Theorem ovnhoi 45991

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 1910	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoi.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelcdmda 7094	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoi.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelcdmda 7094	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 11295	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 45966	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 4018	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
12	1, 11	ovnxrcl 45957	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13		icossxr 13442	. . 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 45970	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
16	13, 15	sselid 3978	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
17		fveq2 6897	. . . . . . . 8 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
18	17	fveq1d 6899	. . . . . . 7 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
19	18	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20		ixpeq1 8927	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21		ixp0x 8945	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
23	20, 22	eqtrd 2768	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
24	23	adantl 481	. . . . . . . . 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 11230	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		mapdm0 8861	. . . . . . . . . . 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 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31		eqimss 4038	. . . . . . . 8 ⊢ (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
33	32	ovn0val 45938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
34	19, 33	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35		0red 11248	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈ ℝ)
36	34, 35	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37		eqidd 2729	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0)
38		fveq2 6897	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
39	38	oveqd 7437	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
40	39	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
41	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
43	42	feq2d 6708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
44	41, 43	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
45	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
46	42	feq2d 6708	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
47	45, 46	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
48	14, 44, 47	hoidmv0val 45971	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
49	40, 48	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0)
50	37, 34, 49	3eqtr4d 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
51	36, 50	eqled 11348	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
52		eqid 2728	. . . . . 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 2732	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
54	53	ifbid 4552	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
55	54	mpteq2dv 5250	. . . . . . 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 5261	. . . . . 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 45989	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
58	57	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
59	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60		neqne 2945	. . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
61	60	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
62	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
63	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
64	14, 59, 61, 62, 63	hoidmvn0val 45972	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
65	64	eqcomd 2734	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵))
66	58, 65	breqtrd 5174	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
67	51, 66	pm2.61dan 812	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
68	49, 35	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ)
69	50	eqcomd 2734	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
70	68, 69	eqled 11348	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71		fveq1 6896	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘))
72	71	fvoveq1d 7442	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
73	72	prodeq2ad 44980	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
74	73	ifeq2d 4549	. . . . . . . . 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 6896	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘))
76	75	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘)))
77	76	fveq2d 6901	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
78	77	prodeq2ad 44980	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
79	78	ifeq2d 4549	. . . . . . . . 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 7515	. . . . . . . 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 7428	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83		eqeq1 2732	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84		prodeq1 15886	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
85	83, 84	ifbieq2d 4555	. . . . . . . 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 7495	. . . . . . 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 2768	. . . . . 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 5261	. . . . 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 2756	. . . 4 ⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
90		eqeq1 2732	. . . . . . . 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 629	. . . . . . 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 3175	. . . . . 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 482	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖)
94	93	fveq1d 6899	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗))
95	94	coeq2d 5865	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
96	95	fveq1d 6899	. . . . . . . . . . . 12 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
97	96	ixpeq2dv 8932	. . . . . . . . . . 11 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
98	97	iuneq2dv 5020	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
99	98	sseq2d 4012	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
100		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖)
101	100	fveq1d 6899	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗))
102	101	coeq2d 5865	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
103	102	fveq1d 6899	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
104	103	fveq2d 6901	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
105	104	prodeq2dv 15900	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
106	105	mpteq2dv 5250	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
107	106	fveq2d 6901	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
108	107	eqeq2d 2739	. . . . . . . . 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 631	. . . . . . . 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 3232	. . . . . . 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 279	. . . . 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 3439	. . . 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 482	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛)
115	114	fveq2d 6901	. . . . . . . . 9 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛))
116	115	fveq1d 6899	. . . . . . . 8 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙))
117	116	fveq2d 6901	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙)))
118	117	mpteq2dva 5248	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
119	118	cbvmptv 5261	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
120	119	mpteq2i 5253	. . . 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 6901	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙)))
122	121	mpteq2dva 5248	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
123	122	cbvmptv 5261	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
124	123	mpteq2i 5253	. . . 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 45990	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
126	70, 125	pm2.61dan 812	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
127	12, 16, 67, 126	xrletrid 13167	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ≠ wne 2937  ∃wrex 3067  {crab 3429  Vcvv 3471   ⊆ wss 3947  ∅c0 4323  ifcif 4529  {csn 4629  ⟨cop 4635  ∪ ciun 4996   class class class wbr 5148   ↦ cmpt 5231   × cxp 5676   ∘ ccom 5682  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420   ∈ cmpo 7422  1^st c1st 7991  2^nd c2nd 7992   ↑_m cmap 8845  Xcixp 8916  Fincfn 8964  ℝcr 11138  0cc0 11139  1c1 11140  +∞cpnf 11276  ℝ^*cxr 11278   ≤ cle 11280  ℕcn 12243  [,)cico 13359  ∏cprod 15882  volcvol 25405  Σ^{^}csumge0 45750  voln*covoln 45924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  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 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9435  df-sup 9466  df-inf 9467  df-oi 9534  df-dju 9925  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-n0 12504  df-z 12590  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-ioo 13361  df-ico 13363  df-icc 13364  df-fz 13518  df-fzo 13661  df-fl 13790  df-seq 14000  df-exp 14060  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-clim 15465  df-rlim 15466  df-sum 15666  df-prod 15883  df-rest 17404  df-topgen 17425  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-top 22809  df-topon 22826  df-bases 22862  df-cmp 23304  df-ovol 25406  df-vol 25407  df-sumge0 45751  df-ovoln 45925

This theorem is referenced by:  vonhoi  46055

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