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Theorem prdstotbnd 35066
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 2821 . . . 4 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
2 eqid 2821 . . . 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 2821 . . . 4 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
6 prdsbnd.s . . . 4 (𝜑𝑆𝑊)
7 prdsbnd.i . . . 4 (𝜑𝐼 ∈ Fin)
8 fvexd 6680 . . . 4 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
9 prdstotbnd.m . . . . 5 ((𝜑𝑥𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
10 totbndmet 35044 . . . . 5 (𝐸 ∈ (TotBnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
119, 10syl 17 . . . 4 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
121, 2, 3, 4, 5, 6, 7, 8, 11prdsmet 22974 . . 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 6719 . . . . . . . 8 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1715, 16sylib 220 . . . . . . 7 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1817oveq2d 7166 . . . . . 6 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
1914, 18syl5eq 2868 . . . . 5 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2019fveq2d 6669 . . . 4 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2113, 20syl5eq 2868 . . 3 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
22 prdsbnd.b . . . . 5 𝐵 = (Base‘𝑌)
2319fveq2d 6669 . . . . 5 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2422, 23syl5eq 2868 . . . 4 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2524fveq2d 6669 . . 3 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
2612, 21, 253eltr4d 2928 . 2 (𝜑𝐷 ∈ (Met‘𝐵))
277adantr 483 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → 𝐼 ∈ Fin)
28 istotbnd3 35043 . . . . . . . . . . 11 (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
2928simprbi 499 . . . . . . . . . 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 3208 . . . . . . . 8 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
32 df-rex 3144 . . . . . . . . 9 (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
33 rexv 3521 . . . . . . . . 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 280 . . . . . . . 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 220 . . . . . . 7 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3635an32s 650 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3736ralrimiva 3182 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → ∀𝑥𝐼𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
38 eleq1 2900 . . . . . . 7 (𝑤 = (𝑓𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin)))
39 iuneq1 4928 . . . . . . . 8 (𝑤 = (𝑓𝑥) → 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟))
4039eqeq1d 2823 . . . . . . 7 (𝑤 = (𝑓𝑥) → ( 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))
4138, 40anbi12d 632 . . . . . 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 8756 . . . . 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 586 . . . 4 ((𝜑𝑟 ∈ ℝ+) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
44 elfpw 8820 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓𝑥) ⊆ 𝑉 ∧ (𝑓𝑥) ∈ Fin))
4544simplbi 500 . . . . . . . . . . 11 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ⊆ 𝑉)
4645adantr 483 . . . . . . . . . 10 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ⊆ 𝑉)
4746ralimi 3160 . . . . . . . . 9 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
4847ad2antll 727 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
49 ss2ixp 8468 . . . . . . . 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 8790 . . . . . . . . . . 11 ((𝑅 Fn 𝐼𝐼 ∈ Fin) → 𝑅 ∈ Fin)
5215, 7, 51syl2anc 586 . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
53 fndm 6450 . . . . . . . . . . 11 (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
5415, 53syl 17 . . . . . . . . . 10 (𝜑 → dom 𝑅 = 𝐼)
5514, 6, 52, 22, 54prdsbas 16724 . . . . . . . . 9 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
563rgenw 3150 . . . . . . . . . 10 𝑥𝐼 𝑉 = (Base‘(𝑅𝑥))
57 ixpeq2 8469 . . . . . . . . . 10 (∀𝑥𝐼 𝑉 = (Base‘(𝑅𝑥)) → X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥)))
5856, 57ax-mp 5 . . . . . . . . 9 X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥))
5955, 58syl6eqr 2874 . . . . . . . 8 (𝜑𝐵 = X𝑥𝐼 𝑉)
6059ad2antrr 724 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 = X𝑥𝐼 𝑉)
6150, 60sseqtrrd 4008 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵)
6227adantr 483 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin)
6344simprbi 499 . . . . . . . . . 10 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ∈ Fin)
6463adantr 483 . . . . . . . . 9 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ∈ Fin)
6564ralimi 3160 . . . . . . . 8 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
6665ad2antll 727 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
67 ixpfi 8815 . . . . . . 7 ((𝐼 ∈ Fin ∧ ∀𝑥𝐼 (𝑓𝑥) ∈ Fin) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
6862, 66, 67syl2anc 586 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
69 elfpw 8820 . . . . . 6 (X𝑥𝐼 (𝑓𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵X𝑥𝐼 (𝑓𝑥) ∈ Fin))
7061, 68, 69sylanbrc 585 . . . . 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 22938 . . . . . . . . . . 11 (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
7226, 71syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ (∞Met‘𝐵))
73 rpxr 12392 . . . . . . . . . 10 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
74 blssm 23022 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
75743expa 1114 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7675an32s 650 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7776ralrimiva 3182 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7872, 73, 77syl2an 597 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7978adantr 483 . . . . . . . 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 4033 . . . . . . . 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 4962 . . . . . . 7 ( 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
8381, 82sylibr 236 . . . . . 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 483 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝐼 ∈ Fin)
8560eleq2d 2898 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵𝑔X𝑥𝐼 𝑉))
86 vex 3498 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
8786elixp 8462 . . . . . . . . . . . . . . 15 (𝑔X𝑥𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉))
8887simprbi 499 . . . . . . . . . . . . . 14 (𝑔X𝑥𝐼 𝑉 → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
89 df-rex 3144 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
90 eliun 4916 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))
91 rexv 3521 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
9289, 90, 913bitr4i 305 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
93 eleq2 2901 . . . . . . . . . . . . . . . . . . 19 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ 𝑉))
9492, 93syl5bbr 287 . . . . . . . . . . . . . . . . . 18 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔𝑥) ∈ 𝑉))
9594biimprd 250 . . . . . . . . . . . . . . . . 17 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9695adantl 484 . . . . . . . . . . . . . . . 16 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9796ral2imi 3156 . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . 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 242 . . . . . . . . . . . 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 409 . . . . . . . . . . 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 2900 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → (𝑧 ∈ (𝑓𝑥) ↔ (𝑦𝑥) ∈ (𝑓𝑥)))
103 oveq1 7157 . . . . . . . . . . . . . 14 (𝑧 = (𝑦𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦𝑥)(ball‘𝐸)𝑟))
104103eleq2d 2898 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → ((𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
105102, 104anbi12d 632 . . . . . . . . . . . 12 (𝑧 = (𝑦𝑥) → ((𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))))
106105ac6sfi 8756 . . . . . . . . . . 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 586 . . . . . . . . . 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 6509 . . . . . . . . . . . . . . . . 17 (𝑦:𝐼⟶V → 𝑦 Fn 𝐼)
109 simpl 485 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑦𝑥) ∈ (𝑓𝑥))
110109ralimi 3160 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥))
111108, 110anim12i 614 . . . . . . . . . . . . . . . 16 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
112 vex 3498 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
113112elixp 8462 . . . . . . . . . . . . . . . 16 (𝑦X𝑥𝐼 (𝑓𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
114111, 113sylibr 236 . . . . . . . . . . . . . . 15 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → 𝑦X𝑥𝐼 (𝑓𝑥))
115114adantl 484 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝑔X𝑥𝐼 𝑉)
117 ixpfn 8461 . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
121120ralimi 3160 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
122121ad2antll 727 . . . . . . . . . . . . . . . 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 8462 . . . . . . . . . . . . . . . 16 (𝑔X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
124119, 122, 123sylanbrc 585 . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . 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 3968 . . . . . . . . . . . . . . . . 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 2916 . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . 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 6665 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
132131cbvmptv 5162 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
133132oveq2i 7161 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
13419, 133syl6eqr 2874 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
135134fveq2d 6669 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
13613, 135syl5eq 2868 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
137136fveq2d 6669 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
138137oveqdr 7178 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
139 eqid 2821 . . . . . . . . . . . . . . . . . 18 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
140 eqid 2821 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
1416adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
1427adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
143 fvexd 6680 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑅𝑥) ∈ V)
144 metxmet 22938 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
14511, 144syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
146145adantlr 713 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
147 simprl 769 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦𝐵)
148134fveq2d 6669 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
14922, 148syl5eq 2868 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
150149adantr 483 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
151147, 150eleqtrd 2915 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15273ad2antll 727 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
153 rpgt0 12395 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ ℝ+ → 0 < 𝑟)
154153ad2antll 727 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
155133, 139, 3, 4, 140, 141, 142, 143, 146, 151, 152, 154prdsbl 23095 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
156138, 155eqtrd 2856 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
157125, 129, 130, 156syl12anc 834 . . . . . . . . . . . . . . 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 2916 . . . . . . . . . . . . . 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 514 . . . . . . . . . . . . 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 415 . . . . . . . . . . . 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 1914 . . . . . . . . . . 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 3144 . . . . . . . . . . 11 (∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
163161, 162syl6ibr 254 . . . . . . . . . 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 415 . . . . . . . 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 4916 . . . . . . . 8 (𝑔 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
167165, 166syl6ibr 254 . . . . . . 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 3973 . . . . . 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 3984 . . . . 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 4928 . . . . . . 7 (𝑣 = X𝑥𝐼 (𝑓𝑥) → 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟))
171170eqeq1d 2823 . . . . . 6 (𝑣 = X𝑥𝐼 (𝑓𝑥) → ( 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵))
172171rspcev 3623 . . . . 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 586 . . . 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 1932 . . 3 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
175174ralrimiva 3182 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
176 istotbnd3 35043 . 2 (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵))
17726, 175, 176sylanbrc 585 1 (𝜑𝐷 ∈ (TotBnd‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wex 1776  wcel 2110  wral 3138  wrex 3139  Vcvv 3495  cin 3935  wss 3936  𝒫 cpw 4539   ciun 4912   class class class wbr 5059  cmpt 5139   × cxp 5548  dom cdm 5550  cres 5552   Fn wfn 6345  wf 6346  cfv 6350  (class class class)co 7150  Xcixp 8455  Fincfn 8503  0cc0 10531  *cxr 10668   < clt 10669  +crp 12383  Basecbs 16477  distcds 16568  Xscprds 16713  ∞Metcxmet 20524  Metcmet 20525  ballcbl 20526  TotBndctotbnd 35038
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-rp 12384  df-xneg 12501  df-xadd 12502  df-xmul 12503  df-icc 12739  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-prds 16715  df-psmet 20531  df-xmet 20532  df-met 20533  df-bl 20534  df-totbnd 35040
This theorem is referenced by:  prdsbnd2  35067
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