ovnhoilem2 - Mathbox for Glauco Siliprandi

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Theorem ovnhoilem2 41610

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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
ovnhoilem2.s	⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((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 2898	. . . . . . . . 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 3326	. . . . . . . . 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 267	. . . . . . . 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 208	. . . . . . 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 491	. . . . . 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 475	. . . . 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 1169	. . . . . . . . . 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 1169	. . . . . . . . . 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 1169	. . . . . . . . . 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 8144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
16	15	ffvelrnda 6608	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
17		elmapi 8144	. . . . . . . . . . . . . . . . . 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 6608	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
20		xp1st 7460	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ) → (1^st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
22	21	fmpttd 6634	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
23		reex 10343	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
24	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
25		1nn 11363	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
26	25	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 1 ∈ ℕ)
27	15, 26	ffvelrnd 6609	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
28		elmapex 8143	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
29	28	simprd 491	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
32		elmapg 8135	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
33	24, 31, 32	syl2anc 581	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
34	22, 33	mpbird 249	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
35	34	fmpttd 6634	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((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 11357	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
38	37	mptex 6742	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
39	38	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
40		ovnhoilem2.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
41	40	fvmpt2 6538	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
42	36, 39, 41	syl2anc 581	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
43	42	feq1d 6263	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ((𝐹‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋)))
44	35, 43	mpbird 249	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
45	44	3ad2ant2 1170	. . . . . . . . . 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 7461	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
47	19, 46	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
48	47	fmpttd 6634	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
49		elmapg 8135	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
50	24, 31, 49	syl2anc 581	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
51	48, 50	mpbird 249	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
52	51	fmpttd 6634	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋))
53	37	mptex 6742	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
54	53	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
55		ovnhoilem2.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
56	55	fvmpt2 6538	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
57	36, 54, 56	syl2anc 581	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
58	57	feq1d 6263	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ((𝑆‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋) ↔ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋)))
59	52, 58	mpbird 249	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
60	59	3ad2ant2 1170	. . . . . . . . . 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 1174	. . . . . . . . . . . 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 6433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
65	64	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → ((𝑖‘𝑗)‘𝑘) = ((𝑖‘𝑛)‘𝑘))
66	65	fveq2d 6437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (1^st ‘((𝑖‘𝑗)‘𝑘)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
67	65	fveq2d 6437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
68	66, 67	oveq12d 6923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
69	68	ixpeq2dv 8191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑛 → X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
70	69	cbviunv 4779	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘)))
71	70	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
72	15	ffvelrnda 6608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
73		elmapi 8144	. . . . . . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
76		simpr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
77	75, 76	fvovco 40189	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))
78	77	ixpeq2dva 8190	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))
79	78	iuneq2dv 4762	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))
80		simpl 476	. . . . . . . . . . . . . . . . . . . . . . . . 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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
82	80, 81, 41	syl2anc 581	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
83	82	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
84		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
85		mptexg 6740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
86	30, 85	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
87	86	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
88		eqid 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
89	88	fvmpt2 6538	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
90	84, 87, 89	syl2anc 581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
91	83, 90	eqtrd 2861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
92	91	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
93	92	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
94		eqidd 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
95		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
96	95	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
97	96	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
98		simpr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
99		fvexd 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
100	94, 97, 98, 99	fvmptd 6535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
101	100	adantlr 708	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
102	93, 101	eqtrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
103	57	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
104	103	adantr 474	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
105		mptexg 6740	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
106	30, 105	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
107	106	adantr 474	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
108		eqid 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
109	108	fvmpt2 6538	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
110	84, 107, 109	syl2anc 581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
111	104, 110	eqtrd 2861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
112	111	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
113	112	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
114		eqidd 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
115		2fveq3 6438	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
116	115	adantl 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
117		fvexd 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
118	114, 116, 98, 117	fvmptd 6535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
119	118	adantlr 708	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
120	113, 119	eqtrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
121	102, 120	oveq12d 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
122	121	ixpeq2dva 8190	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
123	122	iuneq2dv 4762	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
124	71, 79, 123	3eqtr4d 2871	. . . . . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 1168	. . . . . . . . . . . . 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 3859	. . . . . . . . . . . 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 224	. . . . . . . . . . 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 1237	. . . . . . . . . 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 41608	. . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → 𝑛 = 𝑗)
132	131	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑛) = (𝑖‘𝑗))
133	132	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑗)‘𝑙))
134	133	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑙)))
135	134	mpteq2dva 4967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
136	135	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
137	136	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
138		eqidd 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
139		2fveq3 6438	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
140	139	adantl 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
141		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
142		fvexd 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (1^st ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
143	138, 140, 141, 142	fvmptd 6535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
144	143	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
145	137, 144	eqtrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
146	133	fveq2d 6437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑙)))
147	146	mpteq2dva 4967	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
148	147	fveq1d 6435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
149	148	adantr 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
150		eqidd 2826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
151		2fveq3 6438	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
152	151	adantl 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
153		fvexd 6448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
154	150, 152, 141, 153	fvmptd 6535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
155	154	adantl 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
156	149, 155	eqtrd 2861	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
157	145, 156	oveq12d 6923	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → (((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)) = ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))
158	157	fveq2d 6437	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))) = (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘)))))
159	158	prodeq2dv 15026	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘)))))
160	159	cbvmptv 4973	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘)))))
161	160	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))))
162	77	eqcomd 2831	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
163	162	fveq2d 6437	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘)))) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
164	163	prodeq2dv 15026	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
165	164	mpteq2dva 4967	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
166	161, 165	eqtrd 2861	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
167	166	fveq2d 6437	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
168	167	3ad2ant2 1170	. . . . . . . . . . 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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))) = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
169	91	adantll 707	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
170	111	adantll 707	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
171	169, 170	oveq12d 6923	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))(𝐿‘𝑋)(𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
172	9	ad2antrr 719	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
173		ovnhoilem2.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
174	173	ad2antrr 719	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
175	19	adantlll 711	. . . . . . . . . . . . . . . . . 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) → (1^st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
177	176	fmpttd 6634	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
178	175, 46	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
179	178	fmpttd 6634	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
180	8, 172, 174, 177, 179	hoidmvn0val 41592	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))(𝐿‘𝑋)(𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))
181	171, 180	eqtrd 2861	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))
182	181	mpteq2dva 4967	. . . . . . . . . . . . 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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)))))
183	182	fveq2d 6437	. . . . . . . . . . . 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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))))
184	183	3adant3 1168	. . . . . . . . . . 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 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))))
185		simp3 1174	. . . . . . . . . . 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 2871	. . . . . . . . . 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 1235	. . . . . . . . 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 4899	. . . . . . . 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 1154	. . . . . . 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 474	. . . . . 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 3239	. . . . 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 3175	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
194		ssrab2 3912	. . . . . 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 3860	. . . . 5 ⊢ 𝑀 ⊆ ℝ^*
196	195	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ^*)
197		icossxr 12546	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ^*
198	8, 9, 11, 13	hoidmvcl 41590	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
199	197, 198	sseldi 3825	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^*)
200		infxrgelb 12453	. . . 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 581	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
202	193, 201	mpbird 249	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ))
203	62	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
204		nfv 2015	. . . . . 6 ⊢ Ⅎ𝑘𝜑
205	11	ffvelrnda 6608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
206	13	ffvelrnda 6608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
207	206	rexrd 10406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
208	204, 205, 207	hoissrrn2 41586	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑_𝑚 𝑋))
209	203, 208	eqsstrd 3864	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑_𝑚 𝑋))
210	9, 173, 209, 1	ovnn0val 41559	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = inf(𝑀, ℝ^, < ))
211	210	eqcomd 2831	. 2 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐼))
212	202, 211	breqtrd 4899	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∀wral 3117 ∃wrex 3118 {crab 3121 Vcvv 3414 ⊆ wss 3798 ∅c0 4144 ifcif 4306 ∪ ciun 4740 class class class wbr 4873 ↦ cmpt 4952 × cxp 5340 ∘ ccom 5346 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 1^st c1st 7426 2^nd c2nd 7427 ↑_𝑚 cmap 8122 Xcixp 8175 Fincfn 8222 infcinf 8616 ℝcr 10251 0cc0 10252 1c1 10253 +∞cpnf 10388 ℝ^*cxr 10390 < clt 10391 ≤ cle 10392 ℕcn 11350 [,)cico 12465 ∏cprod 15008 volcvol 23629 Σ^{^}csumge0 41370 voln*covoln 41544

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794 df-prod 15009 df-rest 16436 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-bases 21121 df-cmp 21561 df-ovol 23630 df-vol 23631 df-sumge0 41371 df-ovoln 41545

This theorem is referenced by: ovnhoi 41611

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