Theorem ovnhoilem1 43768

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 1922	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoilem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelrnda 6893	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoilem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelrnda 6893	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 10866	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 43745	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3929	. . 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 43712	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14		iftrue 4435	. . . . 5 ⊢ (𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
15	14	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = 0)
16		0xr 10863	. . . . . . 7 ⊢ 0 ∈ ℝ*
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*)
18		pnfxr 10870	. . . . . . 7 ⊢ +∞ ∈ ℝ*
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*)
20	4, 1, 6, 8	hoiprodcl3 43747	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞))
21		icogelb 12969	. . . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
24	15, 23	eqbrtrd 5065	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
25		iffalse 4438	. . . . 5 ⊢ (¬ 𝑋 = ∅ → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
26	25	adantl 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) = inf(𝑀, ℝ*, < ))
27		ssrab2 3983	. . . . . . 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 3925	. . . . . 6 ⊢ 𝑀 ⊆ ℝ*
29	28	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30		icossxr 13003	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ*
31	30, 20	sseldi 3889	. . . . . . . . 9 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*)
32	31	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ*)
33		opelxpi 5577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
34	6, 8, 33	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
35		0re 10818	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
36		opelxpi 5577	. . . . . . . . . . . . . . . . . 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 4475	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40	39	fmpttd 6921	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
41		reex 10803	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
42	41, 41	xpex 7527	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ∈ V
43	1, 42	jctil 523	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
44		elmapg 8510	. . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ ((ℝ × ℝ) ↑m 𝑋))
47	46	adantr 484	. . . . . . . . . . . 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 6920	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
50		ovex 7235	. . . . . . . . . . . 12 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
51		nnex 11819	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
52	50, 51	elmap 8541	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53	49, 52	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
54	53	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐻 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
55		eqidd 2735	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
56	34	fmpttd 6921	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
57		iftrue 4435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
58	57	mpteq2dv 5140	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 1 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
59		1nn 11824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
61		mptexg 7026	. . . . . . . . . . . . . . . . . . . . 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 6830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐻‘1) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
64	63	feq1d 6519	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝐻‘1):𝑋⟶(ℝ × ℝ) ↔ (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩):𝑋⟶(ℝ × ℝ)))
65	56, 64	mpbird 260	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐻‘1):𝑋⟶(ℝ × ℝ))
66	65	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
67		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
68	66, 67	fvovco 42357	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
69	34	elexd 3421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
70	63, 69	fvmpt2d 6820	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
71	70	fveq2d 6710	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
72		fvex 6719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴‘𝑘) ∈ V
73		fvex 6719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵‘𝑘) ∈ V
74	72, 73	op1st 7758	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴‘𝑘))
77	70	fveq2d 6710	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
78	72, 73	op2nd 7759	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵‘𝑘))
81	76, 80	oveq12d 7220	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
82	68, 81	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
83	82	ixpeq2dva 8582	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
84	55, 3, 83	3eqtr4d 2784	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
85		fveq2 6706	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝐻‘𝑗) = (𝐻‘1))
86	85	coeq2d 5720	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ([,) ∘ (𝐻‘𝑗)) = ([,) ∘ (𝐻‘1)))
87	86	fveq1d 6708	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (([,) ∘ (𝐻‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
88	87	ixpeq2dv 8583	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89	88	ssiun2s 4947	. . . . . . . . . . . . 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 3945	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
92	91	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
93	82	fveq2d 6710	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
94	93	eqcomd 2740	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
95	94	prodeq2dv 15466	. . . . . . . . . . . 12 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
96	95	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
97		1red 10817	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
98		icossicc 13007	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
99	4, 1, 65	hoiprodcl 43714	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
100	98, 99	sseldi 3889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
101	87	fveq2d 6710	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
102	101	prodeq2ad 42762	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
103	97, 100, 102	sge0snmpt 43550	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
104	103	eqcomd 2740	. . . . . . . . . . . 12 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
105	104	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
106		nfv 1922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
107	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
108		snssi 4711	. . . . . . . . . . . . . 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 1922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
112	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
113		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → 𝜑)
114		elsni 4548	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {1} → 𝑗 = 1)
115	114	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → 𝑗 = 1)
116	65	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘1):𝑋⟶(ℝ × ℝ))
117	85	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘𝑗) = (𝐻‘1))
118	117	feq1d 6519	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = 1) → ((𝐻‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝐻‘1):𝑋⟶(ℝ × ℝ)))
119	116, 118	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = 1) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
120	113, 115, 119	syl2anc 587	. . . . . . . . . . . . . . 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 43714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,)+∞))
123	98, 122	sseldi 3889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,]+∞))
124	38	fmpttd 6921	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
125	124	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ))
126		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → 𝜑)
127		eldifi 4031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ∈ ℕ)
128	127	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . 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 3421	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ V)
131	129, 130	fvmpt2d 6820	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
132	126, 128, 131	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
133		eldifsni 4693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
134	133	neneqd 2940	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
135	134	iffalsed 4440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
136	135	mpteq2dv 5140	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
137	136	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
138	132, 137	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
139	138	feq1d 6519	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → ((𝐻‘𝑗):𝑋⟶(ℝ × ℝ) ↔ (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩):𝑋⟶(ℝ × ℝ)))
140	125, 139	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
141	140	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
142		simpr 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
143	141, 142	fvovco 42357	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))))
144	37	elexi 3420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨0, 0⟩ ∈ V
145	144	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ⟨0, 0⟩ ∈ V)
146	138, 145	fvmpt2d 6820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((𝐻‘𝑗)‘𝑘) = ⟨0, 0⟩)
147	146	fveq2d 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
148	16	elexi 3420	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
149	148, 148	op1st 7758	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨0, 0⟩) = 0
150	149	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨0, 0⟩) = 0)
151	147, 150	eqtrd 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = 0)
152	146	fveq2d 6710	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
153	148, 148	op2nd 7759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨0, 0⟩) = 0
154	153	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨0, 0⟩) = 0)
155	152, 154	eqtrd 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = 0)
156	151, 155	oveq12d 7220	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))) = (0[,)0))
157		0le0 11914	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 0
158		ico0 12964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
159	16, 16, 158	mp2an 692	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)0) = ∅ ↔ 0 ≤ 0)
160	157, 159	mpbir 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (0[,)0) = ∅
161	160	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (0[,)0) = ∅)
162	143, 156, 161	3eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ∅)
163	162	fveq2d 6710	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘∅))
164		vol0 43129	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
165	164	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
166	163, 165	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
167	166	prodeq2dv 15466	. . . . . . . . . . . . . 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 10809	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℂ)
170		fprodconst 15521	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
171	1, 169, 170	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
172	171	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
173		neqne 2943	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
174	173	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
175	1	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
176		hashnncl 13916	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
177	175, 176	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
178	174, 177	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (♯‘𝑋) ∈ ℕ)
179		0exp 13653	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) ∈ ℕ → (0↑(♯‘𝑋)) = 0)
180	178, 179	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (0↑(♯‘𝑋)) = 0)
181	180	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → (0↑(♯‘𝑋)) = 0)
182	168, 172, 181	3eqtrd 2778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
183	106, 107, 110, 123, 182	sge0ss 43579	. . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
185	92, 184	jca 515	. . . . . . . . 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 2900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
187		nfcv 2900	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ℕ
188		nfmpt1 5142	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
189	187, 188	nfmpt 5141	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
190	48, 189	nfcxfr 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐻
191	186, 190	nfeq 2913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = 𝐻
192		fveq1 6705	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐻 → (𝑖‘𝑗) = (𝐻‘𝑗))
193	192	coeq2d 5720	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐻 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐻‘𝑗)))
194	193	fveq1d 6708	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐻 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘𝑗))‘𝑘))
195	194	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐻 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘𝑗))‘𝑘))
196	191, 195	ixpeq2d 42241	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
197	196	iuneq2d 4923	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
198	197	sseq2d 3923	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐻 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘)))
199	194	fveq2d 6710	. . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐻 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
202	201	prodeq2d 15465	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐻 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
203	202	mpteq2dv 5140	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))
204	203	fveq2d 6710	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
205	204	eqeq2d 2745	. . . . . . . . . . 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 634	. . . . . . . . . 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 3530	. . . . . . . . 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 587	. . . . . . . 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 515	. . . . . . 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 2738	. . . . . . . . . 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 632	. . . . . . . . 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 3209	. . . . . . . 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 3595	. . . . . . 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 237	. . . . . 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 2743	. . . . . . 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 2836	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ 𝑀)
218		infxrlb 12907	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
220	26, 219	eqbrtrd 5065	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
221	24, 220	pm2.61dan 813	. 2 ⊢ (𝜑 → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
222	13, 221	eqbrtrd 5065	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2935  ∃wrex 3055  {crab 3058  Vcvv 3401   ∖ cdif 3854   ⊆ wss 3857  ∅c0 4227  ifcif 4429  {csn 4531  ⟨cop 4537  ∪ ciun 4894   class class class wbr 5043   ↦ cmpt 5124   × cxp 5538   ∘ ccom 5544  ⟶wf 6365  ‘cfv 6369  (class class class)co 7202  1^st c1st 7748  2^nd c2nd 7749   ↑_m cmap 8497  Xcixp 8567  Fincfn 8615  infcinf 9046  ℂcc 10710  ℝcr 10711  0cc0 10712  1c1 10713  +∞cpnf 10847  ℝ^*cxr 10849   < clt 10850   ≤ cle 10851  ℕcn 11813  [,)cico 12920  [,]cicc 12921  ↑cexp 13618  ♯chash 13879  ∏cprod 15448  volcvol 24332  Σ^{^}csumge0 43529  voln*covoln 43703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-of 7458  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-er 8380  df-map 8499  df-pm 8500  df-ixp 8568  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-fi 9016  df-sup 9047  df-inf 9048  df-oi 9115  df-dju 9500  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-n0 12074  df-z 12160  df-uz 12422  df-q 12528  df-rp 12570  df-xneg 12687  df-xadd 12688  df-xmul 12689  df-ioo 12922  df-ico 12924  df-icc 12925  df-fz 13079  df-fzo 13222  df-fl 13350  df-seq 13558  df-exp 13619  df-hash 13880  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-clim 15032  df-rlim 15033  df-sum 15233  df-prod 15449  df-rest 16899  df-topgen 16920  df-psmet 20327  df-xmet 20328  df-met 20329  df-bl 20330  df-mopn 20331  df-top 21763  df-topon 21780  df-bases 21815  df-cmp 22256  df-ovol 24333  df-vol 24334  df-sumge0 43530  df-ovoln 43704

This theorem is referenced by:  ovnhoi  43770