Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prdstotbnd Structured version   Visualization version   GIF version

Theorem prdstotbnd 33564
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 2620 . . . 4 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
2 eqid 2620 . . . 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 2620 . . . 4 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
6 prdsbnd.s . . . 4 (𝜑𝑆𝑊)
7 prdsbnd.i . . . 4 (𝜑𝐼 ∈ Fin)
8 fvexd 6190 . . . 4 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
9 prdstotbnd.m . . . . 5 ((𝜑𝑥𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
10 totbndmet 33542 . . . . 5 (𝐸 ∈ (TotBnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
119, 10syl 17 . . . 4 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
121, 2, 3, 4, 5, 6, 7, 8, 11prdsmet 22156 . . 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 6228 . . . . . . . 8 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1715, 16sylib 208 . . . . . . 7 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
1817oveq2d 6651 . . . . . 6 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
1914, 18syl5eq 2666 . . . . 5 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2019fveq2d 6182 . . . 4 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2113, 20syl5eq 2666 . . 3 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
22 prdsbnd.b . . . . 5 𝐵 = (Base‘𝑌)
2319fveq2d 6182 . . . . 5 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2422, 23syl5eq 2666 . . . 4 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2524fveq2d 6182 . . 3 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
2612, 21, 253eltr4d 2714 . 2 (𝜑𝐷 ∈ (Met‘𝐵))
277adantr 481 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → 𝐼 ∈ Fin)
28 istotbnd3 33541 . . . . . . . . . . 11 (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
2928simprbi 480 . . . . . . . . . 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 2929 . . . . . . . 8 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
32 df-rex 2915 . . . . . . . . 9 (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin) 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
33 rexv 3215 . . . . . . . . 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 267 . . . . . . . 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 208 . . . . . . 7 (((𝜑𝑥𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3635an32s 845 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ 𝑥𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
3736ralrimiva 2963 . . . . 5 ((𝜑𝑟 ∈ ℝ+) → ∀𝑥𝐼𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
38 eleq1 2687 . . . . . . 7 (𝑤 = (𝑓𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin)))
39 iuneq1 4525 . . . . . . . 8 (𝑤 = (𝑓𝑥) → 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟))
4039eqeq1d 2622 . . . . . . 7 (𝑤 = (𝑓𝑥) → ( 𝑧𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))
4138, 40anbi12d 746 . . . . . 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 8189 . . . . 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 692 . . . 4 ((𝜑𝑟 ∈ ℝ+) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
44 elfpw 8253 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓𝑥) ⊆ 𝑉 ∧ (𝑓𝑥) ∈ Fin))
4544simplbi 476 . . . . . . . . . . 11 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ⊆ 𝑉)
4645adantr 481 . . . . . . . . . 10 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ⊆ 𝑉)
4746ralimi 2949 . . . . . . . . 9 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
4847ad2antll 764 . . . . . . . 8 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ⊆ 𝑉)
49 ss2ixp 7906 . . . . . . . 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 8223 . . . . . . . . . . 11 ((𝑅 Fn 𝐼𝐼 ∈ Fin) → 𝑅 ∈ Fin)
5215, 7, 51syl2anc 692 . . . . . . . . . 10 (𝜑𝑅 ∈ Fin)
53 fndm 5978 . . . . . . . . . . 11 (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
5415, 53syl 17 . . . . . . . . . 10 (𝜑 → dom 𝑅 = 𝐼)
5514, 6, 52, 22, 54prdsbas 16098 . . . . . . . . 9 (𝜑𝐵 = X𝑥𝐼 (Base‘(𝑅𝑥)))
563rgenw 2921 . . . . . . . . . 10 𝑥𝐼 𝑉 = (Base‘(𝑅𝑥))
57 ixpeq2 7907 . . . . . . . . . 10 (∀𝑥𝐼 𝑉 = (Base‘(𝑅𝑥)) → X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥)))
5856, 57ax-mp 5 . . . . . . . . 9 X𝑥𝐼 𝑉 = X𝑥𝐼 (Base‘(𝑅𝑥))
5955, 58syl6eqr 2672 . . . . . . . 8 (𝜑𝐵 = X𝑥𝐼 𝑉)
6059ad2antrr 761 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 = X𝑥𝐼 𝑉)
6150, 60sseqtr4d 3634 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵)
6227adantr 481 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin)
6344simprbi 480 . . . . . . . . . 10 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓𝑥) ∈ Fin)
6463adantr 481 . . . . . . . . 9 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓𝑥) ∈ Fin)
6564ralimi 2949 . . . . . . . 8 (∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
6665ad2antll 764 . . . . . . 7 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥𝐼 (𝑓𝑥) ∈ Fin)
67 ixpfi 8248 . . . . . . 7 ((𝐼 ∈ Fin ∧ ∀𝑥𝐼 (𝑓𝑥) ∈ Fin) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
6862, 66, 67syl2anc 692 . . . . . 6 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥𝐼 (𝑓𝑥) ∈ Fin)
69 elfpw 8253 . . . . . 6 (X𝑥𝐼 (𝑓𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥𝐼 (𝑓𝑥) ⊆ 𝐵X𝑥𝐼 (𝑓𝑥) ∈ Fin))
7061, 68, 69sylanbrc 697 . . . . 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 22120 . . . . . . . . . . 11 (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
7226, 71syl 17 . . . . . . . . . 10 (𝜑𝐷 ∈ (∞Met‘𝐵))
73 rpxr 11825 . . . . . . . . . 10 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
74 blssm 22204 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
75743expa 1263 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7675an32s 845 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7776ralrimiva 2963 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7872, 73, 77syl2an 494 . . . . . . . . 9 ((𝜑𝑟 ∈ ℝ+) → ∀𝑦𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
7978adantr 481 . . . . . . . 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 3658 . . . . . . . 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 4552 . . . . . . 7 ( 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
8381, 82sylibr 224 . . . . . 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 481 . . . . . . . . . . 11 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝐼 ∈ Fin)
8560eleq2d 2685 . . . . . . . . . . . . 13 (((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔𝐵𝑔X𝑥𝐼 𝑉))
86 vex 3198 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
8786elixp 7900 . . . . . . . . . . . . . . 15 (𝑔X𝑥𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉))
8887simprbi 480 . . . . . . . . . . . . . 14 (𝑔X𝑥𝐼 𝑉 → ∀𝑥𝐼 (𝑔𝑥) ∈ 𝑉)
89 df-rex 2915 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
90 eliun 4515 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓𝑥)(𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))
91 rexv 3215 . . . . . . . . . . . . . . . . . . . 20 (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
9289, 90, 913bitr4i 292 . . . . . . . . . . . . . . . . . . 19 ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
93 eleq2 2688 . . . . . . . . . . . . . . . . . . 19 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ 𝑉))
9492, 93syl5bbr 274 . . . . . . . . . . . . . . . . . 18 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔𝑥) ∈ 𝑉))
9594biimprd 238 . . . . . . . . . . . . . . . . 17 ( 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9695adantl 482 . . . . . . . . . . . . . . . 16 (((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ((𝑔𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
9796ral2imi 2944 . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . 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 230 . . . . . . . . . . . 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 445 . . . . . . . . . . 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 2687 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → (𝑧 ∈ (𝑓𝑥) ↔ (𝑦𝑥) ∈ (𝑓𝑥)))
103 oveq1 6642 . . . . . . . . . . . . . 14 (𝑧 = (𝑦𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦𝑥)(ball‘𝐸)𝑟))
104103eleq2d 2685 . . . . . . . . . . . . 13 (𝑧 = (𝑦𝑥) → ((𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
105102, 104anbi12d 746 . . . . . . . . . . . 12 (𝑧 = (𝑦𝑥) → ((𝑧 ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))))
106105ac6sfi 8189 . . . . . . . . . . 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 692 . . . . . . . . . 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 6032 . . . . . . . . . . . . . . . . 17 (𝑦:𝐼⟶V → 𝑦 Fn 𝐼)
109 simpl 473 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑦𝑥) ∈ (𝑓𝑥))
110109ralimi 2949 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥))
111108, 110anim12i 589 . . . . . . . . . . . . . . . 16 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
112 vex 3198 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
113112elixp 7900 . . . . . . . . . . . . . . . 16 (𝑦X𝑥𝐼 (𝑓𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑦𝑥) ∈ (𝑓𝑥)))
114111, 113sylibr 224 . . . . . . . . . . . . . . 15 ((𝑦:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))) → 𝑦X𝑥𝐼 (𝑓𝑥))
115114adantl 482 . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥𝐼 ((𝑓𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝑧 ∈ (𝑓𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔𝐵) → 𝑔X𝑥𝐼 𝑉)
117 ixpfn 7899 . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . 18 (((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
121120ralimi 2949 . . . . . . . . . . . . . . . . 17 (∀𝑥𝐼 ((𝑦𝑥) ∈ (𝑓𝑥) ∧ (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)) → ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟))
122121ad2antll 764 . . . . . . . . . . . . . . . 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 7900 . . . . . . . . . . . . . . . 16 (𝑔X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑔𝑥) ∈ ((𝑦𝑥)(ball‘𝐸)𝑟)))
124119, 122, 123sylanbrc 697 . . . . . . . . . . . . . . 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 805 . . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . 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 3596 . . . . . . . . . . . . . . . . 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 2702 . . . . . . . . . . . . . . . 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 806 . . . . . . . . . . . . . . . 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 6178 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
132131cbvmptv 4741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
133132oveq2i 6646 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
13419, 133syl6eqr 2672 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
135134fveq2d 6182 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
13613, 135syl5eq 2666 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
137136fveq2d 6182 . . . . . . . . . . . . . . . . . 18 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
138137oveqdr 6659 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
139 eqid 2620 . . . . . . . . . . . . . . . . . 18 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
140 eqid 2620 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
1416adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
1427adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
143 fvexd 6190 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑅𝑥) ∈ V)
144 metxmet 22120 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
14511, 144syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
146145adantlr 750 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
147 simprl 793 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦𝐵)
148134fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
14922, 148syl5eq 2666 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
150149adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
151147, 150eleqtrd 2701 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15273ad2antll 764 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
153 rpgt0 11829 . . . . . . . . . . . . . . . . . . 19 (𝑟 ∈ ℝ+ → 0 < 𝑟)
154153ad2antll 764 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
155133, 139, 3, 4, 140, 141, 142, 143, 146, 151, 152, 154prdsbl 22277 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
156138, 155eqtrd 2654 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐵𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑦𝑥)(ball‘𝐸)𝑟))
157125, 129, 130, 156syl12anc 1322 . . . . . . . . . . . . . . 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 2702 . . . . . . . . . . . . . 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 554 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 1844 . . . . . . . . . . 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 2915 . . . . . . . . . . 11 (∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦X𝑥𝐼 (𝑓𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
163161, 162syl6ibr 242 . . . . . . . . . 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 450 . . . . . . . 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 4515 . . . . . . . 8 (𝑔 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦X 𝑥𝐼 (𝑓𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
167165, 166syl6ibr 242 . . . . . . 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 3601 . . . . . 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 3612 . . . . 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 4525 . . . . . . 7 (𝑣 = X𝑥𝐼 (𝑓𝑥) → 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟))
171170eqeq1d 2622 . . . . . 6 (𝑣 = X𝑥𝐼 (𝑓𝑥) → ( 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 𝑦X 𝑥𝐼 (𝑓𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵))
172171rspcev 3304 . . . . 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 692 . . . 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 1861 . . 3 ((𝜑𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
175174ralrimiva 2963 . 2 (𝜑 → ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
176 istotbnd3 33541 . 2 (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+𝑣 ∈ (𝒫 𝐵 ∩ Fin) 𝑦𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵))
17726, 175, 176sylanbrc 697 1 (𝜑𝐷 ∈ (TotBnd‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  𝒫 cpw 4149   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  dom cdm 5104  cres 5106   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  Xcixp 7893  Fincfn 7940  0cc0 9921  *cxr 10058   < clt 10059  +crp 11817  Basecbs 15838  distcds 15931  Xscprds 16087  ∞Metcxmt 19712  Metcme 19713  ballcbl 19714  TotBndctotbnd 33536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-map 7844  df-pm 7845  df-ixp 7894  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-sup 8333  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-rp 11818  df-xneg 11931  df-xadd 11932  df-xmul 11933  df-icc 12167  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-plusg 15935  df-mulr 15936  df-sca 15938  df-vsca 15939  df-ip 15940  df-tset 15941  df-ple 15942  df-ds 15945  df-hom 15947  df-cco 15948  df-prds 16089  df-psmet 19719  df-xmet 19720  df-met 19721  df-bl 19722  df-totbnd 33538
This theorem is referenced by:  prdsbnd2  33565
  Copyright terms: Public domain W3C validator