prdsbnd2 - Mathbox for Jeff Madsen

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Theorem prdsbnd2 35949

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 = (𝑆X_s𝑅)
prdsbnd.b	⊢ 𝐵 = (Base‘𝑌)
prdsbnd.v	⊢ 𝑉 = (Base‘(𝑅‘𝑥))
prdsbnd.e	⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
prdsbnd.d	⊢ 𝐷 = (dist‘𝑌)
prdsbnd.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsbnd.i	⊢ (𝜑 → 𝐼 ∈ Fin)
prdsbnd.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsbnd2.c	⊢ 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
prdsbnd2.e	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
prdsbnd2.m	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((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.

Step	Hyp	Ref	Expression
1		totbndbnd 35943	. 2 ⊢ (𝐶 ∈ (TotBnd‘𝐴) → 𝐶 ∈ (Bnd‘𝐴))
2		bndmet 35935	. . . . 5 ⊢ (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (Met‘𝐴))
3		0totbnd 35927	. . . . 5 ⊢ (𝐴 = ∅ → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Met‘𝐴)))
4	2, 3	syl5ibr 245	. . . 4 ⊢ (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
5	4	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 = ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
6		n0 4286	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐴)
7		simprr 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (Bnd‘𝐴))
8		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
9		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
10		prdsbnd.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘(𝑅‘𝑥))
11		prdsbnd.e	. . . . . . . . . . . 12 ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
12		eqid 2740	. . . . . . . . . . . 12 ⊢ (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
13		prdsbnd.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
14		prdsbnd.i	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ Fin)
15		fvexd 6786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
16		prdsbnd2.e	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
17	8, 9, 10, 11, 12, 13, 14, 15, 16	prdsmet 23521	. . . . . . . . . . 11 ⊢ (𝜑 → (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) ∈ (Met‘(Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
18		prdsbnd.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝑌)
19		prdsbnd.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (𝑆X_s𝑅)
20		prdsbnd.r	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 Fn 𝐼)
21		dffn5 6825	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
22	20, 21	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
23	22	oveq2d 7287	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆X_s𝑅) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
24	19, 23	eqtrid 2792	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
25	24	fveq2d 6775	. . . . . . . . . . . 12 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
26	18, 25	eqtrid 2792	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
27		prdsbnd.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑌)
28	24	fveq2d 6775	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
29	27, 28	eqtrid 2792	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
30	29	fveq2d 6775	. . . . . . . . . . 11 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
31	17, 26, 30	3eltr4d 2856	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
32	31	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝐷 ∈ (Met‘𝐵))
33		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴)) → 𝐶 ∈ (Bnd‘𝐴))
34		prdsbnd2.c	. . . . . . . . . . . 12 ⊢ 𝐶 = (𝐷 ↾ (𝐴 × 𝐴))
35	34	bnd2lem 35945	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝐶 ∈ (Bnd‘𝐴)) → 𝐴 ⊆ 𝐵)
36	31, 33, 35	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝐴 ⊆ 𝐵)
37		simprl 768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝑎 ∈ 𝐴)
38	36, 37	sseldd 3927	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝑎 ∈ 𝐵)
39	34	ssbnd 35942	. . . . . . . . 9 ⊢ ((𝐷 ∈ (Met‘𝐵) ∧ 𝑎 ∈ 𝐵) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
40	32, 38, 39	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → (𝐶 ∈ (Bnd‘𝐴) ↔ ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)))
41	7, 40	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → ∃𝑟 ∈ ℝ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
42		simprr 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))
43		xpss12 5605	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝑎(ball‘𝐷)𝑟) ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟)) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
44	42, 42, 43	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝐴 × 𝐴) ⊆ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
45	44	resabs1d 5921	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) = (𝐷 ↾ (𝐴 × 𝐴)))
46	45, 34	eqtr4di 2798	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) = 𝐶)
47		simpll 764	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝜑)
48	38	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ 𝐵)
49		simprl 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ)
50	37	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ 𝐴)
51	42, 50	sseldd 3927	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑎 ∈ (𝑎(ball‘𝐷)𝑟))
52	51	ne0d 4275	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → (𝑎(ball‘𝐷)𝑟) ≠ ∅)
53	31	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (Met‘𝐵))
54		metxmet 23485	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐷 ∈ (∞Met‘𝐵))
56	49	rexrd 11026	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ^*)
57		xbln0 23565	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ^*) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
58	55, 48, 56, 57	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝑎(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
59	52, 58	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 0 < 𝑟)
60	49, 59	elrpd 12768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝑟 ∈ ℝ⁺)
61		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))) = (𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))
62		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))))
63		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) = (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥))
64		eqid 2740	. . . . . . . . . . . 12 ⊢ ((dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)))) = ((dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥))))
65		eqid 2740	. . . . . . . . . . . 12 ⊢ (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))))
66	13	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑆 ∈ 𝑊)
67	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝐼 ∈ Fin)
68		ovex 7304	. . . . . . . . . . . . . 14 ⊢ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)) ∈ V
69		fveq2 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥))
70		2fveq3 6776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (dist‘(𝑅‘𝑦)) = (dist‘(𝑅‘𝑥)))
71		2fveq3 6776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑥)))
72	71, 10	eqtr4di 2798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → (Base‘(𝑅‘𝑦)) = 𝑉)
73	72	sqxpeqd 5622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦))) = (𝑉 × 𝑉))
74	70, 73	reseq12d 5891	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → ((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))) = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)))
75	74, 11	eqtr4di 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑥 → ((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))) = 𝐸)
76	75	fveq2d 6775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦))))) = (ball‘𝐸))
77		fveq2 6771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → (𝑎‘𝑦) = (𝑎‘𝑥))
78		eqidd 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑥 → 𝑟 = 𝑟)
79	76, 77, 78	oveq123d 7292	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟) = ((𝑎‘𝑥)(ball‘𝐸)𝑟))
80	69, 79	oveq12d 7289	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)) = ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
81	80	cbvmptv 5192	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))) = (𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
82	68, 81	fnmpti 6574	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))) Fn 𝐼
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))) Fn 𝐼)
84	16	adantlr 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
85		metxmet 23485	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
86	84, 85	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
87	15	ralrimiva 3110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
88	87	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → ∀𝑥 ∈ 𝐼 (𝑅‘𝑥) ∈ V)
89		simprl 768	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑎 ∈ 𝐵)
90	29	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝐵 = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
91	89, 90	eleqtrd 2843	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑎 ∈ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
92	8, 9, 66, 67, 88, 10, 91	prdsbascl 17192	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → ∀𝑥 ∈ 𝐼 (𝑎‘𝑥) ∈ 𝑉)
93	92	r19.21bi 3135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (𝑎‘𝑥) ∈ 𝑉)
94		simplrr 775	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → 𝑟 ∈ ℝ⁺)
95	94	rpred 12771	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → 𝑟 ∈ ℝ)
96		blbnd 35941	. . . . . . . . . . . . . . 15 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎‘𝑥) ∈ 𝑉 ∧ 𝑟 ∈ ℝ) → (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
97	86, 93, 95, 96	syl3anc 1370	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
98		ovex 7304	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎‘𝑥)(ball‘𝐸)𝑟) ∈ V
99		xpeq12 5615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) ∧ 𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟)) → (𝑦 × 𝑦) = (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
100	99	anidms 567	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (𝑦 × 𝑦) = (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
101	100	reseq2d 5890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (𝐸 ↾ (𝑦 × 𝑦)) = (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
102		fveq2 6771	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (TotBnd‘𝑦) = (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
103	101, 102	eleq12d 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟))))
104		fveq2 6771	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (Bnd‘𝑦) = (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
105	101, 104	eleq12d 2835	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟))))
106	103, 105	bibi12d 346	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)) ↔ ((𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))))
107	106	imbi2d 341	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((𝑎‘𝑥)(ball‘𝐸)𝑟) → (((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))) ↔ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟))))))
108		prdsbnd2.m	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))
109	98, 107, 108	vtocl 3497	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟))))
110	109	adantlr 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)) ↔ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (Bnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟))))
111	97, 110	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) ∈ (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
112		eqid 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))) = (𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))
113	80, 112, 68	fvmpt 6872	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝐼 → ((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥) = ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
114	113	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥) = ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
115	114	fveq2d 6775	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) = (dist‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
116		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)) = ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))
117		eqid 2740	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘(𝑅‘𝑥)) = (dist‘(𝑅‘𝑥))
118	116, 117	ressds 17118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎‘𝑥)(ball‘𝐸)𝑟) ∈ V → (dist‘(𝑅‘𝑥)) = (dist‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
119	98, 118	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (dist‘(𝑅‘𝑥)) = (dist‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
120	115, 119	eqtr4di 2798	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) = (dist‘(𝑅‘𝑥)))
121	114	fveq2d 6775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) = (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
122		rpxr 12738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
123	122	ad2antll 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ^*)
124	123	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → 𝑟 ∈ ℝ^*)
125		blssm 23569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎‘𝑥) ∈ 𝑉 ∧ 𝑟 ∈ ℝ^*) → ((𝑎‘𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
126	86, 93, 124, 125	syl3anc 1370	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝑎‘𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉)
127	116, 10	ressbas2 16947	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎‘𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 → ((𝑎‘𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝑎‘𝑥)(ball‘𝐸)𝑟) = (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
129	121, 128	eqtr4d 2783	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) = ((𝑎‘𝑥)(ball‘𝐸)𝑟))
130	129	sqxpeqd 5622	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥))) = (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
131	120, 130	reseq12d 5891	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)))) = ((dist‘(𝑅‘𝑥)) ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
132	11	reseq1i 5886	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) = (((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
133		xpss12 5605	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑎‘𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉 ∧ ((𝑎‘𝑥)(ball‘𝐸)𝑟) ⊆ 𝑉) → (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
134	126, 126, 133	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟)) ⊆ (𝑉 × 𝑉))
135	134	resabs1d 5921	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) = ((dist‘(𝑅‘𝑥)) ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
136	132, 135	eqtrid 2792	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))) = ((dist‘(𝑅‘𝑥)) ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
137	131, 136	eqtr4d 2783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)))) = (𝐸 ↾ (((𝑎‘𝑥)(ball‘𝐸)𝑟) × ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
138	129	fveq2d 6775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (TotBnd‘(Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥))) = (TotBnd‘((𝑎‘𝑥)(ball‘𝐸)𝑟)))
139	111, 137, 138	3eltr4d 2856	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((dist‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) ↾ ((Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)) × (Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥)))) ∈ (TotBnd‘(Base‘((𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))‘𝑥))))
140	61, 62, 63, 64, 65, 66, 67, 83, 139	prdstotbnd 35948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) ∈ (TotBnd‘(Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))))))
141	24	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑌 = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
142		eqidd 2741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))))
143		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))))
144	81	oveq2i 7282	. . . . . . . . . . . . . 14 ⊢ (𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
145	144	fveq2i 6774	. . . . . . . . . . . . 13 ⊢ (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (dist‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))))
146		fvexd 6786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
147	98	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝑎‘𝑥)(ball‘𝐸)𝑟) ∈ V)
148	141, 142, 143, 18, 145, 66, 66, 67, 146, 147	ressprdsds 23522	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (𝐷 ↾ ((Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))))))
149	128	ixpeq2dva 8683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → X𝑥 ∈ 𝐼 ((𝑎‘𝑥)(ball‘𝐸)𝑟) = X𝑥 ∈ 𝐼 (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
150	69	cbvmptv 5192	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)) = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))
151	150	oveq2i 7282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
152	24, 151	eqtr4di 2798	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑌 = (𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
153	152	fveq2d 6775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
154	18, 153	eqtrid 2792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
155	154	fveq2d 6775	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))))
156	155	oveqdr 7299	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝑎(ball‘𝐷)𝑟) = (𝑎(ball‘(dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟))
157		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
158		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
159	152	fveq2d 6775	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
160	27, 159	eqtrid 2792	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
161	160	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝐵 = (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
162	89, 161	eleqtrd 2843	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 𝑎 ∈ (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
163		rpgt0 12741	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ⁺ → 0 < 𝑟)
164	163	ad2antll 726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → 0 < 𝑟)
165	151, 157, 10, 11, 158, 66, 67, 146, 86, 162, 123, 164	prdsbl 23645	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝑎(ball‘(dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟) = X𝑥 ∈ 𝐼 ((𝑎‘𝑥)(ball‘𝐸)𝑟))
166	156, 165	eqtrd 2780	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝑎(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑎‘𝑥)(ball‘𝐸)𝑟))
167		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))) = (𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
168	68	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑥 ∈ 𝐼) → ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)) ∈ V)
169	168	ralrimiva 3110	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → ∀𝑥 ∈ 𝐼 ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)) ∈ V)
170		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))) = (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))
171	167, 143, 66, 67, 169, 170	prdsbas3 17190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) = X𝑥 ∈ 𝐼 (Base‘((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))
172	149, 166, 171	3eqtr4rd 2791	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) = (𝑎(ball‘𝐷)𝑟))
173	172	sqxpeqd 5622	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → ((Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))))) = ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟)))
174	173	reseq2d 5890	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝐷 ↾ ((Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))) × (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟))))))) = (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))))
175	148, 174	eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (dist‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))))
176	144	fveq2i 6774	. . . . . . . . . . . . 13 ⊢ (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (Base‘(𝑆X_s(𝑥 ∈ 𝐼 ↦ ((𝑅‘𝑥) ↾_s ((𝑎‘𝑥)(ball‘𝐸)𝑟)))))
177	176, 172	eqtrid 2792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟))))) = (𝑎(ball‘𝐷)𝑟))
178	177	fveq2d 6775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (TotBnd‘(Base‘(𝑆X_s(𝑦 ∈ 𝐼 ↦ ((𝑅‘𝑦) ↾_s ((𝑎‘𝑦)(ball‘((dist‘(𝑅‘𝑦)) ↾ ((Base‘(𝑅‘𝑦)) × (Base‘(𝑅‘𝑦)))))𝑟)))))) = (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
179	140, 175, 178	3eltr3d 2855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑟 ∈ ℝ⁺)) → (𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ∈ (TotBnd‘(𝑎(ball‘𝐷)𝑟)))
180	47, 48, 60, 179	syl12anc 834	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (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 35931	. . . . . . . . 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))
182	180, 42, 181	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → ((𝐷 ↾ ((𝑎(ball‘𝐷)𝑟) × (𝑎(ball‘𝐷)𝑟))) ↾ (𝐴 × 𝐴)) ∈ (TotBnd‘𝐴))
183	46, 182	eqeltrrd 2842	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) ∧ (𝑟 ∈ ℝ ∧ 𝐴 ⊆ (𝑎(ball‘𝐷)𝑟))) → 𝐶 ∈ (TotBnd‘𝐴))
184	41, 183	rexlimddv 3222	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝐶 ∈ (Bnd‘𝐴))) → 𝐶 ∈ (TotBnd‘𝐴))
185	184	exp32 421	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
186	185	exlimdv 1940	. . . 4 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ 𝐴 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
187	6, 186	syl5bi 241	. . 3 ⊢ (𝜑 → (𝐴 ≠ ∅ → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴))))
188	5, 187	pm2.61dne 3033	. 2 ⊢ (𝜑 → (𝐶 ∈ (Bnd‘𝐴) → 𝐶 ∈ (TotBnd‘𝐴)))
189	1, 188	impbid2 225	1 ⊢ (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∃wex 1786 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 Vcvv 3431 ⊆ wss 3892 ∅c0 4262 class class class wbr 5079 ↦ cmpt 5162 × cxp 5588 ↾ cres 5592 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 Xcixp 8668 Fincfn 8716 ℝcr 10871 0cc0 10872 ℝ^*cxr 11009 < clt 11010 ℝ⁺crp 12729 Basecbs 16910 ↾_s cress 16939 distcds 16969 X_scprds 17154 ∞Metcxmet 20580 Metcmet 20581 ballcbl 20582 TotBndctotbnd 35920 Bndcbnd 35921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-ec 8483 df-map 8600 df-pm 8601 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-icc 13085 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-hom 16984 df-cco 16985 df-prds 17156 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-totbnd 35922 df-bnd 35933

This theorem is referenced by: cnpwstotbnd 35951

W3C validator