Theorem ovnhoilem1 46882

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 1916	. . . . 5 ⊢ Ⅎ𝑘𝜑
5		ovnhoilem1.a	. . . . . 6 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
6	5	ffvelcdmda 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
7		ovnhoilem1.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	ffvelcdmda 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
9	8	rexrd 11184	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ*)
10	4, 6, 9	hoissrrn2 46859	. . . 4 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋))
11	3, 10	eqsstrd 3967	. . 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 46826	. 2 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )))
14		iftrue 4484	. . . . 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 11181	. . . . . . 7 ⊢ 0 ∈ ℝ*
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*)
18		pnfxr 11188	. . . . . . 7 ⊢ +∞ ∈ ℝ*
19	18	a1i 11	. . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*)
20	4, 1, 6, 8	hoiprodcl3 46861	. . . . . 6 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ (0[,)+∞))
21		icogelb 13314	. . . . . 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 1374	. . . . 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 5119	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → if(𝑋 = ∅, 0, inf(𝑀, ℝ*, < )) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
25		iffalse 4487	. . . . 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 3979	. . . . . 6 ⊢ 𝑀 ⊆ ℝ*
29	28	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑀 ⊆ ℝ*)
30		icossxr 13350	. . . . . . . . . 10 ⊢ (0[,)+∞) ⊆ ℝ*
31	30, 20	sselid 3930	. . . . . . . . 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 5660	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
34	6, 8, 33	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ (ℝ × ℝ))
35		0re 11136	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℝ
36		opelxpi 5660	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
37	35, 35, 36	mp2an 693	. . . . . . . . . . . . . . . . 17 ⊢ ⟨0, 0⟩ ∈ (ℝ × ℝ)
38	37	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨0, 0⟩ ∈ (ℝ × ℝ))
39	34, 38	ifcld 4525	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) ∈ (ℝ × ℝ))
40	39	fmpttd 7060	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)):𝑋⟶(ℝ × ℝ))
41		reex 11119	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ∈ V
42	41, 41	xpex 7698	. . . . . . . . . . . . . . . 16 ⊢ (ℝ × ℝ) ∈ V
43	1, 42	jctil 519	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin))
44		elmapg 8778	. . . . . . . . . . . . . . 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 7059	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
50		ovex 7391	. . . . . . . . . . . 12 ⊢ ((ℝ × ℝ) ↑m 𝑋) ∈ V
51		nnex 12153	. . . . . . . . . . . 12 ⊢ ℕ ∈ V
52	50, 51	elmap 8811	. . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
56	34	fmpttd 7060	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩):𝑋⟶(ℝ × ℝ))
57		iftrue 4484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 1 → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
58	57	mpteq2dv 5191	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 1 → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
59		1nn 12158	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℕ
60	59	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 1 ∈ ℕ)
61		mptexg 7167	. . . . . . . . . . . . . . . . . . . . 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 6964	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐻‘1) = (𝑘 ∈ 𝑋 ↦ ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
64	63	feq1d 6643	. . . . . . . . . . . . . . . . . 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 45474	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))))
69	34	elexd 3463	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩ ∈ V)
70	63, 69	fvmpt2d 6954	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐻‘1)‘𝑘) = ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩)
71	70	fveq2d 6837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (1st ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
72		fvex 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴‘𝑘) ∈ V
73		fvex 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵‘𝑘) ∈ V
74	72, 73	op1st 7941	. . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘1)‘𝑘)) = (𝐴‘𝑘))
77	70	fveq2d 6837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (2nd ‘⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩))
78	72, 73	op2nd 7942	. . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘1)‘𝑘)) = (𝐵‘𝑘))
81	76, 80	oveq12d 7376	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘1)‘𝑘))[,)(2nd ‘((𝐻‘1)‘𝑘))) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
82	68, 81	eqtrd 2770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘1))‘𝑘) = ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
83	82	ixpeq2dva 8852	. . . . . . . . . . . . 13 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
84	55, 3, 83	3eqtr4d 2780	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
85		fveq2 6833	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 1 → (𝐻‘𝑗) = (𝐻‘1))
86	85	coeq2d 5810	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 1 → ([,) ∘ (𝐻‘𝑗)) = ([,) ∘ (𝐻‘1)))
87	86	fveq1d 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (([,) ∘ (𝐻‘𝑗))‘𝑘) = (([,) ∘ (𝐻‘1))‘𝑘))
88	87	ixpeq2dv 8853	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘1))‘𝑘))
89	88	ssiun2s 5003	. . . . . . . . . . . . 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 3977	. . . . . . . . . . 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 6837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
94	93	eqcomd 2741	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
95	94	prodeq2dv 15847	. . . . . . . . . . . 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 11135	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ ℝ)
98		icossicc 13354	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
99	4, 1, 65	hoiprodcl 46828	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,)+∞))
100	98, 99	sselid 3930	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)) ∈ (0[,]+∞))
101	87	fveq2d 6837	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 1 → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
102	101	prodeq2ad 45875	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 1 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
103	97, 100, 102	sge0snmpt 46664	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Σ^‘(𝑗 ∈ {1} ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘1))‘𝑘)))
104	103	eqcomd 2741	. . . . . . . . . . . 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 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝜑 ∧ ¬ 𝑋 = ∅)
107	51	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ℕ ∈ V)
108		snssi 4763	. . . . . . . . . . . . . 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 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1})
112	1	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → 𝑋 ∈ Fin)
113		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → 𝜑)
114		elsni 4596	. . . . . . . . . . . . . . . . 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 6643	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ {1}) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
121	120	adantlr 716	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → (𝐻‘𝑗):𝑋⟶(ℝ × ℝ))
122	111, 112, 121	hoiprodcl 46828	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,)+∞))
123	98, 122	sselid 3930	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ {1}) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) ∈ (0[,]+∞))
124	38	fmpttd 7060	. . . . . . . . . . . . . . . . . . . . . 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 3463	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)) ∈ V)
131	129, 130	fvmpt2d 6954	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
132	126, 128, 131	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
133		eldifsni 4745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → 𝑗 ≠ 1)
134	133	neneqd 2936	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → ¬ 𝑗 = 1)
135	134	iffalsed 4489	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (ℕ ∖ {1}) → if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩) = ⟨0, 0⟩)
136	135	mpteq2dv 5191	. . . . . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → (𝐻‘𝑗) = (𝑘 ∈ 𝑋 ↦ ⟨0, 0⟩))
139	138	feq1d 6643	. . . . . . . . . . . . . . . . . . . . 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 45474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))))
144	37	elexi 3462	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⟨0, 0⟩ ∈ V
145	144	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ⟨0, 0⟩ ∈ V)
146	138, 145	fvmpt2d 6954	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((𝐻‘𝑗)‘𝑘) = ⟨0, 0⟩)
147	146	fveq2d 6837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = (1st ‘⟨0, 0⟩))
148	16	elexi 3462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 ∈ V
149	148, 148	op1st 7941	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1st ‘⟨0, 0⟩) = 0
150	149	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘⟨0, 0⟩) = 0)
151	147, 150	eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐻‘𝑗)‘𝑘)) = 0)
152	146	fveq2d 6837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = (2nd ‘⟨0, 0⟩))
153	148, 148	op2nd 7942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2nd ‘⟨0, 0⟩) = 0
154	153	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘⟨0, 0⟩) = 0)
155	152, 154	eqtrd 2770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐻‘𝑗)‘𝑘)) = 0)
156	151, 155	oveq12d 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐻‘𝑗)‘𝑘))[,)(2nd ‘((𝐻‘𝑗)‘𝑘))) = (0[,)0))
157		0le0 12248	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 ≤ 0
158		ico0 13309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((0[,)0) = ∅ ↔ 0 ≤ 0))
159	16, 16, 158	mp2an 693	. . . . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐻‘𝑗))‘𝑘) = ∅)
163	162	fveq2d 6837	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = (vol‘∅))
164		vol0 46240	. . . . . . . . . . . . . . . . 17 ⊢ (vol‘∅) = 0
165	164	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) = 0)
166	163, 165	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
167	166	prodeq2dv 15847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 0)
168	167	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 0)
169		0cnd 11127	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℂ)
170		fprodconst 15903	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ Fin ∧ 0 ∈ ℂ) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
171	1, 169, 170	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
172	171	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 0 = (0↑(♯‘𝑋)))
173		neqne 2939	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
174	173	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
175	1	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin)
176		hashnncl 14291	. . . . . . . . . . . . . . . . 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 14022	. . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑋 = ∅) ∧ 𝑗 ∈ (ℕ ∖ {1})) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)) = 0)
183	106, 107, 110, 123, 182	sge0ss 46693	. . . . . . . . . . 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 2774	. . . . . . . . . 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 2897	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝑖
187		nfcv 2897	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘ℕ
188		nfmpt1 5196	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩))
189	187, 188	nfmpt 5195	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, ⟨(𝐴‘𝑘), (𝐵‘𝑘)⟩, ⟨0, 0⟩)))
190	48, 189	nfcxfr 2895	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘𝐻
191	186, 190	nfeq 2911	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑖 = 𝐻
192		fveq1 6832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝐻 → (𝑖‘𝑗) = (𝐻‘𝑗))
193	192	coeq2d 5810	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝐻 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐻‘𝑗)))
194	193	fveq1d 6835	. . . . . . . . . . . . . . 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 45350	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
197	196	iuneq2d 4976	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘))
198	197	sseq2d 3965	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐻 → (𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐻‘𝑗))‘𝑘)))
199	194	fveq2d 6837	. . . . . . . . . . . . . . . . 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 3233	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝐻 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
202	201	prodeq2d 15846	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝐻 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))
203	202	mpteq2dv 5191	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐻 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘))))
204	203	fveq2d 6837	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐻 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐻‘𝑗))‘𝑘)))))
205	204	eqeq2d 2746	. . . . . . . . . . 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 633	. . . . . . . . . 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 3575	. . . . . . . . 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 585	. . . . . . . 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 2739	. . . . . . . . . 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 631	. . . . . . . . 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 3159	. . . . . . . 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 3645	. . . . . . 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 2744	. . . . . . 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 2837	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ 𝑀)
218		infxrlb 13252	. . . . 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 585	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → inf(𝑀, ℝ*, < ) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
220	26, 219	eqbrtrd 5119	. . 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 5119	1 ⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∃wrex 3059  {crab 3398  Vcvv 3439   ∖ cdif 3897   ⊆ wss 3900  ∅c0 4284  ifcif 4478  {csn 4579  ⟨cop 4585  ∪ ciun 4945   class class class wbr 5097   ↦ cmpt 5178   × cxp 5621   ∘ ccom 5627  ⟶wf 6487  ‘cfv 6491  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8765  Xcixp 8837  Fincfn 8885  infcinf 9346  ℂcc 11026  ℝcr 11027  0cc0 11028  1c1 11029  +∞cpnf 11165  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169  ℕcn 12147  [,)cico 13265  [,]cicc 13266  ↑cexp 13986  ♯chash 14255  ∏cprod 15828  volcvol 25422  Σ^{^}csumge0 46643  voln*covoln 46817

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-inf2 9552  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-map 8767  df-pm 8768  df-ixp 8838  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fi 9316  df-sup 9347  df-inf 9348  df-oi 9417  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-div 11797  df-nn 12148  df-2 12210  df-3 12211  df-n0 12404  df-z 12491  df-uz 12754  df-q 12864  df-rp 12908  df-xneg 13028  df-xadd 13029  df-xmul 13030  df-ioo 13267  df-ico 13269  df-icc 13270  df-fz 13426  df-fzo 13573  df-fl 13714  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-rlim 15414  df-sum 15612  df-prod 15829  df-rest 17344  df-topgen 17365  df-psmet 21303  df-xmet 21304  df-met 21305  df-bl 21306  df-mopn 21307  df-top 22840  df-topon 22857  df-bases 22892  df-cmp 23333  df-ovol 25423  df-vol 25424  df-sumge0 46644  df-ovoln 46818

This theorem is referenced by:  ovnhoi  46884