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Theorem prdsbnd2 33223
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 33217 . 2 (𝐶 ∈ (TotBnd‘𝐴) → 𝐶 ∈ (Bnd‘𝐴))
2 bndmet 33209 . . . . 5 (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (Met‘𝐴))
3 0totbnd 33201 . . . . 5 (𝐴 = ∅ → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Met‘𝐴)))
42, 3syl5ibr 236 . . . 4 (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
54a1i 11 . . 3 (𝜑 → (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
6 n0 3907 . . . 4 (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎𝐴)
7 simprr 795 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (Bnd‘𝐴))
8 eqid 2621 . . . . . . . . . . . 12 (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
9 eqid 2621 . . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
13 prdsbnd.s . . . . . . . . . . . 12 (𝜑𝑆𝑊)
14 prdsbnd.i . . . . . . . . . . . 12 (𝜑𝐼 ∈ Fin)
15 fvex 6158 . . . . . . . . . . . . 13 (𝑅𝑥) ∈ V
1615a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → (𝑅𝑥) ∈ V)
17 prdsbnd2.e . . . . . . . . . . . 12 ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
188, 9, 10, 11, 12, 13, 14, 16, 17prdsmet 22085 . . . . . . . . . . 11 (𝜑 → (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
19 prdsbnd.d . . . . . . . . . . . 12 𝐷 = (dist‘𝑌)
20 prdsbnd.y . . . . . . . . . . . . . 14 𝑌 = (𝑆Xs𝑅)
21 prdsbnd.r . . . . . . . . . . . . . . . 16 (𝜑𝑅 Fn 𝐼)
22 dffn5 6198 . . . . . . . . . . . . . . . 16 (𝑅 Fn 𝐼𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2321, 22sylib 208 . . . . . . . . . . . . . . 15 (𝜑𝑅 = (𝑥𝐼 ↦ (𝑅𝑥)))
2423oveq2d 6620 . . . . . . . . . . . . . 14 (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2520, 24syl5eq 2667 . . . . . . . . . . . . 13 (𝜑𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
2625fveq2d 6152 . . . . . . . . . . . 12 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
2719, 26syl5eq 2667 . . . . . . . . . . 11 (𝜑𝐷 = (dist‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
28 prdsbnd.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑌)
2925fveq2d 6152 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
3028, 29syl5eq 2667 . . . . . . . . . . . 12 (𝜑𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
3130fveq2d 6152 . . . . . . . . . . 11 (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))))
3218, 27, 313eltr4d 2713 . . . . . . . . . 10 (𝜑𝐷 ∈ (Met‘𝐵))
3332adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐷 ∈ (Met‘𝐵))
34 simpr 477 . . . . . . . . . . 11 ((𝑎𝐴𝐶 ∈ (Bnd‘𝐴)) → 𝐶 ∈ (Bnd‘𝐴))
35 prdsbnd2.c . . . . . . . . . . . 12 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
3635bnd2lem 33219 . . . . . . . . . . 11 ((𝐷 ∈ (Met‘𝐵) ∧ 𝐶 ∈ (Bnd‘𝐴)) → 𝐴𝐵)
3732, 34, 36syl2an 494 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐴𝐵)
38 simprl 793 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐴)
3937, 38sseldd 3584 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝑎𝐵)
4035ssbnd 33216 . . . . . . . . 9 ((𝐷 ∈ (Met‘𝐵) ∧ 𝑎𝐵) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
4133, 39, 40syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
427, 41mpbid 222 . . . . . . 7 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
43 simprr 795 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
44 xpss12 5186 . . . . . . . . . . 11 ((𝐴 ⊆ (𝑎(ball‘𝐷)𝑟) ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4543, 43, 44syl2anc 692 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
4645resabs1d 5387 . . . . . . . . 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)))
4746, 35syl6eqr 2673 . . . . . . . 8 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) = 𝐶)
48 simpll 789 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝜑)
4939adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐵)
50 simprl 793 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ)
5138adantr 481 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎𝐴)
5243, 51sseldd 3584 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ (𝑎(ball‘𝐷)𝑟))
53 ne0i 3897 . . . . . . . . . . . . 13 (𝑎 ∈ (𝑎(ball‘𝐷)𝑟) → (𝑎(ball‘𝐷)𝑟) ≠ ∅)
5452, 53syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝑎(ball‘𝐷)𝑟) ≠ ∅)
5532ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (Met‘𝐵))
56 metxmet 22049 . . . . . . . . . . . . . 14 (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
5755, 56syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (∞Met‘𝐵))
5850rexrd 10033 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ*)
59 xbln0 22129 . . . . . . . . . . . . 13 ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑎𝐵𝑟 ∈ ℝ*) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
6057, 49, 58, 59syl3anc 1323 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
6154, 60mpbid 222 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 0 < 𝑟)
6250, 61elrpd 11813 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ+)
63 eqid 2621 . . . . . . . . . . . 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))))
64 eqid 2621 . . . . . . . . . . . 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)))))
65 eqid 2621 . . . . . . . . . . . 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))
66 eqid 2621 . . . . . . . . . . . 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))))
67 eqid 2621 . . . . . . . . . . . 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)))))
6813adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑆𝑊)
6914adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
70 ovex 6632 . . . . . . . . . . . . . 14 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V
71 fveq2 6148 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑅𝑦) = (𝑅𝑥))
7271fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → (dist‘(𝑅𝑦)) = (dist‘(𝑅𝑥)))
7371fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑥)))
7473, 10syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑥 → (Base‘(𝑅𝑦)) = 𝑉)
7574sqxpeqd 5101 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑥 → ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))) = (𝑉 × 𝑉))
7672, 75reseq12d 5357 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)))
7776, 11syl6eqr 2673 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑥 → ((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))) = 𝐸)
7877fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦))))) = (ball‘𝐸))
79 fveq2 6148 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 → (𝑎𝑦) = (𝑎𝑥))
80 eqidd 2622 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥𝑟 = 𝑟)
8178, 79, 80oveq123d 6625 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟) = ((𝑎𝑥)(ball‘𝐸)𝑟))
8271, 81oveq12d 6622 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8382cbvmptv 4710 . . . . . . . . . . . . . 14 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) = (𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
8470, 83fnmpti 5979 . . . . . . . . . . . . 13 (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) Fn 𝐼
8584a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))) Fn 𝐼)
8617adantlr 750 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))
87 metxmet 22049 . . . . . . . . . . . . . . . 16 (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
8886, 87syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝐸 ∈ (∞Met‘𝑉))
8916ralrimiva 2960 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
9089adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑅𝑥) ∈ V)
91 simprl 793 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎𝐵)
9230adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
9391, 92eleqtrd 2700 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))))
948, 9, 68, 69, 90, 10, 93prdsbascl 16064 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 (𝑎𝑥) ∈ 𝑉)
9594r19.21bi 2927 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑎𝑥) ∈ 𝑉)
96 simplrr 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ+)
9796rpred 11816 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ)
98 blbnd 33215 . . . . . . . . . . . . . . 15 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎𝑥) ∈ 𝑉𝑟 ∈ ℝ) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
9988, 95, 97, 98syl3anc 1323 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
100 ovex 6632 . . . . . . . . . . . . . . . 16 ((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V
101 xpeq12 5094 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) ∧ 𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟)) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
102101anidms 676 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝑦 × 𝑦) = (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
103102reseq2d 5356 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (𝐸 ↾ (𝑦 × 𝑦)) = (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
104 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (TotBnd‘𝑦) = (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
105103, 104eleq12d 2692 . . . . . . . . . . . . . . . . . 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))))
106 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑎𝑥)(ball‘𝐸)𝑟) → (Bnd‘𝑦) = (Bnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
107103, 106eleq12d 2692 . . . . . . . . . . . . . . . . . 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))))
108105, 107bibi12d 335 . . . . . . . . . . . . . . . . 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)))))
109108imbi2d 330 . . . . . . . . . . . . . . . 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))))))
110 prdsbnd2.m . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
111100, 109, 110vtocl 3245 . . . . . . . . . . . . . . 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))))
112111adantlr 750 . . . . . . . . . . . . . 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))))
11399, 112mpbird 247 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎𝑥)(ball‘𝐸)𝑟)))
114 eqid 2621 . . . . . . . . . . . . . . . . . . 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)))
11582, 114, 70fvmpt 6239 . . . . . . . . . . . . . . . . . 18 (𝑥𝐼 → ((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
116115adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
117116fveq2d 6152 . . . . . . . . . . . . . . . 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))))
118 eqid 2621 . . . . . . . . . . . . . . . . . 18 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) = ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))
119 eqid 2621 . . . . . . . . . . . . . . . . . 18 (dist‘(𝑅𝑥)) = (dist‘(𝑅𝑥))
120118, 119ressds 15994 . . . . . . . . . . . . . . . . 17 (((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V → (dist‘(𝑅𝑥)) = (dist‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
121100, 120ax-mp 5 . . . . . . . . . . . . . . . 16 (dist‘(𝑅𝑥)) = (dist‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
122117, 121syl6eqr 2673 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (dist‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = (dist‘(𝑅𝑥)))
123116fveq2d 6152 . . . . . . . . . . . . . . . . 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))))
124 rpxr 11784 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
125124ad2antll 764 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
126125adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → 𝑟 ∈ ℝ*)
127 blssm 22133 . . . . . . . . . . . . . . . . . . 19 ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎𝑥) ∈ 𝑉𝑟 ∈ ℝ*) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
12888, 95, 126, 127syl3anc 1323 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
129118, 10ressbas2 15852 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 → ((𝑎𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
130128, 129syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
131123, 130eqtr4d 2658 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (Base‘((𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))‘𝑥)) = ((𝑎𝑥)(ball‘𝐸)𝑟))
132131sqxpeqd 5101 . . . . . . . . . . . . . . 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)))
133122, 132reseq12d 5357 . . . . . . . . . . . . . 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))))
13411reseq1i 5352 . . . . . . . . . . . . . . 15 (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = (((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)))
135 xpss12 5186 . . . . . . . . . . . . . . . . 17 ((((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 ∧ ((𝑎𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
136128, 128, 135syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
137136resabs1d 5387 . . . . . . . . . . . . . . 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))))
138134, 137syl5eq 2667 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝐸 ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))) = ((dist‘(𝑅𝑥)) ↾ (((𝑎𝑥)(ball‘𝐸)𝑟) × ((𝑎𝑥)(ball‘𝐸)𝑟))))
139133, 138eqtr4d 2658 . . . . . . . . . . . . 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))))
140131fveq2d 6152 . . . . . . . . . . . . 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)))
141113, 139, 1403eltr4d 2713 . . . . . . . . . . . 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))))
14263, 64, 65, 66, 67, 68, 69, 85, 141prdstotbnd 33222 . . . . . . . . . . 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)))))))
14325adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑌 = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥))))
144 eqidd 2622 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
145 eqid 2621 . . . . . . . . . . . . 13 (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
14683oveq2i 6615 . . . . . . . . . . . . . 14 (𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
147146fveq2i 6151 . . . . . . . . . . . . 13 (dist‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (dist‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
14815a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → (𝑅𝑥) ∈ V)
149100a1i 11 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑎𝑥)(ball‘𝐸)𝑟) ∈ V)
150143, 144, 145, 19, 147, 68, 68, 69, 148, 149ressprdsds 22086 . . . . . . . . . . . 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))))))))
151130ixpeq2dva 7867 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
15271cbvmptv 4710 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐼 ↦ (𝑅𝑦)) = (𝑥𝐼 ↦ (𝑅𝑥))
153152oveq2i 6615 . . . . . . . . . . . . . . . . . . . . 21 (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))) = (𝑆Xs(𝑥𝐼 ↦ (𝑅𝑥)))
15425, 153syl6eqr 2673 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑌 = (𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
155154fveq2d 6152 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
15619, 155syl5eq 2667 . . . . . . . . . . . . . . . . . 18 (𝜑𝐷 = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
157156fveq2d 6152 . . . . . . . . . . . . . . . . 17 (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))))
158157oveqdr 6628 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟))
159 eqid 2621 . . . . . . . . . . . . . . . . 17 (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
160 eqid 2621 . . . . . . . . . . . . . . . . 17 (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))) = (dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦))))
161154fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16228, 161syl5eq 2667 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
163162adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
16491, 163eleqtrd 2700 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 𝑎 ∈ (Base‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))
165 rpgt0 11788 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+ → 0 < 𝑟)
166165ad2antll 764 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → 0 < 𝑟)
167153, 159, 10, 11, 160, 68, 69, 148, 88, 164, 125, 166prdsbl 22206 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘(dist‘(𝑆Xs(𝑦𝐼 ↦ (𝑅𝑦)))))𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
168158, 167eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝑎(ball‘𝐷)𝑟) = X𝑥𝐼 ((𝑎𝑥)(ball‘𝐸)𝑟))
169 eqid 2621 . . . . . . . . . . . . . . . 16 (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))) = (𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
17070a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) ∧ 𝑥𝐼) → ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V)
171170ralrimiva 2960 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → ∀𝑥𝐼 ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)) ∈ V)
172 eqid 2621 . . . . . . . . . . . . . . . 16 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))) = (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))
173169, 145, 68, 69, 171, 172prdsbas3 16062 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = X𝑥𝐼 (Base‘((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))
174151, 168, 1733eqtr4rd 2666 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟))))) = (𝑎(ball‘𝐷)𝑟))
175174sqxpeqd 5101 . . . . . . . . . . . . 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)))
176175reseq2d 5356 . . . . . . . . . . . 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))))
177150, 176eqtrd 2655 . . . . . . . . . . 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))))
178146fveq2i 6151 . . . . . . . . . . . . 13 (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (Base‘(𝑆Xs(𝑥𝐼 ↦ ((𝑅𝑥) ↾s ((𝑎𝑥)(ball‘𝐸)𝑟)))))
179178, 174syl5eq 2667 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟))))) = (𝑎(ball‘𝐷)𝑟))
180179fveq2d 6152 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (TotBnd‘(Base‘(𝑆Xs(𝑦𝐼 ↦ ((𝑅𝑦) ↾s ((𝑎𝑦)(ball‘((dist‘(𝑅𝑦)) ↾ ((Base‘(𝑅𝑦)) × (Base‘(𝑅𝑦)))))𝑟)))))) = (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
181142, 177, 1803eltr3d 2712 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑟 ∈ ℝ+)) → (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
18248, 49, 62, 181syl12anc 1321 . . . . . . . . 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)))
183 totbndss 33205 . . . . . . . . 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))
184182, 43, 183syl2anc 692 . . . . . . . 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))
18547, 184eqeltrrd 2699 . . . . . . 7 (((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐶 ∈ (TotBnd‘𝐴))
18642, 185rexlimddv 3028 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (TotBnd‘𝐴))
187186exp32 630 . . . . 5 (𝜑 → (𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
188187exlimdv 1858 . . . 4 (𝜑 → (∃𝑎 𝑎𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1896, 188syl5bi 232 . . 3 (𝜑 → (𝐴 ≠ ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
1905, 189pm2.61dne 2876 . 2 (𝜑 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
1911, 190impbid2 216 1 (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  wss 3555  c0 3891   class class class wbr 4613  cmpt 4673   × cxp 5072  cres 5076   Fn wfn 5842  cfv 5847  (class class class)co 6604  Xcixp 7852  Fincfn 7899  cr 9879  0cc0 9880  *cxr 10017   < clt 10018  +crp 11776  Basecbs 15781  s cress 15782  distcds 15871  Xscprds 16027  ∞Metcxmt 19650  Metcme 19651  ballcbl 19652  TotBndctotbnd 33194  Bndcbnd 33195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-icc 12124  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-prds 16029  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-totbnd 33196  df-bnd 33207
This theorem is referenced by:  cnpwstotbnd  33225
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