Theorem ovnhoilem2 43241

Description: The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. Second part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
ovnhoilem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnhoilem2.n	⊢ (𝜑 → 𝑋 ≠ ∅)
ovnhoilem2.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
ovnhoilem2.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
ovnhoilem2.i	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
ovnhoilem2.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
ovnhoilem2.m	⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
ovnhoilem2.f	⊢ 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
ovnhoilem2.s	⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))

Assertion

Ref	Expression
ovnhoilem2	⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑖,𝑘,𝑧   𝐵,𝑎,𝑏,𝑖,𝑘,𝑧   𝑘,𝐹,𝑛   𝐼,𝑎,𝑏,𝑖,𝑛,𝑥,𝑧   𝐿,𝑎,𝑏,𝑖,𝑛,𝑥,𝑧   𝑖,𝑀,𝑧   𝑆,𝑘,𝑛   𝑋,𝑎,𝑏,𝑖,𝑗,𝑘,𝑙,𝑛   𝑥,𝑋,𝑧,𝑗,𝑘   𝜑,𝑎,𝑏,𝑖,𝑘,𝑙,𝑛   𝜑,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑗)   𝐴(𝑥,𝑗,𝑛,𝑙)   𝐵(𝑥,𝑗,𝑛,𝑙)   𝑆(𝑥,𝑧,𝑖,𝑗,𝑎,𝑏,𝑙)   𝐹(𝑥,𝑧,𝑖,𝑗,𝑎,𝑏,𝑙)   𝐼(𝑗,𝑘,𝑙)   𝐿(𝑗,𝑘,𝑙)   𝑀(𝑥,𝑗,𝑘,𝑛,𝑎,𝑏,𝑙)

Proof of Theorem ovnhoilem2

Step	Hyp	Ref	Expression
1		ovnhoilem2.m	. . . . . . . . . 10 ⊢ 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}
2	1	eleq2i 2881	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑀 ↔ 𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))})
3		rabid 3331	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ↔ (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
4	2, 3	bitri 278	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑀 ↔ (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
5	4	biimpi 219	. . . . . . 7 ⊢ (𝑧 ∈ 𝑀 → (𝑧 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
6	5	simprd 499	. . . . . 6 ⊢ (𝑧 ∈ 𝑀 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
7	6	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
8		ovnhoilem2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
9		ovnhoilem2.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
10	9	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝑋 ∈ Fin)
11		ovnhoilem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
12	11	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐴:𝑋⟶ℝ)
13		ovnhoilem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
14	13	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐵:𝑋⟶ℝ)
15		elmapi 8411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
16	15	ffvelrnda 6828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑m 𝑋))
17		elmapi 8411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝑖‘𝑛):𝑋⟶(ℝ × ℝ))
18	16, 17	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛):𝑋⟶(ℝ × ℝ))
19	18	ffvelrnda 6828	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
20		xp1st 7703	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ) → (1st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
22	21	fmpttd 6856	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
23		reex 10617	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
24	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
25		1nn 11636	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
26	25	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 1 ∈ ℕ)
27	15, 26	ffvelrnd 6829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋))
28		elmapex 8410	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
29	28	simprd 499	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑m 𝑋) → 𝑋 ∈ V)
30	27, 29	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑋 ∈ V)
31	30	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
32		elmapg 8402	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
33	24, 31, 32	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
34	22, 33	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋))
35	34	fmpttd 6856	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋))
36		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
37		nnex 11631	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
38	37	mptex 6963	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
40		ovnhoilem2.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
41	40	fvmpt2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
42	36, 39, 41	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
43	42	feq1d 6472	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝐹‘𝑖):ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋)))
44	35, 43	mpbird 260	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝐹‘𝑖):ℕ⟶(ℝ ↑m 𝑋))
45	44	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐹‘𝑖):ℕ⟶(ℝ ↑m 𝑋))
46		xp2nd 7704	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ) → (2nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
47	19, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
48	47	fmpttd 6856	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
49		elmapg 8402	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
50	24, 31, 49	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
51	48, 50	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑m 𝑋))
52	51	fmpttd 6856	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋))
53	37	mptex 6963	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
54	53	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
55		ovnhoilem2.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))
56	55	fvmpt2 6756	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))
57	36, 54, 56	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))
58	57	feq1d 6472	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝑆‘𝑖):ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑m 𝑋)))
59	52, 58	mpbird 260	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑆‘𝑖):ℕ⟶(ℝ ↑m 𝑋))
60	59	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝑆‘𝑖):ℕ⟶(ℝ ↑m 𝑋))
61		simp3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
62		ovnhoilem2.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
63	62	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
64		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
65	64	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → ((𝑖‘𝑗)‘𝑘) = ((𝑖‘𝑛)‘𝑘))
66	65	fveq2d 6649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (1st ‘((𝑖‘𝑗)‘𝑘)) = (1st ‘((𝑖‘𝑛)‘𝑘)))
67	65	fveq2d 6649	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (2nd ‘((𝑖‘𝑗)‘𝑘)) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
68	66, 67	oveq12d 7153	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))) = ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
69	68	ixpeq2dv 8460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))) = X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
70	69	cbviunv 4927	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘)))
71	70	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
72	15	ffvelrnda 6828	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
73		elmapi 8411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
75	74	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
76		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
77	75, 76	fvovco 41821	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))
78	77	ixpeq2dva 8459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))
79	78	iuneq2dv 4905	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))
80		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
81	38	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
82	80, 81, 41	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
83	82	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
84		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
85		mptexg 6961	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
86	30, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
87	86	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
88		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
89	88	fvmpt2 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
90	84, 87, 89	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
91	83, 90	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
92	91	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
93	92	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
94		eqidd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
95		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
96	95	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
97	96	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (1st ‘((𝑖‘𝑛)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑘)))
98		simpr 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
99		fvexd 6660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
100	94, 97, 98, 99	fvmptd 6752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖‘𝑛)‘𝑘)))
101	100	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖‘𝑛)‘𝑘)))
102	93, 101	eqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = (1st ‘((𝑖‘𝑛)‘𝑘)))
103	57	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
104	103	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
105		mptexg 6961	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
106	30, 105	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
107	106	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
108		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
109	108	fvmpt2 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
110	84, 107, 109	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
111	104, 110	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
112	111	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
113	112	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
114		eqidd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
115		2fveq3 6650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2nd ‘((𝑖‘𝑛)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
116	115	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖‘𝑛)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
117		fvexd 6660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
118	114, 116, 98, 117	fvmptd 6752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
119	118	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
120	113, 119	eqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = (2nd ‘((𝑖‘𝑛)‘𝑘)))
121	102, 120	oveq12d 7153	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
122	121	ixpeq2dva 8459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
123	122	iuneq2dv 4905	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1st ‘((𝑖‘𝑛)‘𝑘))[,)(2nd ‘((𝑖‘𝑛)‘𝑘))))
124	71, 79, 123	3eqtr4d 2843	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
125	124	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
126	125	3adant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
127	63, 126	sseq12d 3948	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘))))
128	61, 127	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
129	128	3adant3r 1178	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
130	8, 10, 12, 14, 45, 60, 129	hoidmvle 43239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))))
131		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → 𝑛 = 𝑗)
132	131	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑛) = (𝑖‘𝑗))
133	132	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑗)‘𝑙))
134	133	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑛)‘𝑙)) = (1st ‘((𝑖‘𝑗)‘𝑙)))
135	134	mpteq2dva 5125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))))
136	135	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
137	136	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
138		eqidd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))))
139		2fveq3 6650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑗)‘𝑘)))
140	139	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑗)‘𝑘)))
141		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
142		fvexd 6660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (1st ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
143	138, 140, 141, 142	fvmptd 6752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖‘𝑗)‘𝑘)))
144	143	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1st ‘((𝑖‘𝑗)‘𝑘)))
145	137, 144	eqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1st ‘((𝑖‘𝑗)‘𝑘)))
146	133	fveq2d 6649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑛)‘𝑙)) = (2nd ‘((𝑖‘𝑗)‘𝑙)))
147	146	mpteq2dva 5125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))))
148	147	fveq1d 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
149	148	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
150		eqidd 2799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))))
151		2fveq3 6650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑗)‘𝑘)))
152	151	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑗)‘𝑘)))
153		fvexd 6660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (2nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
154	150, 152, 141, 153	fvmptd 6752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖‘𝑗)‘𝑘)))
155	154	adantl 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2nd ‘((𝑖‘𝑗)‘𝑘)))
156	149, 155	eqtrd 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2nd ‘((𝑖‘𝑗)‘𝑘)))
157	145, 156	oveq12d 7153	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → (((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)) = ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))
158	157	fveq2d 6649	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))) = (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘)))))
159	158	prodeq2dv 15269	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘)))))
160	159	cbvmptv 5133	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘)))))
161	160	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))))
162	77	eqcomd 2804	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
163	162	fveq2d 6649	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
164	163	prodeq2dv 15269	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
165	164	mpteq2dva 5125	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1st ‘((𝑖‘𝑗)‘𝑘))[,)(2nd ‘((𝑖‘𝑗)‘𝑘))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
166	161, 165	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
167	166	fveq2d 6649	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
168	167	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
169	91	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
170	111	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
171	169, 170	oveq12d 7153	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)) = ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))(𝐿‘𝑋)(𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))
172	9	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
173		ovnhoilem2.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
174	173	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
175	19	adantlll 717	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
176	175, 20	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
177	176	fmpttd 6856	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
178	175, 46	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
179	178	fmpttd 6856	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
180	8, 172, 174, 177, 179	hoidmvn0val 43223	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))(𝐿‘𝑋)(𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))
181	171, 180	eqtrd 2833	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))
182	181	mpteq2dva 5125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) → (𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))))
183	182	fveq2d 6649	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))))
184	183	3adant3 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = (Σ^‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))))
185		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
186	168, 184, 185	3eqtr4d 2843	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = 𝑧)
187	186	3adant3l 1177	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (Σ^‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = 𝑧)
188	130, 187	breqtrd 5056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
189	188	3exp 1116	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)))
190	189	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)))
191	190	rexlimdv 3242	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
192	7, 191	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
193	192	ralrimiva 3149	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
194		ssrab2 4007	. . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} ⊆ ℝ*
195	1, 194	eqsstri 3949	. . . . 5 ⊢ 𝑀 ⊆ ℝ*
196	195	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ*)
197		icossxr 12810	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ*
198	8, 9, 11, 13	hoidmvcl 43221	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
199	197, 198	sseldi 3913	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
200		infxrgelb 12716	. . . 4 ⊢ ((𝑀 ⊆ ℝ* ∧ (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*) → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
201	196, 199, 200	syl2anc 587	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
202	193, 201	mpbird 260	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ*, < ))
203	62	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
204		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘𝜑
205	11	ffvelrnda 6828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
206	13	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
207	206	rexrd 10680	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
208	204, 205, 207	hoissrrn2 43217	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
209	203, 208	eqsstrd 3953	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
210	9, 173, 209, 1	ovnn0val 43190	. . 3 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = inf(𝑀, ℝ*, < ))
211	210	eqcomd 2804	. 2 ⊢ (𝜑 → inf(𝑀, ℝ*, < ) = ((voln*‘𝑋)‘𝐼))
212	202, 211	breqtrd 5056	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ifcif 4425  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1^st c1st 7669  2^nd c2nd 7670   ↑_m cmap 8389  Xcixp 8444  Fincfn 8492  infcinf 8889  ℝcr 10525  0cc0 10526  1c1 10527  +∞cpnf 10661  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665  ℕcn 11625  [,)cico 12728  ∏cprod 15251  volcvol 24067  Σ^{^}csumge0 43001  voln*covoln 43175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-prod 15252  df-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cmp 21992  df-ovol 24068  df-vol 24069  df-sumge0 43002  df-ovoln 43176

This theorem is referenced by:  ovnhoi  43242