ovnhoilem2 - Mathbox for Glauco Siliprandi

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Theorem ovnhoilem2 40816

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: 𝑥,𝑘 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 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 2693	. . . . . . . . 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 3116	. . . . . . . . 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 264	. . . . . . . 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 206	. . . . . . 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 479	. . . . . 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 482	. . . . 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 1082	. . . . . . . . . 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 1082	. . . . . . . . . 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 1082	. . . . . . . . . 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 7879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑_𝑚 𝑋))
16	15	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
17		elmapi 7879	. . . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
20		xp1st 7198	. . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))
23	21, 22	fmptd 6385	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
24		reex 10027	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
25	24	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
26		1nn 11031	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
27	26	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → 1 ∈ ℕ)
28	15, 27	ffvelrnd 6360	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
29		elmapex 7878	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
30	29	simprd 479	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
33		elmapg 7870	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
36		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
37	35, 36	fmptd 6385	. . . . . . . . . . . 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 11026	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
40	39	mptex 6486	. . . . . . . . . . . . . . 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 6291	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
45	44	feq1d 6030	. . . . . . . . . . . 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 247	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝐹‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
47	46	3ad2ant2 1083	. . . . . . . . . 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 7199	. . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))
51	49, 50	fmptd 6385	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
52		elmapg 7870	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_𝑚 𝑋))
55		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
56	54, 55	fmptd 6385	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_𝑚 𝑋))
57	39	mptex 6486	. . . . . . . . . . . . . . 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 6291	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
62	61	feq1d 6030	. . . . . . . . . . . 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 247	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑆‘𝑖):ℕ⟶(ℝ ↑_𝑚 𝑋))
64	63	3ad2ant2 1083	. . . . . . . . . 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 1063	. . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
69	68	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → ((𝑖‘𝑗)‘𝑘) = ((𝑖‘𝑛)‘𝑘))
70	69	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (1^st ‘((𝑖‘𝑗)‘𝑘)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
71	69	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
72	70, 71	oveq12d 6668	. . . . . . . . . . . . . . . . . . 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 7924	. . . . . . . . . . . . . . . . . 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 4559	. . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑_𝑚 𝑋))
77		elmapi 7879	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
80		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
81	79, 80	fvovco 39381	. . . . . . . . . . . . . . . . . 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 7923	. . . . . . . . . . . . . . . . 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 4542	. . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
87	86	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
89		mptexg 6484	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
92	36	fvmpt2 6291	. . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
95	94	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 2623	. . . . . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
99	98	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
100	99	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
101		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
102		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
103	97, 100, 101, 102	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
104	103	adantlr 751	. . . . . . . . . . . . . . . . . . . 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)))
105	96, 104	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
106	61	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
107	106	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
108		mptexg 6484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑋 ∈ V → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
109	31, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
110	109	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
111	55	fvmpt2 6291	. . . . . . . . . . . . . . . . . . . . . . . 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))))
112	88, 110, 111	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 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))))
113	107, 112	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
114	113	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
115	114	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))
116		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
117		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
118	117	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
119	118	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
120		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
121	116, 119, 101, 120	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
122	121	adantlr 751	. . . . . . . . . . . . . . . . . . . 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)))
123	115, 122	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑖)‘𝑛)‘𝑘) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
124	105, 123	oveq12d 6668	. . . . . . . . . . . . . . . . . 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))))
125	124	ixpeq2dva 7923	. . . . . . . . . . . . . . . . 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))))
126	125	iuneq2dv 4542	. . . . . . . . . . . . . . . 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))))
127	75, 83, 126	3eqtr4d 2666	. . . . . . . . . . . . . . 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)))
128	127	adantl 482	. . . . . . . . . . . . . 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)))
129	128	3adant3 1081	. . . . . . . . . . . . 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)))
130	67, 129	sseq12d 3634	. . . . . . . . . . . 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))))
131	65, 130	mpbid 222	. . . . . . . . . . 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)))
132	131	3adant3r 1323	. . . . . . . . . 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)))
133	8, 10, 12, 14, 47, 64, 132	hoidmvle 40814	. . . . . . . . 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)))))
134		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → 𝑛 = 𝑗)
135	134	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑛) = (𝑖‘𝑗))
136	135	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑗)‘𝑙))
137	136	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑙)))
138	137	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
139	138	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
140	139	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
141		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
142		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 = 𝑘 → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑗)‘𝑘))
143	142	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
144	143	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
145		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
146		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (1^st ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
147	141, 144, 145, 146	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
148	147	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
149	140, 148	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
150	136	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑙)))
151	150	mpteq2dva 4744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
152	151	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
153	152	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
154		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
155	142	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
156	155	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
157		fvexd 6203	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
158	154, 156, 145, 157	fvmptd 6288	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
159	158	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
160	153, 159	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
161	149, 160	oveq12d 6668	. . . . . . . . . . . . . . . . . 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))))
162	161	fveq2d 6195	. . . . . . . . . . . . . . . . 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)))))
163	162	prodeq2dv 14653	. . . . . . . . . . . . . . . 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)))))
164	163	cbvmptv 4750	. . . . . . . . . . . . . . 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)))))
165	164	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))))))
166	81	eqcomd 2628	. . . . . . . . . . . . . . . . 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))
167	166	fveq2d 6195	. . . . . . . . . . . . . . . 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)))
168	167	prodeq2dv 14653	. . . . . . . . . . . . . . 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)))
169	168	mpteq2dva 4744	. . . . . . . . . . . . . 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))))
170	165, 169	eqtrd 2656	. . . . . . . . . . . . 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))))
171	170	fveq2d 6195	. . . . . . . . . . . 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)))))
172	171	3ad2ant2 1083	. . . . . . . . . . 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)))))
173	94	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
174	113	adantll 750	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
175	173, 174	oveq12d 6668	. . . . . . . . . . . . . . 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)))))
176	9	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
177		ovnhoilem2.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
178	177	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
179	19	adantlll 754	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
180	179, 20	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
181	180, 22	fmptd 6385	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
182	179, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) ∈ ℝ)
183	182, 50	fmptd 6385	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_𝑚 𝑋) ↑_𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
184	8, 176, 178, 181, 183	hoidmvn0val 40798	. . . . . . . . . . . . . . 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))))
185	175, 184	eqtrd 2656	. . . . . . . . . . . . . 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))))
186	185	mpteq2dva 4744	. . . . . . . . . . . . 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)))))
187	186	fveq2d 6195	. . . . . . . . . . . 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))))))
188	187	3adant3 1081	. . . . . . . . . . 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))))))
189		simp3 1063	. . . . . . . . . . 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)))))
190	172, 188, 189	3eqtr4d 2666	. . . . . . . . . 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)
191	190	3adant3l 1322	. . . . . . . . 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)
192	133, 191	breqtrd 4679	. . . . . . . 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)
193	192	3exp 1264	. . . . . . 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)))
194	193	adantr 481	. . . . . 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)))
195	194	rexlimdv 3030	. . . . 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))
196	7, 195	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
197	196	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
198		ssrab2 3687	. . . . . 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^*
199	1, 198	eqsstri 3635	. . . . 5 ⊢ 𝑀 ⊆ ℝ^*
200	199	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ^*)
201		icossxr 12258	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ^*
202	8, 9, 11, 13	hoidmvcl 40796	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
203	201, 202	sseldi 3601	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^*)
204		infxrgelb 12165	. . . 4 ⊢ ((𝑀 ⊆ ℝ^* ∧ (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^) → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
205	200, 203, 204	syl2anc 693	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
206	197, 205	mpbird 247	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ))
207	66	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
208		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑘𝜑
209	11	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
210	13	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
211	210	rexrd 10089	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
212	208, 209, 211	hoissrrn2 40792	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑_𝑚 𝑋))
213	207, 212	eqsstrd 3639	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑_𝑚 𝑋))
214	9, 177, 213, 1	ovnn0val 40765	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = inf(𝑀, ℝ^, < ))
215	214	eqcomd 2628	. 2 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐼))
216	206, 215	breqtrd 4679	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 ↑_𝑚 cmap 7857 Xcixp 7908 Fincfn 7955 infcinf 8347 ℝcr 9935 0cc0 9936 1c1 9937 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 [,)cico 12177 ∏cprod 14635 volcvol 23232 Σ^{^}csumge0 40579 voln*covoln 40750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-prod 14636 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-sumge0 40580 df-ovoln 40751

This theorem is referenced by: ovnhoi 40817

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