Theorem ovncvr2 39501

Description: 𝐵 and 𝑇 are the left and right side of a cover of 𝐴. This cover is made of n-dimensional half open intervals, and approximates the n-dimensional Lebesgue outer volume of 𝐴. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
ovncvr2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovncvr2.a	⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
ovncvr2.e	⊢ (𝜑 → 𝐸 ∈ ℝ+)
ovncvr2.c	⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
ovncvr2.l	⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
ovncvr2.d	⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))
ovncvr2.i	⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸))
ovncvr2.b	⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
ovncvr2.t	⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))

Assertion

Ref	Expression
ovncvr2	⊢ (𝜑 → (((𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))

Distinct variable groups:   𝐴,𝑎,𝑖,𝑟   𝐴,𝑙,𝑎   𝐵,ℎ   𝐶,𝑎,𝑖,𝑟   𝑖,𝐸,𝑟   ℎ,𝐼,𝑗,𝑘   𝑖,𝐼,𝑗   𝐼,𝑙,𝑗,𝑘   𝐿,𝑎,𝑖,𝑟   𝑇,ℎ   𝑋,𝑎,𝑖,𝑗,𝑟   ℎ,𝑋,𝑘   𝑋,𝑙   𝑘,𝑎,𝜑,𝑗   𝜑,ℎ   𝜑,𝑟

Allowed substitution hints:   𝜑(𝑖,𝑙)   𝐴(ℎ,𝑗,𝑘)   𝐵(𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝐶(ℎ,𝑗,𝑘,𝑙)   𝐷(ℎ,𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝑇(𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝐸(ℎ,𝑗,𝑘,𝑎,𝑙)   𝐼(𝑟,𝑎)   𝐿(ℎ,𝑗,𝑘,𝑙)

Proof of Theorem ovncvr2

Step	Hyp	Ref	Expression
1		ovncvr2.c	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
2	1	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}))
3		sseq1 3589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
4	3	rabbidv 3164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐴 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
5	4	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
6		ovncvr2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑𝑚 𝑋))
7		ovex 6577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ ↑𝑚 𝑋) ∈ V
8	7	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
9	8, 6	ssexd 4733	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
10		elpwg 4116	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋)))
11	9, 10	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋)))
12	6, 11	mpbird 246	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
13		ovex 6577	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∈ V
14	13	rabex 4740	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V)
16	2, 5, 12, 15	fvmptd 6197	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
17		ssrab2 3650	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
19	16, 18	eqsstrd 3602	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘𝐴) ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
20		ovncvr2.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸))
21		ovncvr2.d	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))
22	21	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)})))
23		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝐴 → (𝐶‘𝑎) = (𝐶‘𝐴))
24	23	eleq2d 2673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝐴 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝐴)))
25		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
26	25	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
27	26	breq2d 4595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝐴 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)))
28	24, 27	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝐴 → ((𝑖 ∈ (𝐶‘𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)) ↔ (𝑖 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))))
29	28	rabbidva2 3162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝐴 → {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)})
30	29	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝐴 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31	30	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
32		rpex 38503	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ+ ∈ V
33	32	mptex 6390	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V
34	33	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V)
35	22, 31, 12, 34	fvmptd 6197	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
36		oveq2 6557	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
37	36	breq2d 4595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
38	37	rabbidv 3164	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = 𝐸 → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
39	38	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑟 = 𝐸) → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
40		ovncvr2.e	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
41		fvex 6113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶‘𝐴) ∈ V
42	41	rabex 4740	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V)
44	35, 39, 40, 43	fvmptd 6197	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐷‘𝐴)‘𝐸) = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
45	20, 44	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
46		fveq1 6102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
47	46	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝐼 → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(𝐼‘𝑗)))
48	47	mpteq2dv 4673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))
49	48	fveq2d 6107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
50	49	breq1d 4593	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
51	50	elrab 3331	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ↔ (𝐼 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
52	45, 51	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
53	52	simpld 474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ (𝐶‘𝐴))
54	19, 53	sseldd 3569	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
55		elmapi 7765	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
56	54, 55	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
58		simpr 476	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
59	57, 58	ffvelrnd 6268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
60		elmapi 7765	. . . . . . . . . 10 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
61	59, 60	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
62	61	ffvelrnda 6267	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ))
63		xp1st 7089	. . . . . . . 8 ⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
64	62, 63	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
65		eqid 2610	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))
66	64, 65	fmptd 6292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)
67		reex 9906	. . . . . . . . 9 ⊢ ℝ ∈ V
68	67	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
69		ovncvr2.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
70		elmapg 7757	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
71	68, 69, 70	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
72	71	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
73	66, 72	mpbird 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋))
74		eqid 2610	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
75	73, 74	fmptd 6292	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋))
76		ovncvr2.b	. . . . . 6 ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
77	76	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))))
78	77	feq1d 5943	. . . 4 ⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
79	75, 78	mpbird 246	. . 3 ⊢ (𝜑 → 𝐵:ℕ⟶(ℝ ↑𝑚 𝑋))
80		xp2nd 7090	. . . . . . . 8 ⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
81	62, 80	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
82		eqid 2610	. . . . . . 7 ⊢ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))
83	81, 82	fmptd 6292	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)
84		elmapg 7757	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
85	68, 69, 84	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
86	85	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
87	83, 86	mpbird 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋))
88		eqid 2610	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
89	87, 88	fmptd 6292	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋))
90		ovncvr2.t	. . . . . 6 ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
91	90	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))))
92	91	feq1d 5943	. . . 4 ⊢ (𝜑 → (𝑇:ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
93	89, 92	mpbird 246	. . 3 ⊢ (𝜑 → 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋))
94	79, 93	jca 553	. 2 ⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)))
95	16	idi 2	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
96	53, 95	eleqtrd 2690	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
97		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐼 → (𝑙‘𝑗) = (𝐼‘𝑗))
98	97	coeq2d 5206	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐼 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
99	98	fveq1d 6105	. . . . . . . . . 10 ⊢ (𝑙 = 𝐼 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
100	99	ixpeq2dv 7810	. . . . . . . . 9 ⊢ (𝑙 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
101	100	adantr 480	. . . . . . . 8 ⊢ ((𝑙 = 𝐼 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
102	101	iuneq2dv 4478	. . . . . . 7 ⊢ (𝑙 = 𝐼 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
103	102	sseq2d 3596	. . . . . 6 ⊢ (𝑙 = 𝐼 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
104	103	elrab 3331	. . . . 5 ⊢ (𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ↔ (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
105	96, 104	sylib 207	. . . 4 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
106	105	simprd 478	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
107	61	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
108		simpr 476	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
109	107, 108	fvovco 38376	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
110		mptexg 6389	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
111	69, 110	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
112	111	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
113	77, 112	fvmpt2d 6202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
114		fvex 6113	. . . . . . . . . 10 ⊢ (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ V
115	114	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ V)
116	113, 115	fvmpt2d 6202	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) = (1st ‘((𝐼‘𝑗)‘𝑘)))
117	116	eqcomd 2616	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = ((𝐵‘𝑗)‘𝑘))
118		mptexg 6389	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
119	69, 118	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
120	119	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
121	91, 120	fvmpt2d 6202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
122		fvex 6113	. . . . . . . . . 10 ⊢ (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ V
123	122	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ V)
124	121, 123	fvmpt2d 6202	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) = (2nd ‘((𝐼‘𝑗)‘𝑘)))
125	124	eqcomd 2616	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = ((𝑇‘𝑗)‘𝑘))
126	117, 125	oveq12d 6567	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
127	109, 126	eqtrd 2644	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
128	127	ixpeq2dva 7809	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
129	128	iuneq2dv 4478	. . 3 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
130	106, 129	sseqtrd 3604	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
131		ovncvr2.l	. . . . . . . 8 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
132	131	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))))
133		coeq2 5202	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝐼‘𝑗)))
134	133	fveq1d 6105	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐼‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
135	134	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
136	135	adantllr 751	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
137	109	adantlr 747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
138	126	adantlr 747	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
139	136, 137, 138	3eqtrd 2648	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
140	139	fveq2d 6107	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
141	140	prodeq2dv 14492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
142	69	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
143	76	fvmpt2 6200	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
144	58, 112, 143	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
145	144	feq1d 5943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐵‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
146	66, 145	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗):𝑋⟶ℝ)
147	146	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑗):𝑋⟶ℝ)
148	147, 108	ffvelrnd 6268	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) ∈ ℝ)
149	90	fvmpt2 6200	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
150	58, 120, 149	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
151	150	feq1d 5943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑇‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
152	83, 151	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗):𝑋⟶ℝ)
153	152	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑇‘𝑗):𝑋⟶ℝ)
154	153, 108	ffvelrnd 6268	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) ∈ ℝ)
155		volicore 39471	. . . . . . . . 9 ⊢ ((((𝐵‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇‘𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
156	148, 154, 155	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
157	142, 156	fprodrecl 14522	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
158	132, 141, 59, 157	fvmptd 6197	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
159	158	eqcomd 2616	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) = (𝐿‘(𝐼‘𝑗)))
160	159	mpteq2dva 4672	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))
161	160	fveq2d 6107	. . 3 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
162	52	simprd 478	. . 3 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
163	161, 162	eqbrtrd 4605	. 2 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
164	94, 130, 163	jca31 555	1 ⊢ (𝜑 → (((𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  Xcixp 7794  Fincfn 7841  ℝcr 9814   ≤ cle 9954  ℕcn 10897  ℝ⁺crp 11708   +_𝑒 cxad 11820  [,)cico 12048  ∏cprod 14474  volcvol 23039  Σ^{^}csumge0 39255  voln*covoln 39426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-prod 14475  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-0g 15925  df-topgen 15927  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041

This theorem is referenced by:  hspmbllem3  39518