ovnhoilem2 - Mathbox for Glauco Siliprandi

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Theorem ovnhoilem2 39492

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 2680	. . . . . . . . 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 3095	. . . . . . . . 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 263	. . . . . . . 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 205	. . . . . . 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 478	. . . . . 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 481	. . . . 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 1075	. . . . . . . . . 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 1075	. . . . . . . . . 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 1075	. . . . . . . . . 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 7765	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
16	15	ffvelrnda 6267	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
17		elmapi 7765	. . . . . . . . . . . . . . . . . 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 6267	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
20		xp1st 7089	. . . . . . . . . . . . . . . 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		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))
23	21, 22	fmptd 6292	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
24		reex 9906	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
26		1nn 10908	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
27	26	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 1 ∈ ℕ)
28	15, 27	ffvelrnd 6268	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
29		elmapex 7764	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
30	29	simprd 478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → 𝑋 ∈ V)
31	28, 30	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 𝑋 ∈ V)
32	31	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
33		elmapg 7757	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
34	25, 32, 33	syl2anc 691	. . . . . . . . . . . . . 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))
35	23, 34	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
36		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
37	35, 36	fmptd 6292	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋))
38		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
39		nnex 10903	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
40	39	mptex 6390	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
42		ovnhoilem2.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
43	42	fvmpt2 6200	. . . . . . . . . . . . . 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)))))
44	38, 41, 43	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
45	44	feq1d 5943	. . . . . . . . . . . 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)))
46	37, 45	mpbird 246	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
47	46	3ad2ant2 1076	. . . . . . . . . 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))
48		xp2nd 7090	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
49	19, 48	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
50		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))
51	49, 50	fmptd 6292	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
52		elmapg 7757	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∈ V ∧ 𝑋 ∈ V) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ))
53	25, 32, 52	syl2anc 691	. . . . . . . . . . . . . 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))
54	51, 53	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
55		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
56	54, 55	fmptd 6292	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋))
57	39	mptex 6390	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V
58	57	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
59		ovnhoilem2.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
60	59	fvmpt2 6200	. . . . . . . . . . . . . 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)))))
61	38, 58, 60	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
62	61	feq1d 5943	. . . . . . . . . . . 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)))
63	56, 62	mpbird 246	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
64	63	3ad2ant2 1076	. . . . . . . . . 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))
65		simp3 1056	. . . . . . . . . . . 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))
66		ovnhoilem2.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
67	66	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)))
68		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
69	68	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → ((𝑖‘𝑗)‘𝑘) = ((𝑖‘𝑛)‘𝑘))
70	69	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (1^st ‘((𝑖‘𝑗)‘𝑘)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
71	69	fveq2d 6107	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
72	70, 71	oveq12d 6567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑛 → ((1^st ‘((𝑖‘𝑗)‘𝑘))[,)(2^nd ‘((𝑖‘𝑗)‘𝑘))) = ((1^st ‘((𝑖‘𝑛)‘𝑘))[,)(2^nd ‘((𝑖‘𝑛)‘𝑘))))
73	72	ixpeq2dv 7810	. . . . . . . . . . . . . . . . . 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))))
74	73	cbviunv 4495	. . . . . . . . . . . . . . . . 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)))
75	74	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))))
76	15	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
77		elmapi 7765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
78	76, 77	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
79	78	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
80		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
81	79, 80	fvovco 38376	. . . . . . . . . . . . . . . . . 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))))
82	81	ixpeq2dva 7809	. . . . . . . . . . . . . . . . 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))))
83	82	iuneq2dv 4478	. . . . . . . . . . . . . . . 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))))
84		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ))
85	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) ∈ V)
86	84, 85, 43	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
87	86	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
88		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
89		mptexg 6389	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
90	31, 89	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
91	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
92	36	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . 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))))
93	88, 91, 92	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 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))))
94	87, 93	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
95	94	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
96	95	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
97		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
98		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
99	98	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
100	99	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
101		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
102		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1^st ‘((𝑖‘𝑛)‘𝑘)) ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
104	97, 100, 101, 103	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
105	104	adantlr 747	. . . . . . . . . . . . . . . . . . . 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)))
106	96, 105	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
107	61	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
108	107	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
109		mptexg 6389	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
110	31, 109	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
111	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
112	55	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . 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))))
113	88, 111, 112	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 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))))
114	108, 113	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
115	114	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
116	115	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
117		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
118		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
119	118	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
120	119	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
121		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2^nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V
122	121	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
123	117, 120, 101, 122	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
124	123	adantlr 747	. . . . . . . . . . . . . . . . . . . 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)))
125	116, 124	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
126	106, 125	oveq12d 6567	. . . . . . . . . . . . . . . . . 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))))
127	126	ixpeq2dva 7809	. . . . . . . . . . . . . . . . 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))))
128	127	iuneq2dv 4478	. . . . . . . . . . . . . . . 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))))
129	75, 83, 128	3eqtr4d 2654	. . . . . . . . . . . . . . 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)))
130	129	adantl 481	. . . . . . . . . . . . . 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)))
131	130	3adant3 1074	. . . . . . . . . . . . 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)))
132	67, 131	sseq12d 3597	. . . . . . . . . . . 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))))
133	65, 132	mpbid 221	. . . . . . . . . . 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)))
134	133	3adant3r 1315	. . . . . . . . . 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)))
135	8, 10, 12, 14, 47, 64, 134	hoidmvle 39490	. . . . . . . . 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)))))
136		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → 𝑛 = 𝑗)
137	136	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑛) = (𝑖‘𝑗))
138	137	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑗)‘𝑙))
139	138	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑙)))
140	139	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
141	140	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
142	141	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
143		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
144		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑗)‘𝑘))
145	144	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
146	145	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
147		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
148		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1^st ‘((𝑖‘𝑗)‘𝑘)) ∈ V
149	148	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (1^st ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
150	143, 146, 147, 149	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
151	150	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
152	142, 151	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
153	138	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑙)))
154	153	mpteq2dva 4672	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
155	154	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
156	155	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
157		eqidd 2611	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
158	144	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
159	158	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
160		fvex 6113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (2^nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V
161	160	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
162	157, 159, 147, 161	fvmptd 6197	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
163	162	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
164	156, 163	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
165	152, 164	oveq12d 6567	. . . . . . . . . . . . . . . . . 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))))
166	165	fveq2d 6107	. . . . . . . . . . . . . . . . 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)))))
167	166	prodeq2dv 14492	. . . . . . . . . . . . . . . 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)))))
168	167	cbvmptv 4678	. . . . . . . . . . . . . . 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)))))
169	168	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))))))
170	81	eqcomd 2616	. . . . . . . . . . . . . . . . 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))
171	170	fveq2d 6107	. . . . . . . . . . . . . . . 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)))
172	171	prodeq2dv 14492	. . . . . . . . . . . . . . 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)))
173	172	mpteq2dva 4672	. . . . . . . . . . . . . 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))))
174	169, 173	eqtrd 2644	. . . . . . . . . . . . 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))))
175	174	fveq2d 6107	. . . . . . . . . . . 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)))))
176	175	3ad2ant2 1076	. . . . . . . . . . 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)))))
177	94	adantll 746	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
178	114	adantll 746	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
179	177, 178	oveq12d 6567	. . . . . . . . . . . . . . 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)))))
180	9	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
181		ovnhoilem2.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
182	181	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
183	19	adantlll 750	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
184	183, 20	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
185	184, 22	fmptd 6292	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
186	183, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
187	186, 50	fmptd 6292	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
188	8, 180, 182, 185, 187	hoidmvn0val 39474	. . . . . . . . . . . . . . 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))))
189	179, 188	eqtrd 2644	. . . . . . . . . . . . . 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))))
190	189	mpteq2dva 4672	. . . . . . . . . . . . 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)))))
191	190	fveq2d 6107	. . . . . . . . . . . 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))))))
192	191	3adant3 1074	. . . . . . . . . . 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))))))
193		simp3 1056	. . . . . . . . . . 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)))))
194	176, 192, 193	3eqtr4d 2654	. . . . . . . . . 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)
195	194	3adant3l 1314	. . . . . . . . 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)
196	135, 195	breqtrd 4609	. . . . . . . 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)
197	196	3exp 1256	. . . . . . 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)))
198	197	adantr 480	. . . . . 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)))
199	198	rexlimdv 3012	. . . . 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))
200	7, 199	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
201	200	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
202		ssrab2 3650	. . . . . 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^*
203	1, 202	eqsstri 3598	. . . . 5 ⊢ 𝑀 ⊆ ℝ^*
204	203	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ^*)
205		icossxr 12129	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ^*
206	8, 9, 11, 13	hoidmvcl 39472	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
207	205, 206	sseldi 3566	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^*)
208		infxrgelb 12037	. . . 4 ⊢ ((𝑀 ⊆ ℝ^* ∧ (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^) → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
209	204, 207, 208	syl2anc 691	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
210	201, 209	mpbird 246	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ))
211	66	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
212		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑘𝜑
213	11	ffvelrnda 6267	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
214	13	ffvelrnda 6267	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
215	214	rexrd 9968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
216	212, 213, 215	hoissrrn2 39468	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑_𝑚 𝑋))
217	211, 216	eqsstrd 3602	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑_𝑚 𝑋))
218	9, 181, 217, 1	ovnn0val 39441	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = inf(𝑀, ℝ^, < ))
219	218	eqcomd 2616	. 2 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐼))
220	210, 219	breqtrd 4609	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 ↑_𝑚 cmap 7744 Xcixp 7794 Fincfn 7841 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕcn 10897 [,)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

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-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-n0 11170 df-z 11255 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-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041 df-sumge0 39256 df-ovoln 39427

This theorem is referenced by: ovnhoi 39493

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