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Theorem prdsbnd2 34202
Description: If balls are totally bounded in each factor, then balls are bounded in a metric product. (Contributed by Mario Carneiro, 16-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)
prdsbnd2.c 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
prdsbnd2.e ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
prdsbnd2.m ((𝜑𝑥𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
Assertion
Ref Expression
prdsbnd2 (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))
Distinct variable groups:   𝑦,𝐷   𝑥,𝑦,𝑅   𝑥,𝐵,𝑦   𝑦,𝐸   𝜑,𝑥,𝑦   𝑥,𝐼,𝑦   𝑥,𝑆   𝑦,𝑉   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥)   𝑆(𝑦)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥,𝑦)   𝑌(𝑦)

Proof of Theorem prdsbnd2
Dummy variables 𝑟 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 totbndbnd 34196 . 2 (𝐶 ∈ (TotBnd‘𝐴) → 𝐶 ∈ (Bnd‘𝐴))
2 bndmet 34188 . . . . 5 (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (Met‘𝐴))
3 0totbnd 34180 . . . . 5 (𝐴 = ∅ → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Met‘𝐴)))
42, 3syl5ibr 238 . . . 4 (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
54a1i 11 . . 3 (𝜑 → (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
6 n0 4158 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
7 simprr 763 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (Bnd‘𝐴))
8 eqid 2777 . . . . . . . . . . . 12 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
9 eqid 2777 . . . . . . . . . . . 12 (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
10 prdsbnd.v . . . . . . . . . . . 12 𝑉 = (Base‘(𝑅𝑥))
11 prdsbnd.e . . . . . . . . . . . 12 𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))
12 eqid 2777 . . . . . . . . . . . 12 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
13 prdsbnd.s . . . . . . . . . . . 12 (𝜑𝑆𝑊)
14 prdsbnd.i . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
15 fvexd 6461 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
16 prdsbnd2.e . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
178, 9, 10, 11, 12, 13, 14, 15, 16prdsmet 22583 . . . . . . . . . . 11 (𝜑 → (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
18 prdsbnd.d . . . . . . . . . . . 12 𝐷 = (dist‘𝑌)
19 prdsbnd.y . . . . . . . . . . . . . 14 𝑌 = (𝑆Xs𝑅)
20 prdsbnd.r . . . . . . . . . . . . . . . 16 (𝜑𝑅 Fn 𝐼)
21 dffn5 6501 . . . . . . . . . . . . . . . 16 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2220, 21sylib 210 . . . . . . . . . . . . . . 15 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2322oveq2d 6938 . . . . . . . . . . . . . 14 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2419, 23syl5eq 2825 . . . . . . . . . . . . 13 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2524fveq2d 6450 . . . . . . . . . . . 12 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2618, 25syl5eq 2825 . . . . . . . . . . 11 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
27 prdsbnd.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑌)
2824fveq2d 6450 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2927, 28syl5eq 2825 . . . . . . . . . . . 12 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
3029fveq2d 6450 . . . . . . . . . . 11 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
3117, 26, 303eltr4d 2873 . . . . . . . . . 10 (𝜑𝐷 ∈ (Met‘𝐵))
3231adantr 474 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐷 ∈ (Met‘𝐵))
33 simpr 479 . . . . . . . . . . 11 ((𝑎𝐴𝐶 ∈ (Bnd‘𝐴)) → 𝐶 ∈ (Bnd‘𝐴))
34 prdsbnd2.c . . . . . . . . . . . 12 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
3534bnd2lem 34198 . . . . . . . . . . 11 ((𝐷 ∈ (Met‘𝐵) ∧ 𝐶 ∈ (Bnd‘𝐴)) → 𝐴𝐵)
3631, 33, 35syl2an 589 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐴𝐵)
37 simprl 761 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐴)
3836, 37sseldd 3821 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐵)
3934ssbnd 34195 . . . . . . . . 9 ((𝐷 ∈ (Met‘𝐵) ∧ 𝑎𝐵) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
4032, 38, 39syl2anc 579 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
417, 40mpbid 224 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
42 simprr 763 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
43 xpss12 5370 . . . . . . . . . . 11 ((𝐴 ⊆ (𝑎(ball‘𝐷)𝑟) ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4442, 42, 43syl2anc 579 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4544resabs1d 5677 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) = (𝐷 ↾ (𝐴 × 𝐴)))
4645, 34syl6eqr 2831 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) = 𝐶)
47 simpll 757 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝜑)
4838adantr 474 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐵)
49 simprl 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ)
5037adantr 474 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐴)
5142, 50sseldd 3821 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ (𝑎(ball‘𝐷)𝑟))
5251ne0d 4149 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝑎(ball‘𝐷)𝑟) ≠ ∅)
5331ad2antrr 716 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (Met‘𝐵))
54 metxmet 22547 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
5553, 54syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (∞Met‘𝐵))
5649rexrd 10426 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ*)
57 xbln0 22627 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑎𝐵𝑟 ∈ ℝ*) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
5855, 48, 56, 57syl3anc 1439 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
5952, 58mpbid 224 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 0 < 𝑟)
6049, 59elrpd 12178 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ+)
61 eqid 2777 . . . . . . . . . . . 12 (𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))) = (𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))
62 eqid 2777 . . . . . . . . . . . 12 (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))))
63 eqid 2777 . . . . . . . . . . . 12 (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥))
64 eqid 2777 . . . . . . . . . . . 12 ((dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)))) = ((dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥))))
65 eqid 2777 . . . . . . . . . . . 12 (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))))
6613adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
6714adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
68 ovex 6954 . . . . . . . . . . . . . 14 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V
69 fveq2 6446 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
70 2fveq3 6451 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (dist‘(𝑅𝑦)) = (dist‘(𝑅𝑥)))
71 2fveq3 6451 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑥)))
7271, 10syl6eqr 2831 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = 𝑉)
7372sqxpeqd 5387 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))) = (𝑉 × 𝑉))
7470, 73reseq12d 5643 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)))
7574, 11syl6eqr 2831 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = 𝐸)
7675fveq2d 6450 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))))) = (ball‘𝐸))
77 fveq2 6446 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑎𝑦) = (𝑎𝑥))
78 eqidd 2778 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥𝑟 = 𝑟)
7976, 77, 78oveq123d 6943 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟) = ((𝑎𝑥)(ball‘𝐸)𝑟))
8069, 79oveq12d 6940 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8180cbvmptv 4985 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) = (𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8268, 81fnmpti 6268 . . . . . . . . . . . . 13 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) Fn 𝐼
8382a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) Fn 𝐼)
8416adantlr 705 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
85 metxmet 22547 . . . . . . . . . . . . . . . 16 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
8684, 85syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
8715ralrimiva 3147 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
8887adantr 474 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
89 simprl 761 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎𝐵)
9029adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
9189, 90eleqtrd 2860 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
928, 9, 66, 67, 88, 10, 91prdsbascl 16529 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑎𝑥) ∈ 𝑉)
9392r19.21bi 3113 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑎𝑥) ∈ 𝑉)
94 simplrr 768 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ+)
9594rpred 12181 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ)
96 blbnd 34194 . . . . . . . . . . . . . . 15 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎𝑥) ∈ 𝑉𝑟 ∈ ℝ) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
9786, 93, 95, 96syl3anc 1439 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
98 ovex 6954 . . . . . . . . . . . . . . . 16 ((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V
99 xpeq12 5380 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) ∧ 𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟)) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
10099anidms 562 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
101100reseq2d 5642 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝐸 ↾ (𝑦 × 𝑦)) = (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
102 fveq2 6446 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (TotBnd‘𝑦) = (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
103101, 102eleq12d 2852 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟))))
104 fveq2 6446 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (Bnd‘𝑦) = (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
105101, 104eleq12d 2852 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟))))
106103, 105bibi12d 337 . . . . . . . . . . . . . . . . 17 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)) ↔ ((𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))))
107106imbi2d 332 . . . . . . . . . . . . . . . 16 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (((𝜑𝑥𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))) ↔ ((𝜑𝑥𝐼) → ((𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟))))))
108 prdsbnd2.m . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
10998, 107, 108vtocl 3459 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐼) → ((𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟))))
110109adantlr 705 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟))))
11197, 110mpbird 249 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
112 eqid 2777 . . . . . . . . . . . . . . . . . . 19 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) = (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))
11380, 112, 68fvmpt 6542 . . . . . . . . . . . . . . . . . 18 (𝑥𝐼 → ((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
114113adantl 475 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
115114fveq2d 6450 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (dist‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
116 eqid 2777 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))
117 eqid 2777 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑅𝑥)) = (dist‘(𝑅𝑥))
118116, 117ressds 16459 . . . . . . . . . . . . . . . . 17 (((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V → (dist‘(𝑅𝑥)) = (dist‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
11998, 118ax-mp 5 . . . . . . . . . . . . . . . 16 (dist‘(𝑅𝑥)) = (dist‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
120115, 119syl6eqr 2831 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (dist‘(𝑅𝑥)))
121114fveq2d 6450 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
122 rpxr 12148 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
123122ad2antll 719 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
124123adantr 474 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ*)
125 blssm 22631 . . . . . . . . . . . . . . . . . . 19 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎𝑥) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
12686, 93, 124, 125syl3anc 1439 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
127116, 10ressbas2 16327 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 → ((𝑎𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
128126, 127syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
129121, 128eqtr4d 2816 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = ((𝑎𝑥)(ball‘𝐸)𝑟))
130129sqxpeqd 5387 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥))) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
131120, 130reseq12d 5643 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)))) = ((dist‘(𝑅𝑥)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
13211reseq1i 5638 . . . . . . . . . . . . . . 15 (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = (((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
133 xpss12 5370 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 ∧ ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
134126, 126, 133syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
135134resabs1d 5677 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = ((dist‘(𝑅𝑥)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
136132, 135syl5eq 2825 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = ((dist‘(𝑅𝑥)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
137131, 136eqtr4d 2816 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)))) = (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
138129fveq2d 6450 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (TotBnd‘(Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥))) = (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
139111, 137, 1383eltr4d 2873 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)))) ∈ (TotBnd‘(Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥))))
14061, 62, 63, 64, 65, 66, 67, 83, 139prdstotbnd 34201 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) ∈ (TotBnd‘(Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))))))
14124adantr 474 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
142 eqidd 2778 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
143 eqid 2777 . . . . . . . . . . . . 13 (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
14481oveq2i 6933 . . . . . . . . . . . . . 14 (𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
145144fveq2i 6449 . . . . . . . . . . . . 13 (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
146 fvexd 6461 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑅𝑥) ∈ V)
14798a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V)
148141, 142, 143, 18, 145, 66, 66, 67, 146, 147ressprdsds 22584 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (𝐷 ↾ ((Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))))))
149128ixpeq2dva 8209 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
15069cbvmptv 4985 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
151150oveq2i 6933 . . . . . . . . . . . . . . . . . . . . 21 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
15224, 151syl6eqr 2831 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
153152fveq2d 6450 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15418, 153syl5eq 2825 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
155154fveq2d 6450 . . . . . . . . . . . . . . . . 17 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
156155oveqdr 6950 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
157 eqid 2777 . . . . . . . . . . . . . . . . 17 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
158 eqid 2777 . . . . . . . . . . . . . . . . 17 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
159152fveq2d 6450 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16027, 159syl5eq 2825 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
161160adantr 474 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16289, 161eleqtrd 2860 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
163 rpgt0 12151 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+ → 0 < 𝑟)
164163ad2antll 719 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
165151, 157, 10, 11, 158, 66, 67, 146, 86, 162, 123, 164prdsbl 22704 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
166156, 165eqtrd 2813 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
167 eqid 2777 . . . . . . . . . . . . . . . 16 (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
16868a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V)
169168ralrimiva 3147 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V)
170 eqid 2777 . . . . . . . . . . . . . . . 16 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
171167, 143, 66, 67, 169, 170prdsbas3 16527 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
172149, 166, 1713eqtr4rd 2824 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (𝑎(ball‘𝐷)𝑟))
173172sqxpeqd 5387 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ((Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))) = ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
174173reseq2d 5642 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝐷 ↾ ((Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))))) = (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))))
175148, 174eqtrd 2813 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))))
176144fveq2i 6449 . . . . . . . . . . . . 13 (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
177176, 172syl5eq 2825 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (𝑎(ball‘𝐷)𝑟))
178177fveq2d 6450 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (TotBnd‘(Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))))) = (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
179140, 175, 1783eltr3d 2872 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
18047, 48, 60, 179syl12anc 827 . . . . . . . . 9 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
181 totbndss 34184 . . . . . . . . 9 (((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)) ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) ∈ (TotBnd‘𝐴))
182180, 42, 181syl2anc 579 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) ∈ (TotBnd‘𝐴))
18346, 182eqeltrrd 2859 . . . . . . 7 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐶 ∈ (TotBnd‘𝐴))
18441, 183rexlimddv 3217 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (TotBnd‘𝐴))
185184exp32 413 . . . . 5 (𝜑 → (𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
186185exlimdv 1976 . . . 4 (𝜑 → (∃𝑎 𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1876, 186syl5bi 234 . . 3 (𝜑 → (𝐴 ≠ ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1885, 187pm2.61dne 3055 . 2 (𝜑 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
1891, 188impbid2 218 1 (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wex 1823  wcel 2106  wne 2968  wral 3089  wrex 3090  Vcvv 3397  wss 3791  c0 4140   class class class wbr 4886  cmpt 4965   × cxp 5353  cres 5357   Fn wfn 6130  cfv 6135  (class class class)co 6922  Xcixp 8194  Fincfn 8241  cr 10271  0cc0 10272  *cxr 10410   < clt 10411  +crp 12137  Basecbs 16255  s cress 16256  distcds 16347  Xscprds 16492  ∞Metcxmet 20127  Metcmet 20128  ballcbl 20129  TotBndctotbnd 34173  Bndcbnd 34174
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 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
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 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-ec 8028  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-icc 12494  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-ip 16356  df-tset 16357  df-ple 16358  df-ds 16360  df-hom 16362  df-cco 16363  df-prds 16494  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-totbnd 34175  df-bnd 34186
This theorem is referenced by:  cnpwstotbnd  34204
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