Theorem ovnhoi 44141

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 1917	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoi.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelrnda 6961	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoi.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelrnda 6961	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 11025	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 44116	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3959	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
12	1, 11	ovnxrcl 44107	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13		icossxr 13164	. . 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 44120	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
16	13, 15	sselid 3919	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
17		fveq2 6774	. . . . . . . 8 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
18	17	fveq1d 6776	. . . . . . 7 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
19	18	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20		ixpeq1 8696	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21		ixp0x 8714	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
23	20, 22	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
24	23	adantl 482	. . . . . . . . 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 10962	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		mapdm0 8630	. . . . . . . . . . 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 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31		eqimss 3977	. . . . . . . 8 ⊢ (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
33	32	ovn0val 44088	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
34	19, 33	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35		0red 10978	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈ ℝ)
36	34, 35	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37		eqidd 2739	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0)
38		fveq2 6774	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
39	38	oveqd 7292	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
40	39	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
41	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
43	42	feq2d 6586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
44	41, 43	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
45	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
46	42	feq2d 6586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
47	45, 46	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
48	14, 44, 47	hoidmv0val 44121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
49	40, 48	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0)
50	37, 34, 49	3eqtr4d 2788	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
51	36, 50	eqled 11078	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
52		eqid 2738	. . . . . 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 2742	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
54	53	ifbid 4482	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
55	54	mpteq2dv 5176	. . . . . . 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 5187	. . . . . 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 44139	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
58	57	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
59	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60		neqne 2951	. . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
61	60	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
62	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
63	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
64	14, 59, 61, 62, 63	hoidmvn0val 44122	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
65	64	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵))
66	58, 65	breqtrd 5100	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
67	51, 66	pm2.61dan 810	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
68	49, 35	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ)
69	50	eqcomd 2744	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
70	68, 69	eqled 11078	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71		fveq1 6773	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘))
72	71	fvoveq1d 7297	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
73	72	prodeq2ad 43133	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
74	73	ifeq2d 4479	. . . . . . . . 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 6773	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘))
76	75	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘)))
77	76	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
78	77	prodeq2ad 43133	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
79	78	ifeq2d 4479	. . . . . . . . 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 7370	. . . . . . . 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 7283	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83		eqeq1 2742	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84		prodeq1 15619	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
85	83, 84	ifbieq2d 4485	. . . . . . . 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 7350	. . . . . . 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 2778	. . . . . 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 5187	. . . . 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 2766	. . . 4 ⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
90		eqeq1 2742	. . . . . . . 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 3226	. . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖)
94	93	fveq1d 6776	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗))
95	94	coeq2d 5771	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
96	95	fveq1d 6776	. . . . . . . . . . . 12 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
97	96	ixpeq2dv 8701	. . . . . . . . . . 11 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
98	97	iuneq2dv 4948	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
99	98	sseq2d 3953	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
100		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖)
101	100	fveq1d 6776	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗))
102	101	coeq2d 5771	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
103	102	fveq1d 6776	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
104	103	fveq2d 6778	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
105	104	prodeq2dv 15633	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
106	105	mpteq2dv 5176	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
107	106	fveq2d 6778	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
108	107	eqeq2d 2749	. . . . . . . . 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 3384	. . . . . . 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 278	. . . . 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 3426	. . . 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 483	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛)
115	114	fveq2d 6778	. . . . . . . . 9 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛))
116	115	fveq1d 6776	. . . . . . . 8 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙))
117	116	fveq2d 6778	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙)))
118	117	mpteq2dva 5174	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
119	118	cbvmptv 5187	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
120	119	mpteq2i 5179	. . . 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 6778	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙)))
122	121	mpteq2dva 5174	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
123	122	cbvmptv 5187	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
124	123	mpteq2i 5179	. . . 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 44140	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
126	70, 125	pm2.61dan 810	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
127	12, 16, 67, 126	xrletrid 12889	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3065  {crab 3068  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  ifcif 4459  {csn 4561  ⟨cop 4567  ∪ ciun 4924   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830   ↑_m cmap 8615  Xcixp 8685  Fincfn 8733  ℝcr 10870  0cc0 10871  1c1 10872  +∞cpnf 11006  ℝ^*cxr 11008   ≤ cle 11010  ℕcn 11973  [,)cico 13081  ∏cprod 15615  volcvol 24627  Σ^{^}csumge0 43900  voln*covoln 44074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-sum 15398  df-prod 15616  df-rest 17133  df-topgen 17154  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-ovol 24628  df-vol 24629  df-sumge0 43901  df-ovoln 44075

This theorem is referenced by:  vonhoi  44205