Theorem ovnhoi 43242

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 1915	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoi.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelrnda 6828	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoi.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelrnda 6828	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 10680	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 43217	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3953	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
12	1, 11	ovnxrcl 43208	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13		icossxr 12810	. . 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 43221	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
16	13, 15	sseldi 3913	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
17		fveq2 6645	. . . . . . . 8 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
18	17	fveq1d 6647	. . . . . . 7 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
19	18	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20		ixpeq1 8455	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21		ixp0x 8473	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
23	20, 22	eqtrd 2833	. . . . . . . . . 10 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
24	23	adantl 485	. . . . . . . . 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 10617	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		mapdm0 8404	. . . . . . . . . . 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 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑m ∅))
31		eqimss 3971	. . . . . . . 8 ⊢ (𝐼 = (ℝ ↑m ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m ∅))
33	32	ovn0val 43189	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
34	19, 33	eqtrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35		0red 10633	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈ ℝ)
36	34, 35	eqeltrd 2890	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37		eqidd 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0)
38		fveq2 6645	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
39	38	oveqd 7152	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
40	39	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
41	5	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
42		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
43	42	feq2d 6473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
44	41, 43	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
45	7	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
46	42	feq2d 6473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
47	45, 46	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
48	14, 44, 47	hoidmv0val 43222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
49	40, 48	eqtrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0)
50	37, 34, 49	3eqtr4d 2843	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
51	36, 50	eqled 10732	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
52		eqid 2798	. . . . . 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 2802	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
54	53	ifbid 4447	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
55	54	mpteq2dv 5126	. . . . . . 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 5133	. . . . . 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 43240	. . . . 5 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
58	57	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
59	1	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
60		neqne 2995	. . . . . . 7 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
61	60	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
62	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
63	7	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
64	14, 59, 61, 62, 63	hoidmvn0val 43223	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
65	64	eqcomd 2804	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵))
66	58, 65	breqtrd 5056	. . 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 2890	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ)
69	50	eqcomd 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
70	68, 69	eqled 10732	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71		fveq1 6644	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘))
72	71	fvoveq1d 7157	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
73	72	prodeq2ad 42234	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
74	73	ifeq2d 4444	. . . . . . . . 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 6644	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘))
76	75	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘)))
77	76	fveq2d 6649	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
78	77	prodeq2ad 42234	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
79	78	ifeq2d 4444	. . . . . . . . 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 7228	. . . . . . . 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 7143	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m 𝑦))
83		eqeq1 2802	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
84		prodeq1 15255	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
85	83, 84	ifbieq2d 4450	. . . . . . . 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 7208	. . . . . . 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 2833	. . . . . 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 5133	. . . . 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 2821	. . . 4 ⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
90		eqeq1 2802	. . . . . . . 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 631	. . . . . . 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 3256	. . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖)
94	93	fveq1d 6647	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗))
95	94	coeq2d 5697	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
96	95	fveq1d 6647	. . . . . . . . . . . 12 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
97	96	ixpeq2dv 8460	. . . . . . . . . . 11 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
98	97	iuneq2dv 4905	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
99	98	sseq2d 3947	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
100		simpl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖)
101	100	fveq1d 6647	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗))
102	101	coeq2d 5697	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
103	102	fveq1d 6647	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
104	103	fveq2d 6649	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
105	104	prodeq2dv 15269	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
106	105	mpteq2dv 5126	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
107	106	fveq2d 6649	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
108	107	eqeq2d 2809	. . . . . . . . 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 633	. . . . . . . 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 3397	. . . . . . 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 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 486	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛)
115	114	fveq2d 6649	. . . . . . . . 9 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛))
116	115	fveq1d 6647	. . . . . . . 8 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙))
117	116	fveq2d 6649	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙)))
118	117	mpteq2dva 5125	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
119	118	cbvmptv 5133	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
120	119	mpteq2i 5122	. . . 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 6649	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙)))
122	121	mpteq2dva 5125	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
123	122	cbvmptv 5133	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
124	123	mpteq2i 5122	. . . 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 43241	. . 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 12536	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ifcif 4425  {csn 4525  ⟨cop 4531  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110   × cxp 5517   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  1^st c1st 7669  2^nd c2nd 7670   ↑_m cmap 8389  Xcixp 8444  Fincfn 8492  ℝcr 10525  0cc0 10526  1c1 10527  +∞cpnf 10661  ℝ^*cxr 10663   ≤ cle 10665  ℕcn 11625  [,)cico 12728  ∏cprod 15251  volcvol 24067  Σ^{^}csumge0 43001  voln*covoln 43175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-prod 15252  df-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cmp 21992  df-ovol 24068  df-vol 24069  df-sumge0 43002  df-ovoln 43176

This theorem is referenced by:  vonhoi  43306