtrd 2700 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
197 simprrr 804 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → (𝑝𝑘) ∈ 𝑢)
19855mopni2 22221 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
199192, 196, 197, 198syl3anc 1323 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
20018ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
201 simprrl 803 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑝𝐵)
202201adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝𝐵)
203 rpxr 11792 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
204203ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
205156blopn 22228 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
206200, 202, 204, 205syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
207 simprl 793 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
208 blcntr 22141 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
209200, 202, 207, 208syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
210 blssm 22146 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
211200, 202, 204, 210syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
212 simplrr 800 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
213 simplll 797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
214 rpgt0 11796 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑟 ∈ ℝ+ → 0 < 𝑟)
215214ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
216213, 202, 204, 215, 97syl121anc 1328 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
217216eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)))
218217biimpa 501 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
219 vex 3192 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑤 ∈ V
220219elixp 7867 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
221220simprbi 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) → ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
222218, 221syl 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))
223 simp-4r 806 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘𝐼)
224 rsp 2924 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟) → (𝑘𝐼 → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
225222, 223, 224sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
226212, 225sseldd 3588 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ 𝑢)
227211, 226ssrabdv 3665 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢})
228227, 189syl6sseqr 3636 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))
229 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝𝑦𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
230 sseq1 3610 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
231229, 230anbi12d 746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑝(ball‘𝐷)𝑟) → ((𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)) ↔ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
232231rspcev 3298 . . . . . . . . . . . . . . . . . . . . 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)))
233206, 209, 228, 232syl12anc 1321 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
234199, 233rexlimddv 3029 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
235234expr 642 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
236190, 235syl5bi 232 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
237236ralrimiv 2960 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
238158ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (MetOpen‘𝐷) ∈ Top)
239 eltop2 20703 . . . . . . . . . . . . . . . . 17 ((MetOpen‘𝐷) ∈ Top → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
240238, 239syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
241237, 240mpbird 247 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242185, 241eqeltrrd 2699 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
243242ralrimiva 2961 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
244160raleqdv 3136 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)))
245243, 244mpbird 247 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
246245ralrimiva 2961 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
247 fveq2 6153 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
248 fveq2 6153 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑤𝑘) = (𝑤𝑚))
249248mpteq2dv 4710 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
250249cnveqd 5263 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚(𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
251250imaeq1d 5429 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
252251eleq1d 2683 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
253247, 252raleqbidv 3144 . . . . . . . . . . . 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)))
254253cbvralv 3162 . . . . . . . . . . 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))
255246, 254sylib 208 . . . . . . . . . 10 (𝜑 → ∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
256 eqid 2621 . . . . . . . . . . 11 (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) = (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
257256fmpt2x 7188 . . . . . . . . . 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))
258255, 257sylib 208 . . . . . . . . 9 (𝜑 → (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
259 frn 6015 . . . . . . . . 9 ((𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷) → ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
260258, 259syl 17 . . . . . . . 8 (𝜑 → ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
261181, 260unssd 3772 . . . . . . 7 (𝜑 → ({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷))
262 fiss 8282 . . . . . . 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)))
263158, 261, 262syl2anc 692 . . . . . 6 (𝜑 → (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
264155, 263eqsstrd 3623 . . . . 5 (𝜑𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
265 fitop 20637 . . . . . . 7 ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
266158, 265syl 17 . . . . . 6 (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
267156mopnval 22166 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
26818, 267syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
269 tgdif0 20720 . . . . . . 7 (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
270268, 269syl6eqr 2673 . . . . . 6 (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
271266, 270eqtrd 2655 . . . . 5 (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
272264, 271sseqtrd 3625 . . . 4 (𝜑𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
273 2basgen 20718 . . . 4 (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
274138, 272, 273syl2anc 692 . . 3 (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
27512, 274eqtr4d 2658 . 2 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
276 prdsxms.j . . 3 𝐽 = (TopOpen‘𝑌)
27714, 15, 1, 5, 276prdstopn 21354 . 2 (𝜑𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
278275, 277, 2703eqtr4d 2665 1 (𝜑𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3189  cdif 3556  cun 3557  wss 3559  c0 3896  {csn 4153   cuni 4407   ciun 4490   class class class wbr 4618  cmpt 4678   × cxp 5077  ccnv 5078  ran crn 5080  cres 5081  cima 5082  ccom 5083   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  Xcixp 7860  Fincfn 7907  ficfi 8268  0cc0 9888  *cxr 10025   < clt 10026  +crp 11784  Basecbs 15792  distcds 15882  TopOpenctopn 16014  topGenctg 16030  tcpt 16031  Xscprds 16038  ∞Metcxmt 19663  ballcbl 19665  MetOpencmopn 19668  Topctop 20630  TopOnctopon 20647  TopSpctps 20660  ∞MetSpcxme 22045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fi 8269  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-icc 12132  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-plusg 15886  df-mulr 15887  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-hom 15898  df-cco 15899  df-rest 16015  df-topn 16016  df-topgen 16036  df-pt 16037  df-prds 16040  df-psmet 19670  df-xmet 19671  df-bl 19673  df-mopn 19674  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-xms 22048
This theorem is referenced by:  prdsxms  22258
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