Theorem ovncvr2 47590

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	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
2		sseq1 3956	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)))
3	2	rabbidv 3420	. . . . . . . . . . . . . . . 16 ⊢ (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)})
4		ovncvr2.a	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋))
5		ovexd 7453	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ ↑m 𝑋) ∈ V)
6	5, 4	ssexd 5286	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ V)
7		elpwg 4560	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
8	6, 7	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑m 𝑋) ↔ 𝐴 ⊆ (ℝ ↑m 𝑋)))
9	4, 8	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
10		ovex 7451	. . . . . . . . . . . . . . . . . 18 ⊢ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
11	10	rabex 5300	. . . . . . . . . . . . . . . . 17 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V
12	11	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V)
13	1, 3, 9, 12	fvmptd3 7015	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
14		ssrab2 4028	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)
15	14	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))
16	13, 15	eqsstrd 3965	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘𝐴) ⊆ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
17		ovncvr2.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸))
18		ovncvr2.d	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))
19		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝐴 → (𝐶‘𝑎) = (𝐶‘𝐴))
20	19	eleq2d 2847	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝐴 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝐴)))
21		fveq2 6883	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
22	21	oveq1d 7433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
23	22	breq2d 5115	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = 𝐴 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)))
24	20, 23	anbi12d 644	. . . . . . . . . . . . . . . . . . . . 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))))
25	24	rabbidva2 3415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)})
26	25	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . 19 ⊢ (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)}))
27		rpex 46327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ+ ∈ V
28	27	mptex 7227	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V)
30	18, 26, 9, 29	fvmptd3 7015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐷‘𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31		oveq2 7426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
32	31	breq2d 5115	. . . . . . . . . . . . . . . . . . . 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)))
33	32	rabbidv 3420	. . . . . . . . . . . . . . . . . . 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)})
34	33	adantl 487	. . . . . . . . . . . . . . . . . 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)})
35		ovncvr2.e	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
36		fvex 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐶‘𝐴) ∈ V
37	36	rabex 5300	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
38	37	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V)
39	30, 34, 35, 38	fvmptd 6999	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐷‘𝐴)‘𝐸) = {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
40	17, 39	eleqtrd 2863	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
41		fveq1 6882	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗))
42	41	fveq2d 6887	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝐼 → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(𝐼‘𝑗)))
43	42	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))
44	43	fveq2d 6887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
45	44	breq1d 5113	. . . . . . . . . . . . . . . . 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)))
46	45	elrab 3645	. . . . . . . . . . . . . . . 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)))
47	40, 46	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ∈ (𝐶‘𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
48	47	simpld 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ (𝐶‘𝐴))
49	16, 48	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
50		elmapi 8862	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
51	49, 50	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
52	51	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
54	52, 53	ffvelcdmd 7083	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
55		elmapi 8862	. . . . . . . . . 10 ⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
56	54, 55	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
57	56	ffvelcdmda 7082	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ))
58		xp1st 8031	. . . . . . . 8 ⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
59	57, 58	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
60	59	fmpttd 7113	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)
61		reex 11284	. . . . . . . . 9 ⊢ ℝ ∈ V
62	61	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ∈ V)
63		ovncvr2.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
64		elmapg 8852	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
65	62, 63, 64	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
66	65	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
67	60, 66	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
68	67	fmpttd 7113	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋))
69		ovncvr2.b	. . . . . 6 ⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
70	69	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))))
71	70	feq1d 6689	. . . 4 ⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
72	68, 71	mpbird 260	. . 3 ⊢ (𝜑 → 𝐵:ℕ⟶(ℝ ↑m 𝑋))
73		xp2nd 8032	. . . . . . . 8 ⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
74	57, 73	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ)
75	74	fmpttd 7113	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)
76		elmapg 8852	. . . . . . . 8 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
77	62, 63, 76	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
78	77	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
79	75, 78	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
80	79	fmpttd 7113	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋))
81		ovncvr2.t	. . . . . 6 ⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
82	81	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))))
83	82	feq1d 6689	. . . 4 ⊢ (𝜑 → (𝑇:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
84	80, 83	mpbird 260	. . 3 ⊢ (𝜑 → 𝑇:ℕ⟶(ℝ ↑m 𝑋))
85	72, 84	jca 521	. 2 ⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)))
86	48, 13	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)})
87		fveq1 6882	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐼 → (𝑙‘𝑗) = (𝐼‘𝑗))
88	87	coeq2d 5840	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐼 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝐼‘𝑗)))
89	88	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑙 = 𝐼 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
90	89	ixpeq2dv 8934	. . . . . . . . 9 ⊢ (𝑙 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
91	90	adantr 486	. . . . . . . 8 ⊢ ((𝑙 = 𝐼 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
92	91	iuneq2dv 4976	. . . . . . 7 ⊢ (𝑙 = 𝐼 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
93	92	sseq2d 3963	. . . . . 6 ⊢ (𝑙 = 𝐼 → (𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
94	93	elrab 3645	. . . . 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)))
95	86, 94	sylib 221	. . . 4 ⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)))
96	95	simprd 501	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))
97	56	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ × ℝ))
98		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
99	97, 98	fvovco 46177	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
100		mptexg 7225	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
101	63, 100	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
102	101	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
103	70, 102	fvmpt2d 7005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
104		fvexd 6898	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ V)
105	103, 104	fvmpt2d 7005	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) = (1st ‘((𝐼‘𝑗)‘𝑘)))
106	105	eqcomd 2767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = ((𝐵‘𝑗)‘𝑘))
107		mptexg 7225	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
108	63, 107	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
109	108	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V)
110	82, 109	fvmpt2d 7005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
111		fvexd 6898	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ V)
112	110, 111	fvmpt2d 7005	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) = (2nd ‘((𝐼‘𝑗)‘𝑘)))
113	112	eqcomd 2767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = ((𝑇‘𝑗)‘𝑘))
114	106, 113	oveq12d 7436	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
115	99, 114	eqtrd 2796	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
116	115	ixpeq2dva 8933	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
117	116	iuneq2dv 4976	. . 3 ⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
118	96, 117	sseqtrd 3967	. 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
119		ovncvr2.l	. . . . . . . 8 ⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)))
120	119	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐿 = (ℎ ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘))))
121		coeq2 5836	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝐼‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝐼‘𝑗)))
122	121	fveq1d 6885	. . . . . . . . . . . 12 ⊢ (ℎ = (𝐼‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
123	122	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
124	123	adantllr 732	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘))
125	99	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))))
126	114	adantlr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
127	124, 125, 126	3eqtrd 2800	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))
128	127	fveq2d 6887	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
129	128	prodeq2dv 16083	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
130	63	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
131	69	fvmpt2 7003	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
132	53, 102, 131	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))
133	132	feq1d 6689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐵‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
134	60, 133	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗):𝑋⟶ℝ)
135	134	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑗):𝑋⟶ℝ)
136	135, 98	ffvelcdmd 7083	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) ∈ ℝ)
137	81	fvmpt2 7003	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
138	53, 109, 137	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))
139	138	feq1d 6689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑇‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ))
140	75, 139	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗):𝑋⟶ℝ)
141	140	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑇‘𝑗):𝑋⟶ℝ)
142	141, 98	ffvelcdmd 7083	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) ∈ ℝ)
143		volicore 47560	. . . . . . . . 9 ⊢ ((((𝐵‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇‘𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
144	136, 142, 143	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
145	130, 144	fprodrecl 16113	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ)
146	120, 129, 54, 145	fvmptd 6999	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))
147	146	eqcomd 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) = (𝐿‘(𝐼‘𝑗)))
148	147	mpteq2dva 5198	. . . 4 ⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))
149	148	fveq2d 6887	. . 3 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))))
150	47	simprd 501	. . 3 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
151	149, 150	eqbrtrd 5127	. 2 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
152	85, 118, 151	jca31 524	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ⊆ wss 3899  𝒫 cpw 4557  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649   ∘ ccom 5655  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  1^st c1st 7997  2^nd c2nd 7998   ↑_m cmap 8840  Xcixp 8918  Fincfn 8966  ℝcr 11192   ≤ cle 11337  ℕcn 12328  ℝ⁺crp 13113   +_𝑒 cxad 13232  [,)cico 13471  ∏cprod 16065  volcvol 25777  Σ^{^}csumge0 47341  voln*covoln 47515

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fi 9396  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-z 12687  df-uz 12959  df-q 13069  df-rp 13114  df-xneg 13234  df-xadd 13235  df-xmul 13236  df-ioo 13473  df-ico 13475  df-icc 13476  df-fz 13633  df-fzo 13782  df-fl 13925  df-seq 14138  df-exp 14198  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-rlim 15649  df-sum 15847  df-prod 16066  df-rest 17586  df-topgen 17607  df-psmet 21663  df-xmet 21664  df-met 21665  df-bl 21666  df-mopn 21667  df-top 23205  df-topon 23222  df-bases 23257  df-cmp 23698  df-ovol 25778  df-vol 25779

This theorem is used by:  hspmbllem3  47607