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Theorem prdsbnd2 37401
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 37395 . 2 (𝐶 ∈ (TotBnd‘𝐴) → 𝐶 ∈ (Bnd‘𝐴))
2 bndmet 37387 . . . . 5 (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (Met‘𝐴))
3 0totbnd 37379 . . . . 5 (𝐴 = ∅ → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Met‘𝐴)))
42, 3imbitrrid 245 . . . 4 (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
54a1i 11 . . 3 (𝜑 → (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
6 n0 4346 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
7 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (Bnd‘𝐴))
8 eqid 2725 . . . . . . . . . . . 12 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
9 eqid 2725 . . . . . . . . . . . 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 2725 . . . . . . . . . . . 12 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
13 prdsbnd.s . . . . . . . . . . . 12 (𝜑𝑆𝑊)
14 prdsbnd.i . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
15 fvexd 6911 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
16 prdsbnd2.e . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
178, 9, 10, 11, 12, 13, 14, 15, 16prdsmet 24325 . . . . . . . . . . 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 6956 . . . . . . . . . . . . . . . 16 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2220, 21sylib 217 . . . . . . . . . . . . . . 15 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2322oveq2d 7435 . . . . . . . . . . . . . 14 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2419, 23eqtrid 2777 . . . . . . . . . . . . 13 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2524fveq2d 6900 . . . . . . . . . . . 12 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2618, 25eqtrid 2777 . . . . . . . . . . 11 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
27 prdsbnd.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑌)
2824fveq2d 6900 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2927, 28eqtrid 2777 . . . . . . . . . . . 12 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
3029fveq2d 6900 . . . . . . . . . . 11 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
3117, 26, 303eltr4d 2840 . . . . . . . . . 10 (𝜑𝐷 ∈ (Met‘𝐵))
3231adantr 479 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐷 ∈ (Met‘𝐵))
33 simpr 483 . . . . . . . . . . 11 ((𝑎𝐴𝐶 ∈ (Bnd‘𝐴)) → 𝐶 ∈ (Bnd‘𝐴))
34 prdsbnd2.c . . . . . . . . . . . 12 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
3534bnd2lem 37397 . . . . . . . . . . 11 ((𝐷 ∈ (Met‘𝐵) ∧ 𝐶 ∈ (Bnd‘𝐴)) → 𝐴𝐵)
3631, 33, 35syl2an 594 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐴𝐵)
37 simprl 769 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐴)
3836, 37sseldd 3977 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐵)
3934ssbnd 37394 . . . . . . . . 9 ((𝐷 ∈ (Met‘𝐵) ∧ 𝑎𝐵) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
4032, 38, 39syl2anc 582 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
417, 40mpbid 231 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
42 simprr 771 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
43 xpss12 5693 . . . . . . . . . . 11 ((𝐴 ⊆ (𝑎(ball‘𝐷)𝑟) ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4442, 42, 43syl2anc 582 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4544resabs1d 6013 . . . . . . . . 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, 34eqtr4di 2783 . . . . . . . 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 765 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝜑)
4838adantr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐵)
49 simprl 769 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ)
5037adantr 479 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐴)
5142, 50sseldd 3977 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ (𝑎(ball‘𝐷)𝑟))
5251ne0d 4335 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝑎(ball‘𝐷)𝑟) ≠ ∅)
5331ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (Met‘𝐵))
54 metxmet 24289 . . . . . . . . . . . . . 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 11301 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ*)
57 xbln0 24369 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑎𝐵𝑟 ∈ ℝ*) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
5855, 48, 56, 57syl3anc 1368 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
5952, 58mpbid 231 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 0 < 𝑟)
6049, 59elrpd 13053 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ+)
61 eqid 2725 . . . . . . . . . . . 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 2725 . . . . . . . . . . . 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 2725 . . . . . . . . . . . 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 2725 . . . . . . . . . . . 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 2725 . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
6714adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
68 ovex 7452 . . . . . . . . . . . . . 14 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V
69 fveq2 6896 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
70 2fveq3 6901 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (dist‘(𝑅𝑦)) = (dist‘(𝑅𝑥)))
71 2fveq3 6901 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑥)))
7271, 10eqtr4di 2783 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = 𝑉)
7372sqxpeqd 5710 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))) = (𝑉 × 𝑉))
7470, 73reseq12d 5986 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)))
7574, 11eqtr4di 2783 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = 𝐸)
7675fveq2d 6900 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))))) = (ball‘𝐸))
77 fveq2 6896 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑎𝑦) = (𝑎𝑥))
78 eqidd 2726 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥𝑟 = 𝑟)
7976, 77, 78oveq123d 7440 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟) = ((𝑎𝑥)(ball‘𝐸)𝑟))
8069, 79oveq12d 7437 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8180cbvmptv 5262 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) = (𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8268, 81fnmpti 6699 . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
85 metxmet 24289 . . . . . . . . . . . . . . . 16 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
8684, 85syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
8715ralrimiva 3135 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
8887adantr 479 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
89 simprl 769 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎𝐵)
9029adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
9189, 90eleqtrd 2827 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
928, 9, 66, 67, 88, 10, 91prdsbascl 17473 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑎𝑥) ∈ 𝑉)
9392r19.21bi 3238 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑎𝑥) ∈ 𝑉)
94 simplrr 776 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ+)
9594rpred 13056 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ)
96 blbnd 37393 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
98 ovex 7452 . . . . . . . . . . . . . . . 16 ((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V
99 xpeq12 5703 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) ∧ 𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟)) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
10099anidms 565 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
101100reseq2d 5985 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝐸 ↾ (𝑦 × 𝑦)) = (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
102 fveq2 6896 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (TotBnd‘𝑦) = (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
103101, 102eleq12d 2819 . . . . . . . . . . . . . . . . . 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 6896 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (Bnd‘𝑦) = (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
105101, 104eleq12d 2819 . . . . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . . . . 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 339 . . . . . . . . . . . . . . . 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 3536 . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
112 eqid 2725 . . . . . . . . . . . . . . . . . . 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 7004 . . . . . . . . . . . . . . . . . 18 (𝑥𝐼 → ((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
114113adantl 480 . . . . . . . . . . . . . . . . 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 6900 . . . . . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))
117 eqid 2725 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑅𝑥)) = (dist‘(𝑅𝑥))
118116, 117ressds 17399 . . . . . . . . . . . . . . . . 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, 119eqtr4di 2783 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (dist‘(𝑅𝑥)))
121114fveq2d 6900 . . . . . . . . . . . . . . . . 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 13023 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
123122ad2antll 727 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
124123adantr 479 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ*)
125 blssm 24373 . . . . . . . . . . . . . . . . . . 19 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎𝑥) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
12686, 93, 124, 125syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
127116, 10ressbas2 17226 . . . . . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = ((𝑎𝑥)(ball‘𝐸)𝑟))
130129sqxpeqd 5710 . . . . . . . . . . . . . . 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 5986 . . . . . . . . . . . . . 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 5981 . . . . . . . . . . . . . . 15 (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = (((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
133 xpss12 5693 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 ∧ ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
134126, 126, 133syl2anc 582 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
135134resabs1d 6013 . . . . . . . . . . . . . . 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, 135eqtrid 2777 . . . . . . . . . . . . . 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 2768 . . . . . . . . . . . . 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 6900 . . . . . . . . . . . . 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 2840 . . . . . . . . . . . 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 37400 . . . . . . . . . . 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 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
142 eqidd 2726 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
143 eqid 2725 . . . . . . . . . . . . 13 (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
14481oveq2i 7430 . . . . . . . . . . . . . 14 (𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
145144fveq2i 6899 . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . 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 24326 . . . . . . . . . . . 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 8931 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
15069cbvmptv 5262 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
151150oveq2i 7430 . . . . . . . . . . . . . . . . . . . . 21 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
15224, 151eqtr4di 2783 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
153152fveq2d 6900 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15418, 153eqtrid 2777 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
155154fveq2d 6900 . . . . . . . . . . . . . . . . 17 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
156155oveqdr 7447 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
157 eqid 2725 . . . . . . . . . . . . . . . . 17 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
158 eqid 2725 . . . . . . . . . . . . . . . . 17 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
159152fveq2d 6900 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16027, 159eqtrid 2777 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
161160adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16289, 161eleqtrd 2827 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
163 rpgt0 13026 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+ → 0 < 𝑟)
164163ad2antll 727 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
165151, 157, 10, 11, 158, 66, 67, 146, 86, 162, 123, 164prdsbl 24449 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
166156, 165eqtrd 2765 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
167 eqid 2725 . . . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V)
170 eqid 2725 . . . . . . . . . . . . . . . 16 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
171167, 143, 66, 67, 169, 170prdsbas3 17471 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
172149, 166, 1713eqtr4rd 2776 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (𝑎(ball‘𝐷)𝑟))
173172sqxpeqd 5710 . . . . . . . . . . . . 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 5985 . . . . . . . . . . . 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 2765 . . . . . . . . . . 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 6899 . . . . . . . . . . . . 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, 172eqtrid 2777 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (𝑎(ball‘𝐷)𝑟))
178177fveq2d 6900 . . . . . . . . . . 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 2839 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
18047, 48, 60, 179syl12anc 835 . . . . . . . . 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 37383 . . . . . . . . 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 582 . . . . . . . 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 2826 . . . . . . 7 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐶 ∈ (TotBnd‘𝐴))
18441, 183rexlimddv 3150 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (TotBnd‘𝐴))
185184exp32 419 . . . . 5 (𝜑 → (𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
186185exlimdv 1928 . . . 4 (𝜑 → (∃𝑎 𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1876, 186biimtrid 241 . . 3 (𝜑 → (𝐴 ≠ ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1885, 187pm2.61dne 3017 . 2 (𝜑 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
1891, 188impbid2 225 1 (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  wrex 3059  Vcvv 3461  wss 3944  c0 4322   class class class wbr 5149  cmpt 5232   × cxp 5676  cres 5680   Fn wfn 6544  cfv 6549  (class class class)co 7419  Xcixp 8916  Fincfn 8964  cr 11144  0cc0 11145  *cxr 11284   < clt 11285  +crp 13014  Basecbs 17188  s cress 17217  distcds 17250  Xscprds 17435  ∞Metcxmet 21286  Metcmet 21287  ballcbl 21288  TotBndctotbnd 37372  Bndcbnd 37373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-ec 8727  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-icc 13371  df-fz 13525  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17189  df-ress 17218  df-plusg 17254  df-mulr 17255  df-sca 17257  df-vsca 17258  df-ip 17259  df-tset 17260  df-ple 17261  df-ds 17263  df-hom 17265  df-cco 17266  df-prds 17437  df-psmet 21293  df-xmet 21294  df-met 21295  df-bl 21296  df-totbnd 37374  df-bnd 37385
This theorem is referenced by:  cnpwstotbnd  37403
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