ovnhoilem2 - Mathbox for Glauco Siliprandi

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Theorem ovnhoilem2 46053

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 2817	. . . . . . . . 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 3440	. . . . . . . . 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 274	. . . . . . . 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 215	. . . . . . 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 494	. . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑀) → ∃𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
8		ovnhoilem2.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
9		ovnhoilem2.x	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ Fin)
10	9	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝑋 ∈ Fin)
11		ovnhoilem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
12	11	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐴:𝑋⟶ℝ)
13		ovnhoilem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
14	13	3ad2ant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → 𝐵:𝑋⟶ℝ)
15		elmapi 8866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → 𝑖:ℕ⟶((ℝ × ℝ) ↑_m 𝑋))
16	15	ffvelcdmda 7091	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑖‘𝑛) ∈ ((ℝ × ℝ) ↑_m 𝑋))
17		elmapi 8866	. . . . . . . . . . . . . . . . . 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	ffvelcdmda 7091	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) ∈ (ℝ × ℝ))
20		xp1st 8024	. . . . . . . . . . . . . . . 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 7122	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
23		reex 11229	. . . . . . . . . . . . . . . 16 ⊢ ℝ ∈ V
24	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ℝ ∈ V)
25		1nn 12253	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℕ
26	25	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → 1 ∈ ℕ)
27	15, 26	ffvelcdmd 7092	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝑖‘1) ∈ ((ℝ × ℝ) ↑_m 𝑋))
28		elmapex 8865	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖‘1) ∈ ((ℝ × ℝ) ↑_m 𝑋) → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ V))
29	28	simprd 494	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ V)
32		elmapg 8856	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_m 𝑋))
35	34	fmpttd 7122	. . . . . . . . . . . 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 12248	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
38	37	mptex 7233	. . . . . . . . . . . . . . 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 7013	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
43	42	feq1d 6706	. . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝐹‘𝑖):ℕ⟶(ℝ ↑_m 𝑋))
45	44	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐹‘𝑖):ℕ⟶(ℝ ↑_m 𝑋))
46		xp2nd 8025	. . . . . . . . . . . . . . . 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 7122	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))):𝑋⟶ℝ)
49		elmapg 8856	. . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ (ℝ ↑_m 𝑋))
52	51	fmpttd 7122	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))):ℕ⟶(ℝ ↑_m 𝑋))
53	37	mptex 7233	. . . . . . . . . . . . . . 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 7013	. . . . . . . . . . . . . 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 582	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝑆‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))))
58	57	feq1d 6706	. . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → (𝑆‘𝑖):ℕ⟶(ℝ ↑_m 𝑋))
60	59	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝑆‘𝑖):ℕ⟶(ℝ ↑_m 𝑋))
61		simp3 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
62		ovnhoilem2.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
63	62	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
64		fveq2 6894	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 𝑛 → (𝑖‘𝑗) = (𝑖‘𝑛))
65	64	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 𝑛 → ((𝑖‘𝑗)‘𝑘) = ((𝑖‘𝑛)‘𝑘))
66	65	fveq2d 6898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (1^st ‘((𝑖‘𝑗)‘𝑘)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
67	65	fveq2d 6898	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 𝑛 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
68	66, 67	oveq12d 7435	. . . . . . . . . . . . . . . . . . 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 8930	. . . . . . . . . . . . . . . . . 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 5043	. . . . . . . . . . . . . . . . 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	ffvelcdmda 7091	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑗 ∈ ℕ) → (𝑖‘𝑗) ∈ ((ℝ × ℝ) ↑_m 𝑋))
73		elmapi 8866	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑖‘𝑗):𝑋⟶(ℝ × ℝ))
76		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
77	75, 76	fvovco 44630	. . . . . . . . . . . . . . . . . 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 8929	. . . . . . . . . . . . . . . . 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 5020	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑖) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))))
83	82	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
85		mptexg 7231	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
88		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
89	88	fvmpt2 7013	. . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
92	91	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 𝑙 = 𝑘)
96	95	fveq2d 6898	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑛)‘𝑘))
97	96	fveq2d 6898	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
98		simpr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
99		fvexd 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
100	94, 97, 98, 99	fvmptd 7009	. . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐹‘𝑖)‘𝑛)‘𝑘) = (1^st ‘((𝑖‘𝑛)‘𝑘)))
103	57	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → ((𝑆‘𝑖)‘𝑛) = ((𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))‘𝑛))
104	103	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 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 7231	. . . . . . . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) ∈ V)
108		eqid 2725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
109	108	fvmpt2 7013	. . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑛 ∈ ℕ) → ((𝑆‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))))
112	111	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
116	115	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑛)‘𝑘)))
117		fvexd 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑘 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑘)) ∈ V)
118	114, 116, 98, 117	fvmptd 7009	. . . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . . . . 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 8929	. . . . . . . . . . . . . . . . 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 5020	. . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
126	125	3adant3 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
127	63, 126	sseq12d 4011	. . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
129	128	3adant3r 1178	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝐹‘𝑖)‘𝑛)‘𝑘)[,)(((𝑆‘𝑖)‘𝑛)‘𝑘)))
130	8, 10, 12, 14, 45, 60, 129	hoidmvle 46051	. . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → 𝑛 = 𝑗)
132	131	fveq2d 6898	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑛) = (𝑖‘𝑗))
133	132	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑛)‘𝑙) = ((𝑖‘𝑗)‘𝑙))
134	133	fveq2d 6898	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (1^st ‘((𝑖‘𝑛)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑙)))
135	134	mpteq2dva 5248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
136	135	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
137	136	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
138		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙))))
139		2fveq3 6899	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (1^st ‘((𝑖‘𝑗)‘𝑙)) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
140	139	adantl 480	. . . . . . . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (1^st ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
143	138, 140, 141, 142	fvmptd 7009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
144	143	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
145	137, 144	eqtrd 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (1^st ‘((𝑖‘𝑗)‘𝑘)))
146	133	fveq2d 6898	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑛 = 𝑗 ∧ 𝑙 ∈ 𝑋) → (2^nd ‘((𝑖‘𝑛)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑙)))
147	146	mpteq2dva 5248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
148	147	fveq1d 6896	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
149	148	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘))
150		eqidd 2726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙))))
151		2fveq3 6899	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑘 → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
152	151	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ 𝑋 ∧ 𝑙 = 𝑘) → (2^nd ‘((𝑖‘𝑗)‘𝑙)) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
153		fvexd 6909	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝑋 → (2^nd ‘((𝑖‘𝑗)‘𝑘)) ∈ V)
154	150, 152, 141, 153	fvmptd 7009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑋 → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
155	154	adantl 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑗)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
156	149, 155	eqtrd 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 = 𝑗 ∧ 𝑘 ∈ 𝑋) → ((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘) = (2^nd ‘((𝑖‘𝑗)‘𝑘)))
157	145, 156	oveq12d 7435	. . . . . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . . 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 15899	. . . . . . . . . . . . . . . 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 5261	. . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . 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 6898	. . . . . . . . . . . . . . . 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 15899	. . . . . . . . . . . . . . 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 5248	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . 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 6898	. . . . . . . . . . . 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 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (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 712	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑖)‘𝑛) = (𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙))))
170	111	adantll 712	. . . . . . . . . . . . . . . 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 7435	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
173		ovnhoilem2.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
174	173	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ)) ∧ 𝑛 ∈ ℕ) → 𝑋 ≠ ∅)
175	19	adantlll 716	. . . . . . . . . . . . . . . . . 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 7122	. . . . . . . . . . . . . . . 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 7122	. . . . . . . . . . . . . . . 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 46035	. . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 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 5248	. . . . . . . . . . . . 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 6898	. . . . . . . . . . . 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 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = (Σ^{^}‘(𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝑙 ∈ 𝑋 ↦ (1^st ‘((𝑖‘𝑛)‘𝑙)))‘𝑘)[,)((𝑙 ∈ 𝑋 ↦ (2^nd ‘((𝑖‘𝑛)‘𝑙)))‘𝑘))))))
185		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
186	168, 184, 185	3eqtr4d 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = 𝑧)
187	186	3adant3l 1177	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (Σ^{^}‘(𝑛 ∈ ℕ ↦ (((𝐹‘𝑖)‘𝑛)(𝐿‘𝑋)((𝑆‘𝑖)‘𝑛)))) = 𝑧)
188	130, 187	breqtrd 5174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
189	188	3exp 1116	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (((ℝ × ℝ) ↑_m 𝑋) ↑_m ℕ) → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^{^}‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) → (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)))
190	189	adantr 479	. . . . . 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 3143	. . . . 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 3136	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧)
194		ssrab2 4074	. . . . . 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 4012	. . . . 5 ⊢ 𝑀 ⊆ ℝ^*
196	195	a1i 11	. . . 4 ⊢ (𝜑 → 𝑀 ⊆ ℝ^*)
197		icossxr 13441	. . . . 5 ⊢ (0[,)+∞) ⊆ ℝ^*
198	8, 9, 11, 13	hoidmvcl 46033	. . . . 5 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
199	197, 198	sselid 3975	. . . 4 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ^*)
200		infxrgelb 13346	. . . 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 582	. . 3 ⊢ (𝜑 → ((𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐴(𝐿‘𝑋)𝐵) ≤ 𝑧))
202	193, 201	mpbird 256	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ inf(𝑀, ℝ^*, < ))
203	62	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
204		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑘𝜑
205	11	ffvelcdmda 7091	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
206	13	ffvelcdmda 7091	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
207	206	rexrd 11294	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
208	204, 205, 207	hoissrrn2 46029	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑_m 𝑋))
209	203, 208	eqsstrd 4016	. . . 4 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑_m 𝑋))
210	9, 173, 209, 1	ovnn0val 46002	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = inf(𝑀, ℝ^, < ))
211	210	eqcomd 2731	. 2 ⊢ (𝜑 → inf(𝑀, ℝ^, < ) = ((voln‘𝑋)‘𝐼))
212	202, 211	breqtrd 5174	1 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {crab 3419 Vcvv 3463 ⊆ wss 3945 ∅c0 4323 ifcif 4529 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 × cxp 5675 ∘ ccom 5681 ⟶wf 6543 ‘cfv 6547 (class class class)co 7417 ∈ cmpo 7419 1^st c1st 7990 2^nd c2nd 7991 ↑_m cmap 8843 Xcixp 8914 Fincfn 8962 infcinf 9464 ℝcr 11137 0cc0 11138 1c1 11139 +∞cpnf 11275 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279 ℕcn 12242 [,)cico 13358 ∏cprod 15881 volcvol 25422 Σ^{^}csumge0 45813 voln*covoln 45987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-isom 6556 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7683 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-prod 15882 df-rest 17403 df-topgen 17424 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-top 22826 df-topon 22843 df-bases 22879 df-cmp 23321 df-ovol 25423 df-vol 25424 df-sumge0 45814 df-ovoln 45988

This theorem is referenced by: ovnhoi 46054

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