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Theorem prdsxmslem2 24558
Description: Lemma for prdsxms 24559. 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 17472 . . . . 5 TopOpen Fn V
3 prdsxms.r . . . . . . 7 (𝜑𝑅:𝐼⟶∞MetSp)
43ffnd 6738 . . . . . 6 (𝜑𝑅 Fn 𝐼)
5 dffn2 6739 . . . . . 6 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
64, 5sylib 218 . . . . 5 (𝜑𝑅:𝐼⟶V)
7 fnfco 6774 . . . . 5 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
82, 6, 7sylancr 587 . . . 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 23594 . . . 4 ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
111, 8, 10syl2anc 584 . . 3 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
12 eldifsn 4791 . . . . . . . 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 24557 . . . . . . . . . . 11 (𝜑𝐷 ∈ (∞Met‘𝐵))
18 blrn 24435 . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
21 simprl 771 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑝𝐵)
22 simprr 773 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
23 xbln0 24440 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2420, 21, 22, 23syl3anc 1370 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
2513ad2ant1 1132 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
2625mptexd 7244 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∈ V)
27 ovex 7464 . . . . . . . . . . . . . . . . . . 19 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
2827rgenw 3063 . . . . . . . . . . . . . . . . . 18 𝑛𝐼 ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) ∈ V
29 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))
3029fnmpt 6709 . . . . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
3332ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . 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 24482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
3733, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
38 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
39 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
40143ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆𝑊)
4133ralrimiva 3144 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑅𝑘) ∈ ∞MetSp)
42 simp2l 1198 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝𝐵)
4332feqmptd 6977 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘𝐼 ↦ (𝑅𝑘)))
4443oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4513, 44eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘))))
4645fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4716, 46eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4842, 47eleqtrd 2841 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))))
4938, 39, 40, 25, 41, 34, 48prdsbascl 17530 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 (𝑝𝑘) ∈ 𝑉)
5049r19.21bi 3249 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → (𝑝𝑘) ∈ 𝑉)
51 simp2r 1199 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
5251adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝑟 ∈ ℝ*)
53 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 (MetOpen‘𝐸) = (MetOpen‘𝐸)
5453blopn 24529 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝𝑘) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
5537, 50, 52, 54syl3anc 1370 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
56 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → (dist‘(𝑅𝑛)) = (dist‘(𝑅𝑘)))
57 2fveq3 6912 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = (Base‘(𝑅𝑘)))
5857, 34eqtr4di 2793 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = 𝑘 → (Base‘(𝑅𝑛)) = 𝑉)
5958sqxpeqd 5721 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑘 → ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))) = (𝑉 × 𝑉))
6056, 59reseq12d 6001 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = ((dist‘(𝑅𝑘)) ↾ (𝑉 × 𝑉)))
6160, 35eqtr4di 2793 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 𝑘 → ((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))) = 𝐸)
6261fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛))))) = (ball‘𝐸))
63 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘 → (𝑝𝑛) = (𝑝𝑘))
64 eqidd 2736 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 𝑘𝑟 = 𝑟)
6562, 63, 64oveq123d 7452 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟) = ((𝑝𝑘)(ball‘𝐸)𝑟))
66 ovex 7464 . . . . . . . . . . . . . . . . . . . . 21 ((𝑝𝑘)(ball‘𝐸)𝑟) ∈ V
6765, 29, 66fvmpt 7016 . . . . . . . . . . . . . . . . . . . 20 (𝑘𝐼 → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
6867adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
69 fvco3 7008 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅𝑘)))
70 prdsxms.k . . . . . . . . . . . . . . . . . . . . . 22 𝐾 = (TopOpen‘(𝑅𝑘))
7169, 70eqtr4di 2793 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅:𝐼⟶∞MetSp ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7232, 71sylan 580 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
7370, 34, 35xmstopn 24477 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
7433, 73syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
7572, 74eqtrd 2775 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
7655, 68, 753eltr4d 2854 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘𝐼) → ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7776ralrimiva 3144 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘𝐼 ((𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
7832feqmptd 6977 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛𝐼 ↦ (𝑅𝑛)))
7978oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8013, 79eqtrid 2787 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
8180fveq2d 6911 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8215, 81eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
8382fveq2d 6911 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))))
8483oveqd 7448 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟))
85 fveq2 6907 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑘 → (𝑅𝑛) = (𝑅𝑘))
8685cbvmptv 5261 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐼 ↦ (𝑅𝑛)) = (𝑘𝐼 ↦ (𝑅𝑘))
8786oveq2i 7442 . . . . . . . . . . . . . . . . . . 19 (𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))) = (𝑆Xs(𝑘𝐼 ↦ (𝑅𝑘)))
88 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
89 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))) = (dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛))))
9080fveq2d 6911 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9116, 90eqtrid 2787 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
9242, 91eleqtrd 2841 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))
93 simp3 1137 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
9487, 88, 34, 35, 89, 40, 25, 33, 37, 92, 51, 93prdsbl 24520 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛𝐼 ↦ (𝑅𝑛)))))𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
9584, 94eqtrd 2775 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
96 fneq1 6660 . . . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) ∧ 𝑘𝐼) → (𝑔𝑘) = ((𝑝𝑘)(ball‘𝐸)𝑟))
102101ixpeq2dva 8951 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 = (𝑛𝐼 ↦ ((𝑝𝑛)(ball‘((dist‘(𝑅𝑛)) ↾ ((Base‘(𝑅𝑛)) × (Base‘(𝑅𝑛)))))𝑟)) → X𝑘𝐼 (𝑔𝑘) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
103102eqeq2d 2746 . . . . . . . . . . . . . . . . . . . 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 3597 . . . . . . . . . . . . . . . . . 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 1115 . . . . . . . . . . . . . . . . 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 1374 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
1081073expia 1120 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → (0 < 𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
10924, 108sylbid 240 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑔𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘))))
110109adantr 480 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
112111neeq1d 2998 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
113 df-3an 1088 . . . . . . . . . . . . . . . 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)))
114 ral0 4519 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)
115 difeq2 4130 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = 𝐼 → (𝐼𝑧) = (𝐼𝐼))
116 difid 4382 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝐼) = ∅
117115, 116eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = 𝐼 → (𝐼𝑧) = ∅)
118117raleqdv 3324 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)))
119118rspcev 3622 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
1201, 114, 119sylancl 586 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
121120adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑝𝐵𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼𝑧)(𝑔𝑘) = ((TopOpen ∘ 𝑅)‘𝑘))
122121biantrud 531 . . . . . . . . . . . . . . . 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))))
123113, 122bitr4id 290 . . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . 15 (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘𝐼 (𝑔𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 (𝑔𝑘)))
125123, 124bi2anan9 638 . . . . . . . . . . . . . 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 1919 . . . . . . . . . . . . 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 294 . . . . . . . . . . . 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 412 . . . . . . . . . . 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 3211 . . . . . . . . . 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 240 . . . . . . . . 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 410 . . . . . . . 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, 131biimtrid 242 . . . . . . 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 1925 . . . . . 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 4074 . . . . . 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 234 . . . . 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 4047 . . . 4 (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
137 ssv 4020 . . . . . . . . . 10 ∞MetSp ⊆ V
138 fnssres 6692 . . . . . . . . . 10 ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
1392, 137, 138mp2an 692 . . . . . . . . 9 (TopOpen ↾ ∞MetSp) Fn ∞MetSp
140 fvres 6926 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
141 xmstps 24479 . . . . . . . . . . . 12 (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
142 eqid 2735 . . . . . . . . . . . . 13 (TopOpen‘𝑥) = (TopOpen‘𝑥)
143142tpstop 22959 . . . . . . . . . . . 12 (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
144141, 143syl 17 . . . . . . . . . . 11 (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
145140, 144eqeltrd 2839 . . . . . . . . . 10 (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
146145rgen 3061 . . . . . . . . 9 𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
147 ffnfv 7139 . . . . . . . . 9 ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
148139, 146, 147mpbir2an 711 . . . . . . . 8 (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
149 fco2 6763 . . . . . . . 8 (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
150148, 3, 149sylancr 587 . . . . . . 7 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
151 eqid 2735 . . . . . . . 8 X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
1529, 151ptbasfi 23605 . . . . . . 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 584 . . . . . 6 (𝜑𝐶 = (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))))
154 eqid 2735 . . . . . . . . 9 (MetOpen‘𝐷) = (MetOpen‘𝐷)
155154mopntop 24466 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
15617, 155syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) ∈ Top)
15713, 16, 14, 1, 4prdsbas2 17516 . . . . . . . . . . . 12 (𝜑𝐵 = X𝑘𝐼 (Base‘(𝑅𝑘)))
1583, 71sylan 580 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
1593ffvelcdmda 7104 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ ∞MetSp)
160 xmstps 24479 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑘) ∈ ∞MetSp → (𝑅𝑘) ∈ TopSp)
161159, 160syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐼) → (𝑅𝑘) ∈ TopSp)
16234, 70istps 22956 . . . . . . . . . . . . . . . . 17 ((𝑅𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
163161, 162sylib 218 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐼) → 𝐾 ∈ (TopOn‘𝑉))
164158, 163eqeltrd 2839 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
165 toponuni 22936 . . . . . . . . . . . . . . 15 (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
166164, 165syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐼) → 𝑉 = ((TopOpen ∘ 𝑅)‘𝑘))
16734, 166eqtr3id 2789 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → (Base‘(𝑅𝑘)) = ((TopOpen ∘ 𝑅)‘𝑘))
168167ixpeq2dva 8951 . . . . . . . . . . . 12 (𝜑X𝑘𝐼 (Base‘(𝑅𝑘)) = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
169157, 168eqtrd 2775 . . . . . . . . . . 11 (𝜑𝐵 = X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘))
170 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
171170unieqd 4925 . . . . . . . . . . . 12 (𝑘 = 𝑛 ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
172171cbvixpv 8954 . . . . . . . . . . 11 X𝑘𝐼 ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)
173169, 172eqtrdi 2791 . . . . . . . . . 10 (𝜑𝐵 = X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛))
174154mopntopon 24465 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
17517, 174syl 17 . . . . . . . . . . 11 (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
176 toponmax 22948 . . . . . . . . . . 11 ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
177175, 176syl 17 . . . . . . . . . 10 (𝜑𝐵 ∈ (MetOpen‘𝐷))
178173, 177eqeltrrd 2840 . . . . . . . . 9 (𝜑X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
179178snssd 4814 . . . . . . . 8 (𝜑 → {X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
180173mpteq1d 5243 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
181180ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
182181cnveqd 5889 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)))
183182imaeq1d 6079 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢))
184 fveq1 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = 𝑝 → (𝑤𝑘) = (𝑝𝑘))
185184eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑝 → ((𝑤𝑘) ∈ 𝑢 ↔ (𝑝𝑘) ∈ 𝑢))
186 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (𝑤𝐵 ↦ (𝑤𝑘)) = (𝑤𝐵 ↦ (𝑤𝑘))
187186mptpreima 6260 . . . . . . . . . . . . . . . . . . 19 ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) = {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢}
188185, 187elrab2 3698 . . . . . . . . . . . . . . . . . 18 (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ↔ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))
189159, 36syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑘𝐼) → 𝐸 ∈ (∞Met‘𝑉))
190189adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
191 simprl 771 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢𝐾)
192159, 73syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑘𝐼) → 𝐾 = (MetOpen‘𝐸))
193192adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
194191, 193eleqtrd 2841 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
195 simprrr 782 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → (𝑝𝑘) ∈ 𝑢)
19653mopni2 24522 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
197190, 194, 195, 196syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
19817ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
199 simprrl 781 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → 𝑝𝐵)
200199adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝𝐵)
201 rpxr 13042 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
202201ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
203154blopn 24529 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
204198, 200, 202, 203syl3anc 1370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
205 simprl 771 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
206 blcntr 24439 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
207198, 200, 205, 206syl3anc 1370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
208 blssm 24444 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝𝐵𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
209198, 200, 202, 208syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
210 simplrr 778 . . . . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
212 rpgt0 13045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑟 ∈ ℝ+ → 0 < 𝑟)
213212ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
214211, 200, 202, 213, 95syl121anc 1374 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟))
215214eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟)))
216215biimpa 476 . . . . . . . . . . . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑤 ∈ V
218217elixp 8943 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤X𝑘𝐼 ((𝑝𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘𝐼 (𝑤𝑘) ∈ ((𝑝𝑘)(ball‘𝐸)𝑟)))
219218simprbi 496 . . . . . . . . . . . . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘𝐼)
222 rsp 3245 . . . . . . . . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤𝑘) ∈ 𝑢)
225209, 224ssrabdv 4084 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤𝐵 ∣ (𝑤𝑘) ∈ 𝑢})
226225, 187sseqtrrdi 4047 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))
227 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝𝑦𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
228 sseq1 4021 . . . . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
232197, 231rexlimddv 3159 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑘𝐼) ∧ (𝑢𝐾 ∧ (𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
233232expr 456 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑝𝐵 ∧ (𝑝𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
234188, 233biimtrid 242 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢))))
235234ralrimiv 3143 . . . . . . . . . . . . . . . 16 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ∀𝑝 ∈ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝𝑦𝑦 ⊆ ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢)))
236156ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → (MetOpen‘𝐷) ∈ Top)
237 eltop2 22998 . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . 15 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤𝐵 ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
240183, 239eqeltrrd 2840 . . . . . . . . . . . . . 14 (((𝜑𝑘𝐼) ∧ 𝑢𝐾) → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
241240ralrimiva 3144 . . . . . . . . . . . . 13 ((𝜑𝑘𝐼) → ∀𝑢𝐾 ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242241, 158raleqtrrdv 3328 . . . . . . . . . . . 12 ((𝜑𝑘𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
243242ralrimiva 3144 . . . . . . . . . . 11 (𝜑 → ∀𝑘𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
244 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
245 fveq2 6907 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑚 → (𝑤𝑘) = (𝑤𝑚))
246245mpteq2dv 5250 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
247246cnveqd 5889 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚(𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) = (𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)))
248247imaeq1d 6079 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) = ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
249248eleq1d 2824 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
250244, 249raleqbidv 3344 . . . . . . . . . . . 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)))
251250cbvralvw 3235 . . . . . . . . . . 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))
252243, 251sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑚𝐼𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
253 eqid 2735 . . . . . . . . . . 11 (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) = (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))
254253fmpox 8091 . . . . . . . . . 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))
255252, 254sylib 218 . . . . . . . . 9 (𝜑 → (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)): 𝑚𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
256255frnd 6745 . . . . . . . 8 (𝜑 → ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
257179, 256unssd 4202 . . . . . . 7 (𝜑 → ({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷))
258 fiss 9462 . . . . . . 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)))
259156, 257, 258syl2anc 584 . . . . . 6 (𝜑 → (fi‘({X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ ((𝑤X𝑛𝐼 ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
260153, 259eqsstrd 4034 . . . . 5 (𝜑𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
261 fitop 22922 . . . . . . 7 ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
262156, 261syl 17 . . . . . 6 (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
263154mopnval 24464 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
26417, 263syl 17 . . . . . . 7 (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
265 tgdif0 23015 . . . . . . 7 (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
266264, 265eqtr4di 2793 . . . . . 6 (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
267262, 266eqtrd 2775 . . . . 5 (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
268260, 267sseqtrd 4036 . . . 4 (𝜑𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
269 2basgen 23013 . . . 4 (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
270136, 268, 269syl2anc 584 . . 3 (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
27111, 270eqtr4d 2778 . 2 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
272 prdsxms.j . . 3 𝐽 = (TopOpen‘𝑌)
27313, 14, 1, 4, 272prdstopn 23652 . 2 (𝜑𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
274271, 273, 2663eqtr4d 2785 1 (𝜑𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  ccnv 5688  ran crn 5690  cres 5691  cima 5692  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Xcixp 8936  Fincfn 8984  ficfi 9448  0cc0 11153  *cxr 11292   < clt 11293  +crp 13032  Basecbs 17245  distcds 17307  TopOpenctopn 17468  topGenctg 17484  tcpt 17485  Xscprds 17492  ∞Metcxmet 21367  ballcbl 21369  MetOpencmopn 21372  Topctop 22915  TopOnctopon 22932  TopSpctps 22954  ∞MetSpcxms 24343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-icc 13391  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-topgen 17490  df-pt 17491  df-prds 17494  df-psmet 21374  df-xmet 21375  df-bl 21377  df-mopn 21378  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-xms 24346
This theorem is referenced by:  prdsxms  24559
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