Theorem ovnhoi 40817

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:   𝑥,𝑘   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 1843	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoi.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoi.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelrnda 6359	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 10089	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 40792	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑𝑚 𝑋))
11	3, 10	eqsstrd 3639	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑𝑚 𝑋))
12	1, 11	ovnxrcl 40783	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈ ℝ*)
13		icossxr 12258	. . 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 40796	. . 3 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞))
16	13, 15	sseldi 3601	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ*)
17		fveq2 6191	. . . . . . . 8 ⊢ (𝑋 = ∅ → (voln*‘𝑋) = (voln*‘∅))
18	17	fveq1d 6193	. . . . . . 7 ⊢ (𝑋 = ∅ → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
19	18	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼))
20		ixpeq1 7919	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
21		ixp0x 7936	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}
22	21	a1i 11	. . . . . . . . . . 11 ⊢ (𝑋 = ∅ → X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅})
23	20, 22	eqtrd 2656	. . . . . . . . . 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 10027	. . . . . . . . . . 11 ⊢ ℝ ∈ V
27		mapdm0 7872	. . . . . . . . . . 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 2666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑𝑚 ∅))
31		eqimss 3657	. . . . . . . 8 ⊢ (𝐼 = (ℝ ↑𝑚 ∅) → 𝐼 ⊆ (ℝ ↑𝑚 ∅))
32	30, 31	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑𝑚 ∅))
33	32	ovn0val 40764	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘∅)‘𝐼) = 0)
34	19, 33	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0)
35		0red 10041	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈ ℝ)
36	34, 35	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ)
37		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0)
38		fveq2 6191	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
39	38	oveqd 6667	. . . . . . 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 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅)
43	42	feq2d 6031	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
44	41, 43	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
45	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
46	42	feq2d 6031	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
47	45, 46	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
48	14, 44, 47	hoidmv0val 40797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
49	40, 48	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0)
50	37, 34, 49	3eqtr4d 2666	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
51	36, 50	eqled 10140	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
52		eqid 2622	. . . . . 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 2626	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1))
54	53	ifbid 4108	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → if(𝑛 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
55	54	mpteq2dv 4745	. . . . . . 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 4750	. . . . . 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 40815	. . . . 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 2802	. . . . . . 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 40798	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
65	64	eqcomd 2628	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵))
66	58, 65	breqtrd 4679	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
67	51, 66	pm2.61dan 832	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵))
68	49, 35	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ)
69	50	eqcomd 2628	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼))
70	68, 69	eqled 10140	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
71		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘))
72	71	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑏‘𝑘)))
73	72	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
74	73	prodeq2ad 39824	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))
75	74	ifeq2d 4105	. . . . . . . . 9 ⊢ (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))))
76		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘))
77	76	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘)))
78	77	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
79	78	prodeq2ad 39824	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
80	79	ifeq2d 4105	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))
81	75, 80	cbvmpt2v 6735	. . . . . . . 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)))))
82	81	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))))))
83		oveq2 6658	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℝ ↑𝑚 𝑥) = (ℝ ↑𝑚 𝑦))
84		eqeq1 2626	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅))
85		prodeq1 14639	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))
86	84, 85	ifbieq2d 4111	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))
87	83, 83, 86	mpt2eq123dv 6717	. . . . . . 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))))))
88	82, 87	eqtrd 2656	. . . . . 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))))))
89	88	cbvmptv 4750	. . . . 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))))))
90	14, 89	eqtri 2644	. . . 4 ⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑𝑚 𝑦), 𝑑 ∈ (ℝ ↑𝑚 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))))
91		eqeq1 2626	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑤 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))))
92	91	anbi2d 740	. . . . . . 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)))))))
93	92	rexbidv 3052	. . . . . 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)))))))
94		simpl 473	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖)
95	94	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗))
96	95	coeq2d 5284	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
97	96	fveq1d 6193	. . . . . . . . . . . 12 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
98	97	ixpeq2dv 7924	. . . . . . . . . . 11 ⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
99	98	iuneq2dv 4542	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))
100	99	sseq2d 3633	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)))
101		simpl 473	. . . . . . . . . . . . . . . . 17 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖)
102	101	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗))
103	102	coeq2d 5284	. . . . . . . . . . . . . . 15 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗)))
104	103	fveq1d 6193	. . . . . . . . . . . . . 14 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘))
105	104	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
106	105	prodeq2dv 14653	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))
107	106	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))
108	107	fveq2d 6195	. . . . . . . . . 10 ⊢ (ℎ = 𝑖 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))
109	108	eqeq2d 2632	. . . . . . . . 9 ⊢ (ℎ = 𝑖 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
110	100, 109	anbi12d 747	. . . . . . . 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)))))))
111	110	cbvrexv 3172	. . . . . . 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))))))
112	111	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)))))))
113	93, 112	bitrd 268	. . . . 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)))))))
114	113	cbvrabv 3199	. . . 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)))))}
115		simpl 473	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛)
116	115	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛))
117	116	fveq1d 6193	. . . . . . . 8 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙))
118	117	fveq2d 6195	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙)))
119	118	mpteq2dva 4744	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
120	119	cbvmptv 4750	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))
121	120	mpteq2i 4741	. . . 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)))))
122	117	fveq2d 6195	. . . . . . 7 ⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙)))
123	122	mpteq2dva 4744	. . . . . 6 ⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
124	123	cbvmptv 4750	. . . . 5 ⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))
125	124	mpteq2i 4741	. . . 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)))))
126	59, 61, 62, 63, 2, 90, 114, 121, 125	ovnhoilem2 40816	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
127	70, 126	pm2.61dan 832	. 2 ⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼))
128	12, 16, 67, 127	xrletrid 11986	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  Xcixp 7908  Fincfn 7955  ℝcr 9935  0cc0 9936  1c1 9937  +∞cpnf 10071  ℝ^*cxr 10073   ≤ cle 10075  ℕcn 11020  [,)cico 12177  ∏cprod 14635  volcvol 23232  Σ^{^}csumge0 40579  voln*covoln 40750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580  df-ovoln 40751

This theorem is referenced by:  vonhoi  40881