Theorem ovnhoi 47439

Description: The Lebesgue outer measure of a multidimensional half-open interval is its dimensional volume (the product of its length in each dimension, when the dimension is nonzero). Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
ovnhoi.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnhoi.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
ovnhoi.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
ovnhoi.c	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
ovnhoi.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))

Assertion

Ref	Expression
ovnhoi	⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑘   𝐵,𝑎,𝑏,𝑘   𝑋,𝑎,𝑏,𝑘,𝑥   𝜑,𝑎,𝑏,𝑘,𝑥

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐼(𝑥, 𝑘, 𝑎, 𝑏)   𝐿(𝑥, 𝑘, 𝑎, 𝑏)

Proof of Theorem ovnhoi

Dummy variables 𝑐 𝑑 𝑖 𝑗 𝑛 𝑧 𝑦 ℎ 𝑤 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovnhoi.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovnhoi.c	. . . . 5 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
4		nfv 1947	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoi.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoi.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 11287	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 47414	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3968	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
12	1, 11	ovnxrcl 47405	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13		icossxr 13489	. . 3 ⊢ (0[,)+∞) ⊆ ℝ*
14		ovnhoi.l	. . . 4 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
15	14, 1, 5, 7	hoidmvcl 47418	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
16	13, 15	sselid 3932	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
17		fveq2 6882	. . . . . . . 8 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
18	17	fveq1d 6884	. . . . . . 7 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
19	18	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20		ixpeq1 8919	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21		ixp0x 8937	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
23	20, 22	eqtrd 2797	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
24	23	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
25	2	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
26		reex 11219	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		mapdm0 8845	. . . . . . . . . . 11 ⊢ (ℝ ∈ V → (ℝ ↑m ∅) = {∅})
28	26, 27	ax-mp 5	. . . . . . . . . 10 ⊢ (ℝ ↑m ∅) = {∅}
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ ↑m ∅) = {∅})
30	24, 25, 29	3eqtr4d 2807	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31		eqimss 3992	. . . . . . . 8 ⊢ (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
32	30, 31	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
33	32	ovn0val 47386	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
34	19, 33	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35		0red 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈ ℝ)
36	34, 35	eqeltrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37		eqidd 2763	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0)
38		fveq2 6882	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
39	38	oveqd 7434	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
40	39	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
41	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
43	42	feq2d 6690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
44	41, 43	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
45	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
46	42	feq2d 6690	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
47	45, 46	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
48	14, 44, 47	hoidmv0val 47419	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
49	40, 48	eqtrd 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0)
50	37, 34, 49	3eqtr4d 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
51	36, 50	eqled 11341	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
52		eqid 2762	. . . . . 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)))))} = {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)))))}
53		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
54	53	ifbid 4509	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
55	54	mpteq2dv 5203	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (𝑘 ∈ 𝑋 ↦ if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
56	55	cbvmptv 5213	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
57	1, 5, 7, 2, 52, 56	ovnhoilem1 47437	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
58	57	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
59	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60		neqne 2965	. . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
61	60	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
62	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
63	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
64	14, 59, 61, 62, 63	hoidmvn0val 47420	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
65	64	eqcomd 2768	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵))
66	58, 65	breqtrd 5135	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
67	51, 66	pm2.61dan 825	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
68	49, 35	eqeltrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ)
69	50	eqcomd 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
70	68, 69	eqled 11341	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71		fveq1 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘))
72	71	fvoveq1d 7439	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
73	72	prodeq2ad 46430	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
74	73	ifeq2d 4506	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))))
75		fveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘))
76	75	oveq2d 7433	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘)))
77	76	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
78	77	prodeq2ad 46430	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
79	78	ifeq2d 4506	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))
80	74, 79	cbvmpov 7512	. . . . . . . 8 ⊢ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))
81	80	a1i 11	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
82		oveq2 7425	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84		prodeq1 16000	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
85	83, 84	ifbieq2d 4512	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))
86	82, 82, 85	mpoeq123dv 7492	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
87	81, 86	eqtrd 2797	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
88	87	cbvmptv 5213	. . . . 5 ⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
89	14, 88	eqtri 2785	. . . 4 ⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
90		eqeq1 2766	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))))
91	90	anbi2d 642	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))))))
92	91	rexbidv 3188	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))))))
93		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖)
94	93	fveq1d 6884	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗))
95	94	coeq2d 5846	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
96	95	fveq1d 6884	. . . . . . . . . . . 12 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
97	96	ixpeq2dv 8924	. . . . . . . . . . 11 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
98	97	iuneq2dv 4979	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
99	98	sseq2d 3966	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
100		simpl 488	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖)
101	100	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗))
102	101	coeq2d 5846	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
103	102	fveq1d 6884	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
104	103	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
105	104	prodeq2dv 16015	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
106	105	mpteq2dv 5203	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
107	106	fveq2d 6886	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
108	107	eqeq2d 2773	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
109	99, 108	anbi12d 644	. . . . . . . 8 ⊢ (ℎ = 𝑖 → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
110	109	cbvrexvw 3243	. . . . . . 7 ⊢ (∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃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))))))
111	110	a1i 11	. . . . . 6 ⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃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)))))))
112	92, 111	bitrd 282	. . . . 5 ⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃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)))))))
113	112	cbvrabv 3424	. . . 4 ⊢ {𝑤 ∈ ℝ* ∣ ∃ℎ ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘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)))))}
114		simpl 488	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛)
115	114	fveq2d 6886	. . . . . . . . 9 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛))
116	115	fveq1d 6884	. . . . . . . 8 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙))
117	116	fveq2d 6886	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙)))
118	117	mpteq2dva 5202	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
119	118	cbvmptv 5213	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
120	119	mpteq2i 5205	. . . 4 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))))
121	116	fveq2d 6886	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙)))
122	121	mpteq2dva 5202	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
123	122	cbvmptv 5213	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
124	123	mpteq2i 5205	. . . 4 ⊢ (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))))
125	59, 61, 62, 63, 2, 89, 113, 120, 124	ovnhoilem2 47438	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
126	70, 125	pm2.61dan 825	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
127	12, 16, 67, 126	xrletrid 13210	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088  {crab 3414  Vcvv 3453   ⊆ wss 3902  ∅c0 4282  ifcif 4485  {csn 4587  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   ↦ cmpt 5190   × cxp 5657   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  1^st c1st 7988  2^nd c2nd 7989   ↑_m cmap 8830  Xcixp 8908  Fincfn 8956  ℝcr 11127  0cc0 11128  1c1 11129  +∞cpnf 11268  ℝ^*cxr 11270   ≤ cle 11272  ℕcn 12261  [,)cico 13404  ∏cprod 15996  volcvol 25697  Σ^{^}csumge0 47198  voln*covoln 47372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-inf2 9624  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-pre-sup 11206

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-pm 8833  df-ixp 8909  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-fi 9385  df-sup 9416  df-inf 9417  df-oi 9486  df-dju 9910  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-div 11900  df-nn 12262  df-2 12331  df-3 12332  df-n0 12533  df-z 12620  df-uz 12892  df-q 13002  df-rp 13047  df-xneg 13167  df-xadd 13168  df-xmul 13169  df-ioo 13406  df-ico 13408  df-icc 13409  df-fz 13566  df-fzo 13714  df-fl 13857  df-seq 14070  df-exp 14130  df-hash 14399  df-cj 15190  df-re 15191  df-im 15192  df-sqrt 15326  df-abs 15327  df-clim 15579  df-rlim 15580  df-sum 15778  df-prod 15997  df-rest 17513  df-topgen 17534  df-psmet 21583  df-xmet 21584  df-met 21585  df-bl 21586  df-mopn 21587  df-top 23125  df-topon 23142  df-bases 23177  df-cmp 23618  df-ovol 25698  df-vol 25699  df-sumge0 47199  df-ovoln 47373

This theorem is used by:  vonhoi  47503