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Theorem prdstotbnd 34222
Description: The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
prdsbnd.y 𝑌 = (𝑆Xs𝑅)
prdsbnd.b 𝐵 = (Base‘𝑌)
prdsbnd.v 𝑉 = (Base‘(𝑅𝑥))
prdsbnd.e 𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))
prdsbnd.d 𝐷 = (dist‘𝑌)
prdsbnd.s (𝜑𝑆𝑊)
prdsbnd.i (𝜑𝐼 ∈ Fin)
prdsbnd.r (𝜑𝑅 Fn 𝐼)
prdstotbnd.m ((𝜑𝑥𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
Assertion
Ref Expression
prdstotbnd (𝜑𝐷 ∈ (TotBnd‘𝐵))
Distinct variable groups:   𝑥,𝑅   𝑥,𝐵   𝜑,𝑥   𝑥,𝐼   𝑥,𝑆   𝑥,𝑌
Allowed substitution hints:   𝐷(𝑥)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem prdstotbnd
Dummy variables 𝑧 𝑟 𝑓 𝑔 𝑣 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2778 . . . 4 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
2 eqid 2778 . . . 4 (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
3 prdsbnd.v . . . 4 𝑉 = (Base‘(𝑅𝑥))
4 prdsbnd.e . . . 4 𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))
5 eqid 2778 . . . 4 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
6 prdsbnd.s . . . 4 (𝜑𝑆𝑊)
7 prdsbnd.i . . . 4 (𝜑𝐼 ∈ Fin)
8 fvexd 6463 . . . 4 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
9 prdstotbnd.m . . . . 5 ((𝜑𝑥𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
10 totbndmet 34200 . . . . 5 (𝐸 ∈ (TotBnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
119, 10syl 17 . . . 4 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
121, 2, 3, 4, 5, 6, 7, 8, 11prdsmet 22587 . . 3 (𝜑 → (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
13 prdsbnd.d . . . 4 𝐷 = (dist‘𝑌)
14 prdsbnd.y . . . . . 6 𝑌 = (𝑆Xs𝑅)
15 prdsbnd.r . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
16 dffn5 6503 . . . . . . . 8 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1715, 16sylib 210 . . . . . . 7 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1817oveq2d 6940 . . . . . 6 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
1914, 18syl5eq 2826 . . . . 5 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2019fveq2d 6452 . . . 4 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2113, 20syl5eq 2826 . . 3 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
22 prdsbnd.b . . . . 5 𝐵 = (Base‘𝑌)
2319fveq2d 6452 . . . . 5 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2422, 23syl5eq 2826 . . . 4 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2524fveq2d 6452 . . 3 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
2612, 21, 253eltr4d 2874 . 2 (𝜑𝐷 ∈ (Met‘𝐵))
277adantr 474 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → 𝐼 ∈ Fin)
28 istotbnd3 34199 . . . . . . . . . . 11 (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
2928simprbi 492 . . . . . . . . . 10 (𝐸 ∈ (TotBnd‘𝑉) → ∀𝑟 ∈ ℝ+𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
309, 29syl 17 . . . . . . . . 9 ((𝜑𝑥𝐼) → ∀𝑟 ∈ ℝ+𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
3130r19.21bi 3114 . . . . . . . 8 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
32 df-rex 3096 . . . . . . . . 9 (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
33 rexv 3422 . . . . . . . . 9 (∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3432, 33bitr4i 270 . . . . . . . 8 (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3531, 34sylib 210 . . . . . . 7 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3635an32s 642 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3736ralrimiva 3148 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → ∀𝑥𝐼𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
38 eleq1 2847 . . . . . . 7 (𝑤 = (𝑓𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin)))
39 iuneq1 4769 . . . . . . . 8 (𝑤 = (𝑓𝑥) → 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟))
4039eqeq1d 2780 . . . . . . 7 (𝑤 = (𝑓𝑥) → ( 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))
4138, 40anbi12d 624 . . . . . 6 (𝑤 = (𝑓𝑥) → ((𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
4241ac6sfi 8494 . . . . 5 ((𝐼 ∈ Fin ∧ ∀𝑥𝐼𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
4327, 37, 42syl2anc 579 . . . 4 ((𝜑𝑟 ∈ ℝ+) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
44 elfpw 8558 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓𝑥) ⊆ 𝑉 ∧ (𝑓𝑥) ∈ Fin))
4544simplbi 493 . . . . . . . . . . 11 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ⊆ 𝑉)
4645adantr 474 . . . . . . . . . 10 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ⊆ 𝑉)
4746ralimi 3134 . . . . . . . . 9 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
4847ad2antll 719 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
49 ss2ixp 8209 . . . . . . . 8 (∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉X𝑥𝐼 (𝑓𝑥) ⊆ X𝑥𝐼 𝑉)
5048, 49syl 17 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ⊆ X𝑥𝐼 𝑉)
51 fnfi 8528 . . . . . . . . . . 11 ((𝑅 Fn 𝐼𝐼 ∈ Fin) → 𝑅 ∈ Fin)
5215, 7, 51syl2anc 579 . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
53 fndm 6237 . . . . . . . . . . 11 (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
5415, 53syl 17 . . . . . . . . . 10 (𝜑 → dom 𝑅 = 𝐼)
5514, 6, 52, 22, 54prdsbas 16507 . . . . . . . . 9 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
563rgenw 3106 . . . . . . . . . 10 𝑥𝐼 𝑉 = (Base‘(𝑅𝑥))
57 ixpeq2 8210 . . . . . . . . . 10 (∀𝑥𝐼 𝑉 = (Base‘(𝑅𝑥)) → X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥)))
5856, 57ax-mp 5 . . . . . . . . 9 X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥))
5955, 58syl6eqr 2832 . . . . . . . 8 (𝜑𝐵 = X𝑥𝐼 𝑉)
6059ad2antrr 716 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 = X𝑥𝐼 𝑉)
6150, 60sseqtr4d 3861 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵)
6227adantr 474 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin)
6344simprbi 492 . . . . . . . . . 10 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ∈ Fin)
6463adantr 474 . . . . . . . . 9 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ∈ Fin)
6564ralimi 3134 . . . . . . . 8 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
6665ad2antll 719 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
67 ixpfi 8553 . . . . . . 7 ((𝐼 ∈ Fin ∧ ∀𝑥𝐼 (𝑓𝑥) ∈ Fin) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
6862, 66, 67syl2anc 579 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
69 elfpw 8558 . . . . . 6 (X𝑥𝐼 (𝑓𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵X𝑥𝐼 (𝑓𝑥) ∈ Fin))
7061, 68, 69sylanbrc 578 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ∈ (𝒫 𝐵 ∩ Fin))
71 metxmet 22551 . . . . . . . . . . 11 (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
7226, 71syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ (∞Met‘𝐵))
73 rpxr 12152 . . . . . . . . . 10 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
74 blssm 22635 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
75743expa 1108 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7675an32s 642 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7776ralrimiva 3148 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7872, 73, 77syl2an 589 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7978adantr 474 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
80 ssralv 3885 . . . . . . . 8 (X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵 → (∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 → ∀𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵))
8161, 79, 80sylc 65 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
82 iunss 4796 . . . . . . 7 ( 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
8381, 82sylibr 226 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
8462adantr 474 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝐼 ∈ Fin)
8560eleq2d 2845 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵𝑔X𝑥𝐼 𝑉))
86 vex 3401 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
8786elixp 8203 . . . . . . . . . . . . . . 15 (𝑔X𝑥𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉))
8887simprbi 492 . . . . . . . . . . . . . 14 (𝑔X𝑥𝐼 𝑉 → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
89 df-rex 3096 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
90 eliun 4759 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))
91 rexv 3422 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
9289, 90, 913bitr4i 295 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
93 eleq2 2848 . . . . . . . . . . . . . . . . . . 19 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ 𝑉))
9492, 93syl5bbr 277 . . . . . . . . . . . . . . . . . 18 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔𝑥) ∈ 𝑉))
9594biimprd 240 . . . . . . . . . . . . . . . . 17 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9695adantl 475 . . . . . . . . . . . . . . . 16 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9796ral2imi 3129 . . . . . . . . . . . . . . 15 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9897ad2antll 719 . . . . . . . . . . . . . 14 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉 → ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9988, 98syl5 34 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔X𝑥𝐼 𝑉 → ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
10085, 99sylbid 232 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵 → ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
101100imp 397 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
102 eleq1 2847 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → (𝑧 ∈ (𝑓𝑥) ↔ (𝑦𝑥) ∈ (𝑓𝑥)))
103 oveq1 6931 . . . . . . . . . . . . . 14 (𝑧 = (𝑦𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦𝑥)(ball‘𝐸)𝑟))
104103eleq2d 2845 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → ((𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
105102, 104anbi12d 624 . . . . . . . . . . . 12 (𝑧 = (𝑦𝑥) → ((𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))))
106105ac6sfi 8494 . . . . . . . . . . 11 ((𝐼 ∈ Fin ∧ ∀𝑥𝐼𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))))
10784, 101, 106syl2anc 579 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))))
108 ffn 6293 . . . . . . . . . . . . . . . . 17 (𝑦:𝐼⟶V → 𝑦 Fn 𝐼)
109 simpl 476 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑦𝑥) ∈ (𝑓𝑥))
110109ralimi 3134 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥))
111108, 110anim12i 606 . . . . . . . . . . . . . . . 16 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
112 vex 3401 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
113112elixp 8203 . . . . . . . . . . . . . . . 16 (𝑦X𝑥𝐼 (𝑓𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
114111, 113sylibr 226 . . . . . . . . . . . . . . 15 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → 𝑦X𝑥𝐼 (𝑓𝑥))
115114adantl 475 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑦X𝑥𝐼 (𝑓𝑥))
11685biimpa 470 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝑔X𝑥𝐼 𝑉)
117 ixpfn 8202 . . . . . . . . . . . . . . . . . 18 (𝑔X𝑥𝐼 𝑉𝑔 Fn 𝐼)
118116, 117syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝑔 Fn 𝐼)
119118adantr 474 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑔 Fn 𝐼)
120 simpr 479 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
121120ralimi 3134 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
122121ad2antll 719 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
12386elixp 8203 . . . . . . . . . . . . . . . 16 (𝑔X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
124119, 122, 123sylanbrc 578 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑔X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
125 simp-4l 773 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝜑)
12650ad2antrr 716 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → X𝑥𝐼 (𝑓𝑥) ⊆ X𝑥𝐼 𝑉)
127126, 115sseldd 3822 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑦X𝑥𝐼 𝑉)
128125, 59syl 17 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝐵 = X𝑥𝐼 𝑉)
129127, 128eleqtrrd 2862 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑦𝐵)
130 simp-4r 774 . . . . . . . . . . . . . . . 16 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑟 ∈ ℝ+)
131 fveq2 6448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
132131cbvmptv 4987 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
133132oveq2i 6935 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
13419, 133syl6eqr 2832 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
135134fveq2d 6452 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
13613, 135syl5eq 2826 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
137136fveq2d 6452 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
138137oveqdr 6952 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
139 eqid 2778 . . . . . . . . . . . . . . . . . 18 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
140 eqid 2778 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
1416adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
1427adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
143 fvexd 6463 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑅𝑥) ∈ V)
144 metxmet 22551 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
14511, 144syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
146145adantlr 705 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
147 simprl 761 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦𝐵)
148134fveq2d 6452 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
14922, 148syl5eq 2826 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
150149adantr 474 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
151147, 150eleqtrd 2861 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15273ad2antll 719 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
153 rpgt0 12155 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ ℝ+ → 0 < 𝑟)
154153ad2antll 719 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
155133, 139, 3, 4, 140, 141, 142, 143, 146, 151, 152, 154prdsbl 22708 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
156138, 155eqtrd 2814 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
157125, 129, 130, 156syl12anc 827 . . . . . . . . . . . . . . 15 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → (𝑦(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
158124, 157eleqtrrd 2862 . . . . . . . . . . . . . 14 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
159115, 158jca 507 . . . . . . . . . . . . 13 (((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))) → (𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
160159ex 403 . . . . . . . . . . . 12 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → (𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))))
161160eximdv 1960 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → ∃𝑦(𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))))
162 df-rex 3096 . . . . . . . . . . 11 (∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
163161, 162syl6ibr 244 . . . . . . . . . 10 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
164107, 163mpd 15 . . . . . . . . 9 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
165164ex 403 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵 → ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
166 eliun 4759 . . . . . . . 8 (𝑔 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
167165, 166syl6ibr 244 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵𝑔 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟)))
168167ssrdv 3827 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟))
16983, 168eqssd 3838 . . . . 5 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵)
170 iuneq1 4769 . . . . . . 7 (𝑣 = X𝑥𝐼 (𝑓𝑥) → 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟))
171170eqeq1d 2780 . . . . . 6 (𝑣 = X𝑥𝐼 (𝑓𝑥) → ( 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵))
172171rspcev 3511 . . . . 5 ((X𝑥𝐼 (𝑓𝑥) ∈ (𝒫 𝐵 ∩ Fin) ∧ 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
17370, 169, 172syl2anc 579 . . . 4 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
17443, 173exlimddv 1978 . . 3 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
175174ralrimiva 3148 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
176 istotbnd3 34199 . 2 (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵))
17726, 175, 176sylanbrc 578 1 (𝜑𝐷 ∈ (TotBnd‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wex 1823  wcel 2107  wral 3090  wrex 3091  Vcvv 3398  cin 3791  wss 3792  𝒫 cpw 4379   ciun 4755   class class class wbr 4888  cmpt 4967   × cxp 5355  dom cdm 5357  cres 5359   Fn wfn 6132  wf 6133  cfv 6137  (class class class)co 6924  Xcixp 8196  Fincfn 8243  0cc0 10274  *cxr 10412   < clt 10413  +crp 12141  Basecbs 16259  distcds 16351  Xscprds 16496  ∞Metcxmet 20131  Metcmet 20132  ballcbl 20133  TotBndctotbnd 34194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-2o 7846  df-oadd 7849  df-er 8028  df-map 8144  df-pm 8145  df-ixp 8197  df-en 8244  df-dom 8245  df-sdom 8246  df-fin 8247  df-sup 8638  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611  df-div 11035  df-nn 11379  df-2 11442  df-3 11443  df-4 11444  df-5 11445  df-6 11446  df-7 11447  df-8 11448  df-9 11449  df-n0 11647  df-z 11733  df-dec 11850  df-uz 11997  df-rp 12142  df-xneg 12261  df-xadd 12262  df-xmul 12263  df-icc 12498  df-fz 12648  df-struct 16261  df-ndx 16262  df-slot 16263  df-base 16265  df-plusg 16355  df-mulr 16356  df-sca 16358  df-vsca 16359  df-ip 16360  df-tset 16361  df-ple 16362  df-ds 16364  df-hom 16366  df-cco 16367  df-prds 16498  df-psmet 20138  df-xmet 20139  df-met 20140  df-bl 20141  df-totbnd 34196
This theorem is referenced by:  prdsbnd2  34223
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