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Theorem prdsxmslem2 23141
 Description: Lemma for prdsxms 23142. 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 16701 . . . . 5 TopOpen Fn V
3 prdsxms.r . . . . . . 7 (𝜑𝑅:𝐼⟶∞MetSp)
43ffnd 6517 . . . . . 6 (𝜑𝑅 Fn 𝐼)
5 dffn2 6518 . . . . . 6 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
64, 5sylib 220 . . . . 5 (𝜑𝑅:𝐼⟶V)
7 fnfco 6545 . . . . 5 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
82, 6, 7sylancr 589 . . . 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 22180 . . . 4 ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
111, 8, 10syl2anc 586 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
12 eldifsn 4721 . . . . . . . 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 23140 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝐵))
18 blrn 23021 . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
21 simprl 769 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑝𝐵)
22 simprr 771 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
23 xbln0 23026 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2420, 21, 22, 23syl3anc 1367 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2513ad2ant1 1129 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
2625mptexd 6989 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
27 ovex 7191 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
2827rgenw 3152 . . . . . . . . . . . . . . . . . 18 𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
29 eqid 2823 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))
3029fnmpt 6490 . . . . . . . . . . . . . . . . . 18 (∀𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3128, 30mp1i 13 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) Fn 𝐼)
3233ad2ant1 1129 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
3332ffvelrnda 6853 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
34 prdsxms.v . . . . . . . . . . . . . . . . . . . . . 22 𝑉 = (Base‘(𝑅𝑘))
35 prdsxms.e . . . . . . . . . . . . . . . . . . . . . 22 𝐸 = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉))
3634, 35xmsxmet 23068 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
3733, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
38 eqid 2823 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
39 eqid 2823 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
40143ad2ant1 1129 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆𝑊)
4133ralrimiva 3184 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑅𝑘) ∈ ∞MetSp)
42 simp2l 1195 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝𝐵)
4332feqmptd 6735 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘𝐼 ↦ (𝑅𝑘)))
4443oveq2d 7174 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4513, 44syl5eq 2870 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4645fveq2d 6676 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4716, 46syl5eq 2870 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4842, 47eleqtrd 2917 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4938, 39, 40, 25, 41, 34, 48prdsbascl 16758 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑝𝑘) ∈ 𝑉)
5049r19.21bi 3210 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑝𝑘) ∈ 𝑉)
51 simp2r 1196 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
5251adantr 483 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝑟 ∈ ℝ*)
53 eqid 2823 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐸) = (MetOpen‘𝐸)
5453blopn 23112 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝𝑘) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
5537, 50, 52, 54syl3anc 1367 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
56 2fveq3 6677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → (dist‘(𝑅𝑛)) = (dist‘(𝑅𝑘)))
57 2fveq3 6677 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = (Base‘(𝑅𝑘)))
5857, 34syl6eqr 2876 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = 𝑉)
5958sqxpeqd 5589 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))) = (𝑉 × 𝑉))
6056, 59reseq12d 5856 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉)))
6160, 35syl6eqr 2876 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = 𝐸)
6261fveq2d 6676 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))))) = (ball‘𝐸))
63 fveq2 6672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝑝𝑛) = (𝑝𝑘))
64 eqidd 2824 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘𝑟 = 𝑟)
6562, 63, 64oveq123d 7179 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) = ((𝑝𝑘)(ball‘𝐸)𝑟))
66 ovex 7191 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ V
6765, 29, 66fvmpt 6770 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐼 → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
6867adantl 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
69 fvco3 6762 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
70 prdsxms.k . . . . . . . . . . . . . . . . . . . . . 22 𝐾 = (TopOpen‘(𝑅𝑘))
7169, 70syl6eqr 2876 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7232, 71sylan 582 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7370, 34, 35xmstopn 23063 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
7433, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
7572, 74eqtrd 2858 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
7655, 68, 753eltr4d 2930 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7776ralrimiva 3184 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7832feqmptd 6735 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛𝐼 ↦ (𝑅𝑛)))
7978oveq2d 7174 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8013, 79syl5eq 2870 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8180fveq2d 6676 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8215, 81syl5eq 2870 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8382fveq2d 6676 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))))
8483oveqd 7175 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟))
85 fveq2 6672 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑅𝑛) = (𝑅𝑘))
8685cbvmptv 5171 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐼 ↦ (𝑅𝑛)) = (𝑘𝐼 ↦ (𝑅𝑘))
8786oveq2i 7169 . . . . . . . . . . . . . . . . . . 19 (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
88 eqid 2823 . . . . . . . . . . . . . . . . . . 19 (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
89 eqid 2823 . . . . . . . . . . . . . . . . . . 19 (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
9080fveq2d 6676 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9116, 90syl5eq 2870 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9242, 91eleqtrd 2917 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
93 simp3 1134 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
9487, 88, 34, 35, 89, 40, 25, 33, 37, 92, 51, 93prdsbl 23103 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
9584, 94eqtrd 2858 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
96 fneq1 6446 . . . . . . . . . . . . . . . . . . . . 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))
97 fveq1 6671 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → (𝑔𝑘) = ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘))
9897eleq1d 2899 . . . . . . . . . . . . . . . . . . . . . 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)))
9998ralbidv 3199 . . . . . . . . . . . . . . . . . . . . 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)))
10096, 99anbi12d 632 . . . . . . . . . . . . . . . . . . . 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))))
10197, 67sylan9eq 2878 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∧ 𝑘𝐼) → (𝑔𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
102101ixpeq2dva 8478 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → X𝑘𝐼 (𝑔𝑘) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
103102eqeq2d 2834 . . . . . . . . . . . . . . . . . . . 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)))
104100, 103anbi12d 632 . . . . . . . . . . . . . . . . . . 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))))
105104spcegv 3599 . . . . . . . . . . . . . . . . . 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))))
1061053impib 1112 . . . . . . . . . . . . . . . . 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)))
10726, 31, 77, 95, 106syl121anc 1371 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
1081073expia 1117 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → (0 < 𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
10924, 108sylbid 242 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
110109adantr 483 . . . . . . . . . . . . 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))))
111 simpr 487 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
112111neeq1d 3077 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
113 ral0 4458 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)
114 difeq2 4095 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝐼 → (𝐼𝑧) = (𝐼𝐼))
115 difid 4332 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝐼) = ∅
116114, 115syl6eq 2874 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝐼 → (𝐼𝑧) = ∅)
117116raleqdv 3417 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)))
118117rspcev 3625 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
1191, 113, 118sylancl 588 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
120119adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
121120biantrud 534 . . . . . . . . . . . . . . . 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))))
122 df-3an 1085 . . . . . . . . . . . . . . . 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)))
123121, 122syl6rbbr 292 . . . . . . . . . . . . . . 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))))
124 eqeq1 2827 . . . . . . . . . . . . . . 15 (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘𝐼 (𝑔𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
125123, 124bi2anan9 637 . . . . . . . . . . . . . 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))))
126125exbidv 1922 . . . . . . . . . . . . 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))))
127110, 112, 1263imtr4d 296 . . . . . . . . . . . 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))))
128127ex 415 . . . . . . . . . . 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)))))
129128rexlimdvva 3296 . . . . . . . . . 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)))))
13019, 129sylbid 242 . . . . . . . . 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)))))
131130impd 413 . . . . . . . 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))))
13212, 131syl5bi 244 . . . . . . 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))))
133132alrimiv 1928 . . . . . 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))))
134 ssab 4043 . . . . . 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))))
135133, 134sylibr 236 . . . . 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))})
136135, 9sseqtrrdi 4020 . . . 4 (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
137 ssv 3993 . . . . . . . . . 10 ∞MetSp ⊆ V
138 fnssres 6472 . . . . . . . . . 10 ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
1392, 137, 138mp2an 690 . . . . . . . . 9 (TopOpen ↾ ∞MetSp) Fn ∞MetSp
140 fvres 6691 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
141 xmstps 23065 . . . . . . . . . . . 12 (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
142 eqid 2823 . . . . . . . . . . . . 13 (TopOpen‘𝑥) = (TopOpen‘𝑥)
143142tpstop 21547 . . . . . . . . . . . 12 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
144141, 143syl 17 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
145140, 144eqeltrd 2915 . . . . . . . . . 10 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
146145rgen 3150 . . . . . . . . 9 𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
147 ffnfv 6884 . . . . . . . . 9 ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
148139, 146, 147mpbir2an 709 . . . . . . . 8 (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
149 fco2 6535 . . . . . . . 8 (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
150148, 3, 149sylancr 589 . . . . . . 7 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
151 eqid 2823 . . . . . . . 8 X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
1529, 151ptbasfi 22191 . . . . . . 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)))))
1531, 150, 152syl2anc 586 . . . . . 6 (𝜑𝐶 = (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))))
154 eqid 2823 . . . . . . . . 9 (MetOpen‘𝐷) = (MetOpen‘𝐷)
155154mopntop 23052 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
15617, 155syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ∈ Top)
15713, 16, 14, 1, 4prdsbas2 16744 . . . . . . . . . . . 12 (𝜑𝐵 = X𝑘𝐼 (Base‘(𝑅𝑘)))
1583, 71sylan 582 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
1593ffvelrnda 6853 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
160 xmstps 23065 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑘) ∈ ∞MetSp → (𝑅𝑘) ∈ TopSp)
161159, 160syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopSp)
16234, 70istps 21544 . . . . . . . . . . . . . . . . 17 ((𝑅𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
163161, 162sylib 220 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → 𝐾 ∈ (TopOn‘𝑉))
164158, 163eqeltrd 2915 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
165 toponuni 21524 . . . . . . . . . . . . . . 15 (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
166164, 165syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
16734, 166syl5eqr 2872 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (Base‘(𝑅𝑘)) = ((TopOpen ∘ 𝑅)‘𝑘))
168167ixpeq2dva 8478 . . . . . . . . . . . 12 (𝜑X𝑘𝐼 (Base‘(𝑅𝑘)) = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
169157, 168eqtrd 2858 . . . . . . . . . . 11 (𝜑𝐵 = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
170 fveq2 6672 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
171170unieqd 4854 . . . . . . . . . . . 12 (𝑘 = 𝑛 ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
172171cbvixpv 8481 . . . . . . . . . . 11 X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
173169, 172syl6eq 2874 . . . . . . . . . 10 (𝜑𝐵 = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛))
174154mopntopon 23051 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
17517, 174syl 17 . . . . . . . . . . 11 (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
176 toponmax 21536 . . . . . . . . . . 11 ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
177175, 176syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ (MetOpen‘𝐷))
178173, 177eqeltrrd 2916 . . . . . . . . 9 (𝜑X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
179178snssd 4744 . . . . . . . 8 (𝜑 → {X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
180173mpteq1d 5157 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
181180ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
182181cnveqd 5748 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
183182imaeq1d 5930 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢))
184 fveq1 6671 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑝 → (𝑤𝑘) = (𝑝𝑘))
185184eleq1d 2899 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑝 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑝𝑘) ∈ 𝑢))
186 eqid 2823 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤𝐵 ↦ (𝑤𝑘))
187186mptpreima 6094 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢}
188185, 187elrab2 3685 . . . . . . . . . . . . . . . . . 18 (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))
189159, 36syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
190189adantr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
191 simprl 769 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢𝐾)
192159, 73syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
193192adantr 483 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
194191, 193eleqtrd 2917 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
195 simprrr 780 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → (𝑝𝑘) ∈ 𝑢)
19653mopni2 23105 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
197190, 194, 195, 196syl3anc 1367 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
19817ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
199 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑝𝐵)
200199adantr 483 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝𝐵)
201 rpxr 12401 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
202201ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
203154blopn 23112 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
204198, 200, 202, 203syl3anc 1367 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
205 simprl 769 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
206 blcntr 23025 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
207198, 200, 205, 206syl3anc 1367 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
208 blssm 23030 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
209198, 200, 202, 208syl3anc 1367 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
210 simplrr 776 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
211 simplll 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
212 rpgt0 12404 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑟 ∈ ℝ+ → 0 < 𝑟)
213212ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
214211, 200, 202, 213, 95syl121anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
215214eleq2d 2900 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)))
216215biimpa 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
217 vex 3499 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑤 ∈ V
218217elixp 8470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
219218simprbi 499 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) → ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
220216, 219syl 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))
221 simp-4r 782 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘𝐼)
222 rsp 3207 . . . . . . . . . . . . . . . . . . . . . . . . 25 (∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟) → (𝑘𝐼 → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
223220, 221, 222sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟))
224210, 223sseldd 3970 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ 𝑢)
225209, 224ssrabdv 4052 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢})
226225, 187sseqtrrdi 4020 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))
227 eleq2 2903 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝𝑦𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
228 sseq1 3994 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
229227, 228anbi12d 632 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (𝑝(ball‘𝐷)𝑟) → ((𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)) ↔ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
230229rspcev 3625 . . . . . . . . . . . . . . . . . . . . 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)))
231204, 207, 226, 230syl12anc 834 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
232197, 231rexlimddv 3293 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
233232expr 459 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
234188, 233syl5bi 244 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
235234ralrimiv 3183 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
236156ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (MetOpen‘𝐷) ∈ Top)
237 eltop2 21585 . . . . . . . . . . . . . . . . 17 ((MetOpen‘𝐷) ∈ Top → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
238236, 237syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
239235, 238mpbird 259 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
240183, 239eqeltrrd 2916 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
241240ralrimiva 3184 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242158raleqdv 3417 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)))
243241, 242mpbird 259 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
244243ralrimiva 3184 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
245 fveq2 6672 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
246 fveq2 6672 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑤𝑘) = (𝑤𝑚))
247246mpteq2dv 5164 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
248247cnveqd 5748 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚(𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
249248imaeq1d 5930 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
250249eleq1d 2899 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
251245, 250raleqbidv 3403 . . . . . . . . . . . 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)))
252251cbvralvw 3451 . . . . . . . . . . 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))
253244, 252sylib 220 . . . . . . . . . 10 (𝜑 → ∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
254 eqid 2823 . . . . . . . . . . 11 (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) = (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
255254fmpox 7767 . . . . . . . . . 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))
256253, 255sylib 220 . . . . . . . . 9 (𝜑 → (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
257256frnd 6523 . . . . . . . 8 (𝜑 → ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
258179, 257unssd 4164 . . . . . . 7 (𝜑 → ({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷))
259 fiss 8890 . . . . . . 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)))
260156, 258, 259syl2anc 586 . . . . . 6 (𝜑 → (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
261153, 260eqsstrd 4007 . . . . 5 (𝜑𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
262 fitop 21510 . . . . . . 7 ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
263156, 262syl 17 . . . . . 6 (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
264154mopnval 23050 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
26517, 264syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
266 tgdif0 21602 . . . . . . 7 (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
267265, 266syl6eqr 2876 . . . . . 6 (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
268263, 267eqtrd 2858 . . . . 5 (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
269261, 268sseqtrd 4009 . . . 4 (𝜑𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
270 2basgen 21600 . . . 4 (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
271136, 269, 270syl2anc 586 . . 3 (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
27211, 271eqtr4d 2861 . 2 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
273 prdsxms.j . . 3 𝐽 = (TopOpen‘𝑌)
27413, 14, 1, 4, 273prdstopn 22238 . 2 (𝜑𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
275272, 274, 2673eqtr4d 2868 1 (𝜑𝐽 = (MetOpen‘𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  {csn 4569  ∪ cuni 4840  ∪ ciun 4921   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  ◡ccnv 5556  ran crn 5558   ↾ cres 5559   “ cima 5560   ∘ ccom 5561   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  Xcixp 8463  Fincfn 8511  ficfi 8876  0cc0 10539  ℝ*cxr 10676   < clt 10677  ℝ+crp 12392  Basecbs 16485  distcds 16576  TopOpenctopn 16697  topGenctg 16713  ∏tcpt 16714  Xscprds 16721  ∞Metcxmet 20532  ballcbl 20534  MetOpencmopn 20537  Topctop 21503  TopOnctopon 21520  TopSpctps 21542  ∞MetSpcxms 22929 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-map 8410  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fi 8877  df-sup 8908  df-inf 8909  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-icc 12748  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-hom 16591  df-cco 16592  df-rest 16698  df-topn 16699  df-topgen 16719  df-pt 16720  df-prds 16723  df-psmet 20539  df-xmet 20540  df-bl 20542  df-mopn 20543  df-top 21504  df-topon 21521  df-topsp 21543  df-bases 21556  df-xms 22932 This theorem is referenced by:  prdsxms  23142
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