Theorem ovnhoilem1 46602

Description: The Lebesgue outer measure of a multidimensional half-open interval is less than or equal to the product of its length in each dimension. First part of the proof of Proposition 115D (b) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
ovnhoilem1.x	⊢ (𝜑 → 𝑋 ∈ Fin)
ovnhoilem1.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
ovnhoilem1.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
ovnhoilem1.c	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
ovnhoilem1.m	⊢ 𝑀 = {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)))))}
ovnhoilem1.h	⊢ 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))

Assertion

Ref	Expression
ovnhoilem1	⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))

Distinct variable groups:   𝐴,𝑖,𝑗,𝑧   𝐵,𝑖,𝑗,𝑧   𝑖,𝐻,𝑗   𝑖,𝐼,𝑧   𝑖,𝑋,𝑗,𝑘,𝑧   𝜑,𝑗,𝑘

Allowed substitution hints:   𝜑(𝑧,𝑖)   𝐴(𝑘)   𝐵(𝑘)   𝐻(𝑧,𝑘)   𝐼(𝑗,𝑘)   𝑀(𝑧,𝑖,𝑗,𝑘)

Proof of Theorem ovnhoilem1

Step	Hyp	Ref	Expression
1		ovnhoilem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		ovnhoilem1.c	. . . . 5 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
3	2	a1i 11	. . . 4 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
4		nfv 1914	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoilem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelcdmda 7018	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoilem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelcdmda 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 11165	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 46579	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3970	. . 3 ⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋))
12		ovnhoilem1.m	. . 3 ⊢ 𝑀 = {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)))))}
13	1, 11, 12	ovnval2 46546	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14		iftrue 4482	. . . . 5 ⊢ (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
15	14	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16		0xr 11162	. . . . . . 7 ⊢ 0 ∈ ℝ*
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*)
18		pnfxr 11169	. . . . . . 7 ⊢ +∞ ∈ ℝ*
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*)
20	4, 1, 6, 8	hoiprodcl3 46581	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞))
21		icogelb 13299	. . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞)) → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
22	17, 19, 20, 21	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
23	22	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
24	15, 23	eqbrtrd 5114	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
25		iffalse 4485	. . . . 5 ⊢ (¬ 𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
26	25	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
27		ssrab2 4031	. . . . . . 7 ⊢ {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)))))} ⊆ ℝ*
28	12, 27	eqsstri 3982	. . . . . 6 ⊢ 𝑀 ⊆ ℝ*
29	28	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30		icossxr 13335	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ*
31	30, 20	sselid 3933	. . . . . . . . 9 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*)
33		opelxpi 5656	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
34	6, 8, 33	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
35		0re 11117	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
36		opelxpi 5656	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
37	35, 35, 36	mp2an 692	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
38	37	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
39	34, 38	ifcld 4523	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40	39	fmpttd 7049	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
41		reex 11100	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
42	41, 41	xpex 7689	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ∈ V
43	1, 42	jctil 519	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
44		elmapg 8766	. . . . . . . . . . . . . . 15 ⊢ (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ)))
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ)))
46	40, 45	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋))
47	46	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋))
48		ovnhoilem1.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
49	47, 48	fmptd 7048	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
50		ovex 7382	. . . . . . . . . . . 12 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
51		nnex 12134	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
52	50, 51	elmap 8798	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53	49, 52	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
54	53	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
55		eqidd 2730	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
56	34	fmpttd 7049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
57		iftrue 4482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
58	57	mpteq2dv 5186	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 1 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
59		1nn 12139	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
61		mptexg 7157	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) ∈ V)
62	1, 61	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) ∈ V)
63	48, 58, 60, 62	fvmptd3 6953	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐻‘1) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
64	63	feq1d 6634	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐻‘1):𝑋⟶(ℝ × ℝ) ↔ (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
65	56, 64	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐻‘1):𝑋⟶(ℝ × ℝ))
66	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
67		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
68	66, 67	fvovco 45191	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
69	34	elexd 3460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
70	63, 69	fvmpt2d 6943	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
71	70	fveq2d 6826	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
72		fvex 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴‘𝑘) ∈ V
73		fvex 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵‘𝑘) ∈ V
74	72, 73	op1st 7932	. . . . . . . . . . . . . . . . . 18 ⊢ (1st ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = (𝐴‘𝑘)
75	74	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = (𝐴‘𝑘))
76	71, 75	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴‘𝑘))
77	70	fveq2d 6826	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
78	72, 73	op2nd 7933	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = (𝐵‘𝑘)
79	78	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩) = (𝐵‘𝑘))
80	77, 79	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵‘𝑘))
81	76, 80	oveq12d 7367	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
82	68, 81	eqtrd 2764	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
83	82	ixpeq2dva 8839	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
84	55, 3, 83	3eqtr4d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
85		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝐻‘𝑗) = (𝐻‘1))
86	85	coeq2d 5805	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ([,) ∘ (𝐻‘𝑗)) = ([,) ∘ (𝐻‘1)))
87	86	fveq1d 6824	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (([,) ∘ (𝐻‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
88	87	ixpeq2dv 8840	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89	88	ssiun2s 4997	. . . . . . . . . . . . 13 ⊢ (1 ∈ ℕ → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
90	59, 89	ax-mp 5	. . . . . . . . . . . 12 ⊢ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘)
91	84, 90	eqsstrdi 3980	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
92	91	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
93	82	fveq2d 6826	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
94	93	eqcomd 2735	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
95	94	prodeq2dv 15829	. . . . . . . . . . . 12 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
96	95	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
97		1red 11116	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
98		icossicc 13339	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
99	4, 1, 65	hoiprodcl 46548	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
100	98, 99	sselid 3933	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
101	87	fveq2d 6826	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
102	101	prodeq2ad 45593	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
103	97, 100, 102	sge0snmpt 46384	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
104	103	eqcomd 2735	. . . . . . . . . . . 12 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
105	104	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
106		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
107	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
108		snssi 4759	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
109	59, 108	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {1} ⊆ ℕ
110	109	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → {1} ⊆ ℕ)
111		nfv 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
112	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
113		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → 𝜑)
114		elsni 4594	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {1} → 𝑗 = 1)
115	114	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → 𝑗 = 1)
116	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
117	85	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘𝑗) = (𝐻‘1))
118	117	feq1d 6634	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = 1) → ((𝐻‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝐻‘1):𝑋⟶(ℝ × ℝ)))
119	116, 118	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
120	113, 115, 119	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
121	120	adantlr 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
122	111, 112, 121	hoiprodcl 46548	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,)+∞))
123	98, 122	sselid 3933	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,]+∞))
124	38	fmpttd 7049	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
125	124	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
126		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → 𝜑)
127		eldifi 4082	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
128	127	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → 𝑗 ∈ ℕ)
129	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐻 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))))
130	47	elexd 3460	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ V)
131	129, 130	fvmpt2d 6943	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
132	126, 128, 131	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
133		eldifsni 4741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
134	133	neneqd 2930	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
135	134	iffalsed 4487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
136	135	mpteq2dv 5186	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
137	136	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
138	132, 137	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
139	138	feq1d 6634	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → ((𝐻‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ)))
140	125, 139	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
141	140	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
142		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
143	141, 142	fvovco 45191	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))))
144	37	elexi 3459	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨0, 0⟩ ∈ V
145	144	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ⟨0, 0⟩ ∈ V)
146	138, 145	fvmpt2d 6943	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((𝐻‘𝑗)‘𝑘) = ⟨0, 0⟩)
147	146	fveq2d 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
148	16	elexi 3459	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
149	148, 148	op1st 7932	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨0, 0⟩) = 0
150	149	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨0, 0⟩) = 0)
151	147, 150	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = 0)
152	146	fveq2d 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
153	148, 148	op2nd 7933	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨0, 0⟩) = 0
154	153	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨0, 0⟩) = 0)
155	152, 154	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = 0)
156	151, 155	oveq12d 7367	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))) = (0[,)0))
157		0le0 12229	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 0
158		ico0 13294	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
159	16, 16, 158	mp2an 692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)0) = ∅ ↔ 0 ≤ 0)
160	157, 159	mpbir 231	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,)0) = ∅
161	160	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (0[,)0) = ∅)
162	143, 156, 161	3eqtrd 2768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ∅)
163	162	fveq2d 6826	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘∅))
164		vol0 45960	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
165	164	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
166	163, 165	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
167	166	prodeq2dv 15829	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 0)
168	167	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 0)
169		0cnd 11108	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℂ)
170		fprodconst 15885	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
171	1, 169, 170	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
172	171	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
173		neqne 2933	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
174	173	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
175	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
176		hashnncl 14273	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
177	175, 176	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
178	174, 177	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (♯‘𝑋) ∈ ℕ)
179		0exp 14004	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
180	178, 179	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (0↑(♯‘𝑋)) = 0)
181	180	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → (0↑(♯‘𝑋)) = 0)
182	168, 172, 181	3eqtrd 2768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
183	106, 107, 110, 123, 182	sge0ss 46413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
184	96, 105, 183	3eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
185	92, 184	jca 511	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))))
186		nfcv 2891	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
187		nfcv 2891	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ℕ
188		nfmpt1 5191	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
189	187, 188	nfmpt 5190	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
190	48, 189	nfcxfr 2889	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐻
191	186, 190	nfeq 2905	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = 𝐻
192		fveq1 6821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐻 → (𝑖‘𝑗) = (𝐻‘𝑗))
193	192	coeq2d 5805	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐻 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐻‘𝑗)))
194	193	fveq1d 6824	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐻 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘𝑗))‘𝑘))
195	194	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐻 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘𝑗))‘𝑘))
196	191, 195	ixpeq2d 45066	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
197	196	iuneq2d 4972	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
198	197	sseq2d 3968	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐻 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘)))
199	194	fveq2d 6826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐻 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
200	199	a1d 25	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐻 → (𝑘 ∈ 𝑋 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))
201	191, 200	ralrimi 3227	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐻 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
202	201	prodeq2d 15828	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐻 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
203	202	mpteq2dv 5186	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))
204	203	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
205	204	eqeq2d 2740	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐻 → (∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))))
206	198, 205	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑖 = 𝐻 → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))))
207	206	rspcev 3577	. . . . . . . . 9 ⊢ ((𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
208	54, 185, 207	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
209	32, 208	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
210		eqeq1 2733	. . . . . . . . . 10 ⊢ (𝑧 = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))
211	210	anbi2d 630	. . . . . . . . 9 ⊢ (𝑧 = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
212	211	rexbidv 3153	. . . . . . . 8 ⊢ (𝑧 = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘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))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
213	212	elrab 3648	. . . . . . 7 ⊢ (∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘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)))))} ↔ (∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))))
214	209, 213	sylibr 234	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘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)))))})
215	12	eqcomi 2738	. . . . . . 7 ⊢ {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)))))} = 𝑀
216	215	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → {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)))))} = 𝑀)
217	214, 216	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ 𝑀)
218		infxrlb 13237	. . . . 5 ⊢ ((𝑀 ⊆ ℝ* ∧ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
219	29, 217, 218	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
220	26, 219	eqbrtrd 5114	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
221	24, 220	pm2.61dan 812	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
222	13, 221	eqbrtrd 5114	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  ifcif 4476  {csn 4577  ⟨cop 4583  ∪ ciun 4941   class class class wbr 5092   ↦ cmpt 5173   × cxp 5617   ∘ ccom 5623  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923   ↑_m cmap 8753  Xcixp 8824  Fincfn 8872  infcinf 9331  ℂcc 11007  ℝcr 11008  0cc0 11009  1c1 11010  +∞cpnf 11146  ℝ^*cxr 11148   < clt 11149   ≤ cle 11150  ℕcn 12128  [,)cico 13250  [,]cicc 13251  ↑cexp 13968  ♯chash 14237  ∏cprod 15810  volcvol 25362  Σ^{^}csumge0 46363  voln*covoln 46537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-pm 8756  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-prod 15811  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-top 22779  df-topon 22796  df-bases 22831  df-cmp 23272  df-ovol 25363  df-vol 25364  df-sumge0 46364  df-ovoln 46538

This theorem is referenced by:  ovnhoi  46604