vonioolem2 - Mathbox for Glauco Siliprandi

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Theorem vonioolem2 47138

Description: The n-dimensional Lebesgue measure of open intervals. This is the first statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
vonioolem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
vonioolem2.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
vonioolem2.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
vonioolem2.n	⊢ (𝜑 → 𝑋 ≠ ∅)
vonioolem2.t	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
vonioolem2.i	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
vonioolem2.c	⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
vonioolem2.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))

**Assertion**
Ref	Expression
vonioolem2	⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Distinct variable groups: 𝐴,𝑘,𝑛 𝐵,𝑘,𝑛 𝐶,𝑘,𝑛 𝐷,𝑛 𝑛,𝐼 𝑘,𝑋,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝐷(𝑘) 𝐼(𝑘)

Proof of Theorem vonioolem2 Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vonioolem2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
2	1	vonmea 47031	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
3		1zzd 12553	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
4		nnuz 12822	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5	1	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
6		eqid 2741	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
7		vonioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
8	7	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
9	8	ffvelcdmda 7029	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
10		nnrecre 12214	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
11	10	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
12	9, 11	readdcld 11169	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
13	12	fmpttd 7060	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
14		vonioolem2.c	. . . . . . . . . 10 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
15	14	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))))
16	1	mptexd 7172	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
17	16	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
18	15, 17	fvmpt2d 6953	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
19	18	feq1d 6641	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
20	13, 19	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
21		vonioolem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
22	21	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
23	5, 6, 20, 22	hoimbl 47088	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
24		vonioolem2.d	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
25	23, 24	fmptd 7059	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
26		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
27		oveq2 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
28	27	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) + (1 / 𝑛)) = ((𝐴‘𝑘) + (1 / 𝑚)))
29	28	mpteq2dv 5169	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
30	29	cbvmptv 5179	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
31	14, 30	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
32		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
33	32	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
34	33	mpteq2dv 5169	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
35		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
36	35	peano2nnd 12186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
37	5	mptexd 7172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
38	31, 34, 36, 37	fvmptd3 6963	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
39		ovexd 7395	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
40	38, 39	fvmpt2d 6953	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
41		1red 11140	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
42		nnre 12176	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
43	42, 41	readdcld 11169	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
44		peano2nn 12181	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
45		nnne0 12206	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
47	41, 43, 46	redivcld 11978	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
48	47	ad2antlr 734	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
49	9, 48	readdcld 11169	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ ℝ)
50	40, 49	eqeltrd 2841	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ)
51	50	rexrd 11190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ^*)
52		ressxr 11184	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
53	21	ffvelcdmda 7029	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
54	52, 53	sselid 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
55	54	adantlr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
56	42	ltp1d 12081	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
57		nnrp 12949	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
58	44	nnrpd 12979	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ⁺)
59	57, 58	ltrecd 12999	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
60	56, 59	mpbid 234	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
61	47, 10, 60	ltled 11289	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
62	61	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
63	48, 11, 9, 62	leadd2dd 11760	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛)))
64		ovexd 7395	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
65	18, 64	fvmpt2d 6953	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
66	40, 65	breq12d 5088	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ↔ ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛))))
67	63, 66	mpbird 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
68	53	adantlr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
69		eqidd 2742	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (𝐵‘𝑘))
70	68, 69	eqled 11244	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘))
71		icossico 13364	. . . . . . 7 ⊢ (((((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) ∧ (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
72	51, 55, 67, 70, 71	syl22anc 845	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
73	26, 72	ixpssixp 45553	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
74	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
75	23	elexd 3456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
76	74, 75	fvmpt2d 6953	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
77		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
78	77	fveq1d 6833	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
79	78	oveq1d 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
80	79	ixpeq2dv 8855	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
81	80	cbvmptv 5179	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
82	24, 81	eqtri 2764	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
83		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
84	83	fveq1d 6833	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
85	84	oveq1d 7375	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
86	85	ixpeq2dv 8855	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
87		ovex 7393	. . . . . . . . . 10 ⊢ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
88	87	rgenw 3059	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
89		ixpexg 8864	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V → X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
90	88, 89	ax-mp 5	. . . . . . . 8 ⊢ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
91	90	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
92	82, 86, 36, 91	fvmptd3 6963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
93	76, 92	sseq12d 3950	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)) ↔ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))))
94	73, 93	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)))
95	1, 6, 7, 21	hoimbl 47088	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
96		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑘𝜑
97	7	ffvelcdmda 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
98	96, 1, 97, 53	vonhoire 47129	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
99		vonioolem2.i	. . . . . . 7 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
100	99	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
101		nftru 1812	. . . . . . . . 9 ⊢ Ⅎ𝑘⊤
102		ioossico 13386	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘))
103	102	a1i 11	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
104	101, 103	ixpssixp 45553	. . . . . . . 8 ⊢ (⊤ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
105	104	mptru 1555	. . . . . . 7 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))
106	105	a1i 11	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
107	100, 106	eqsstrd 3951	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
108	52	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
109	7, 108	fssd 6676	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
110	21, 108	fssd 6676	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
111	1, 6, 109, 110	ioovonmbl 47134	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
112	99, 111	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ dom (voln‘𝑋))
113	2, 95, 98, 107, 112	meassre 46934	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ)
114	2	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas)
115	76, 23	eqeltrd 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ dom (voln‘𝑋))
116	112	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋))
117	52, 97	sselid 3915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
118	117	adantlr 722	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
119	57	rpreccld 12991	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
120	119	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ⁺)
121	9, 120	ltaddrpd 13014	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)))
122		icossioo 13388	. . . . . . . 8 ⊢ ((((𝐴‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) ∧ ((𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
123	118, 55, 121, 70, 122	syl22anc 845	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
124	26, 123	ixpssixp 45553	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
125	65	oveq1d 7375	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
126	125	ixpeq2dva 8854	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
127	76, 126	eqtrd 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
128	99	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
129	127, 128	sseq12d 3950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ 𝐼 ↔ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
130	124, 129	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ 𝐼)
131	114, 6, 115, 116, 130	meassle 46920	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷‘𝑛)) ≤ ((voln‘𝑋)‘𝐼))
132		eqid 2741	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
133	2, 3, 4, 25, 94, 113, 131, 132	meaiuninc2 46939	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
134	96, 1, 97, 54	iunhoiioo 47133	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
135	127	iuneq2dv 4949	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
136	134, 135, 100	3eqtr4d 2786	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
137	136	eqcomd 2747	. . . . 5 ⊢ (𝜑 → 𝐼 = ∪ 𝑛 ∈ ℕ (𝐷‘𝑛))
138	137	fveq2d 6835	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
139	138	eqcomd 2747	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
140	133, 139	breqtrd 5101	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
141		2fveq3 6836	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
142	141	cbvmptv 5179	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚)))
143	142	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))))
144		vonioolem2.n	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
145		vonioolem2.t	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
146	142	eqcomi 2750	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
147		eqcom 2748	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 ↔ 𝑚 = 𝑛)
148	147	imbi1i 351	. . . . . . . . 9 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)))
149		eqcom 2748	. . . . . . . . . 10 ⊢ (((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘) ↔ ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
150	149	imbi2i 338	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
151	148, 150	bitri 277	. . . . . . . 8 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
152	78, 151	mpbi 232	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
153	152	oveq2d 7376	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
154	153	prodeq2ad 46051	. . . . 5 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
155	154	cbvmptv 5179	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
156		eqid 2741	. . . 4 ⊢ inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
157		eqid 2741	. . . 4 ⊢ ((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) = ((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1)
158		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
159		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
160	158, 159	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) − (𝐴‘𝑗)) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
161	160	cbvmptv 5179	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
162	161	rneqi 5886	. . . . . . . . 9 ⊢ ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
163	162	infeq1i 9386	. . . . . . . 8 ⊢ inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
164	163	oveq2i 7371	. . . . . . 7 ⊢ (1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < )) = (1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))
165	164	fveq2i 6834	. . . . . 6 ⊢ (⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) = (⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )))
166	165	oveq1i 7370	. . . . 5 ⊢ ((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1) = ((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1)
167	166	fveq2i 6834	. . . 4 ⊢ (ℤ_≥‘((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1)) = (ℤ_≥‘((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1))
168	1, 7, 21, 144, 145, 14, 24, 146, 155, 156, 157, 167	vonioolem1 47137	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
169	143, 168	eqbrtrd 5097	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
170		climuni 15509	. 2 ⊢ (((𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
171	140, 169, 170	syl2anc 591	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ⊤wtru 1549 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 Vcvv 3433 ⊆ wss 3885 ∅c0 4264 ∪ ciun 4924 class class class wbr 5075 ↦ cmpt 5156 dom cdm 5621 ran crn 5622 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 Xcixp 8839 Fincfn 8887 infcinf 9348 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 − cmin 11372 / cdiv 11802 ℕcn 12169 ℤ_≥cuz 12783 ℝ⁺crp 12937 (,)cioo 13293 [,)cico 13295 ⌊cfl 13744 ⇝ cli 15441 ∏cprod 15863 Meascmea 46906 volncvoln 46995

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 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cc 10352 ax-ac2 10380 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-disj 5043 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-omul 8404 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-dju 9820 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-prod 15864 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-pws 17407 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-rhm 20447 df-subrng 20522 df-subrg 20546 df-drng 20707 df-field 20708 df-abv 20785 df-staf 20815 df-srng 20816 df-lmod 20856 df-lss 20926 df-lmhm 21016 df-lvec 21097 df-sra 21167 df-rgmod 21168 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-cnfld 21352 df-refld 21584 df-phl 21605 df-dsmm 21711 df-frlm 21726 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cn 23214 df-cnp 23215 df-cmp 23374 df-tx 23549 df-hmeo 23742 df-xms 24307 df-ms 24308 df-tms 24309 df-nm 24569 df-ngp 24570 df-tng 24571 df-nrg 24572 df-nlm 24573 df-cncf 24867 df-clm 25052 df-cph 25157 df-tcph 25158 df-rrx 25374 df-ovol 25453 df-vol 25454 df-salg 46766 df-sumge0 46820 df-mea 46907 df-ome 46947 df-caragen 46949 df-ovoln 46994 df-voln 46996

This theorem is referenced by: vonioo 47139

W3C validator