vonioolem2 - Mathbox for Glauco Siliprandi

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Theorem vonioolem2 46672

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 46565	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
3		1zzd 12540	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
4		nnuz 12812	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
6		eqid 2729	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
7		vonioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
8	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
9	8	ffvelcdmda 7038	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
10		nnrecre 12204	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
11	10	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
12	9, 11	readdcld 11179	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
13	12	fmpttd 7069	. . . . . . 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 7180	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
18	15, 17	fvmpt2d 6963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
19	18	feq1d 6652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
20	13, 19	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
21		vonioolem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
23	5, 6, 20, 22	hoimbl 46622	. . . . 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 7068	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
26		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
27		oveq2 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
28	27	oveq2d 7385	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) + (1 / 𝑛)) = ((𝐴‘𝑘) + (1 / 𝑚)))
29	28	mpteq2dv 5196	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
30	29	cbvmptv 5206	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
31	14, 30	eqtri 2752	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
32		oveq2 7377	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
33	32	oveq2d 7385	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
34	33	mpteq2dv 5196	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
35		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
36	35	peano2nnd 12179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
37	5	mptexd 7180	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
38	31, 34, 36, 37	fvmptd3 6973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
39		ovexd 7404	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
40	38, 39	fvmpt2d 6963	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
41		1red 11151	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
42		nnre 12169	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
43	42, 41	readdcld 11179	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
44		peano2nn 12174	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
45		nnne0 12196	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
47	41, 43, 46	redivcld 11986	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
48	47	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
49	9, 48	readdcld 11179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ ℝ)
50	40, 49	eqeltrd 2828	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ)
51	50	rexrd 11200	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ^*)
52		ressxr 11194	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
53	21	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
54	52, 53	sselid 3941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
55	54	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
56	42	ltp1d 12089	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
57		nnrp 12939	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
58	44	nnrpd 12969	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ⁺)
59	57, 58	ltrecd 12989	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
60	56, 59	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
61	47, 10, 60	ltled 11298	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
62	61	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
63	48, 11, 9, 62	leadd2dd 11769	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛)))
64		ovexd 7404	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
65	18, 64	fvmpt2d 6963	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
66	40, 65	breq12d 5115	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
68	53	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
69		eqidd 2730	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (𝐵‘𝑘))
70	68, 69	eqled 11253	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘))
71		icossico 13353	. . . . . . 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 838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
73	26, 72	ixpssixp 45079	. . . . 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 3468	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
76	74, 75	fvmpt2d 6963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
77		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
78	77	fveq1d 6842	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
79	78	oveq1d 7384	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
80	79	ixpeq2dv 8863	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
81	80	cbvmptv 5206	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
82	24, 81	eqtri 2752	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
83		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
84	83	fveq1d 6842	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
85	84	oveq1d 7384	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
86	85	ixpeq2dv 8863	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
87		ovex 7402	. . . . . . . . . 10 ⊢ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
88	87	rgenw 3048	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
89		ixpexg 8872	. . . . . . . . 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 6973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
93	76, 92	sseq12d 3977	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)))
95	1, 6, 7, 21	hoimbl 46622	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
96		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘𝜑
97	7	ffvelcdmda 7038	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
98	96, 1, 97, 53	vonhoire 46663	. . . . 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 1804	. . . . . . . . 9 ⊢ Ⅎ𝑘⊤
102		ioossico 13375	. . . . . . . . . 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 45079	. . . . . . . 8 ⊢ (⊤ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
105	104	mptru 1547	. . . . . . 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 3978	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
108	52	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
109	7, 108	fssd 6687	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
110	21, 108	fssd 6687	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
111	1, 6, 109, 110	ioovonmbl 46668	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
112	99, 111	eqeltrid 2832	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ dom (voln‘𝑋))
113	2, 95, 98, 107, 112	meassre 46468	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ)
114	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas)
115	76, 23	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ dom (voln‘𝑋))
116	112	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋))
117	52, 97	sselid 3941	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
118	117	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
119	57	rpreccld 12981	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
120	119	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ⁺)
121	9, 120	ltaddrpd 13004	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)))
122		icossioo 13377	. . . . . . . 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 838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
124	26, 123	ixpssixp 45079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
125	65	oveq1d 7384	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
126	125	ixpeq2dva 8862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
127	76, 126	eqtrd 2764	. . . . . . 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 3977	. . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ 𝐼)
131	114, 6, 115, 116, 130	meassle 46454	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷‘𝑛)) ≤ ((voln‘𝑋)‘𝐼))
132		eqid 2729	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
133	2, 3, 4, 25, 94, 113, 131, 132	meaiuninc2 46473	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
134	96, 1, 97, 54	iunhoiioo 46667	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
135	127	iuneq2dv 4976	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
136	134, 135, 100	3eqtr4d 2774	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
137	136	eqcomd 2735	. . . . 5 ⊢ (𝜑 → 𝐼 = ∪ 𝑛 ∈ ℕ (𝐷‘𝑛))
138	137	fveq2d 6844	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
139	138	eqcomd 2735	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
140	133, 139	breqtrd 5128	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
141		2fveq3 6845	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
142	141	cbvmptv 5206	. . . 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 2738	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
147		eqcom 2736	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 ↔ 𝑚 = 𝑛)
148	147	imbi1i 349	. . . . . . . . 9 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)))
149		eqcom 2736	. . . . . . . . . 10 ⊢ (((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘) ↔ ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
150	149	imbi2i 336	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
151	148, 150	bitri 275	. . . . . . . 8 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
152	78, 151	mpbi 230	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
153	152	oveq2d 7385	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
154	153	prodeq2ad 45583	. . . . 5 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
155	154	cbvmptv 5206	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
156		eqid 2729	. . . 4 ⊢ inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
157		eqid 2729	. . . 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 6840	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
159		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
160	158, 159	oveq12d 7387	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) − (𝐴‘𝑗)) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
161	160	cbvmptv 5206	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
162	161	rneqi 5890	. . . . . . . . 9 ⊢ ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
163	162	infeq1i 9406	. . . . . . . 8 ⊢ inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
164	163	oveq2i 7380	. . . . . . 7 ⊢ (1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < )) = (1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))
165	164	fveq2i 6843	. . . . . 6 ⊢ (⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) = (⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )))
166	165	oveq1i 7379	. . . . 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 6843	. . . 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 46671	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
169	143, 168	eqbrtrd 5124	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
170		climuni 15494	. 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 584	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3444 ⊆ wss 3911 ∅c0 4292 ∪ ciun 4951 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ran crn 5632 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 Xcixp 8847 Fincfn 8895 infcinf 9368 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 ℤ_≥cuz 12769 ℝ⁺crp 12927 (,)cioo 13282 [,)cico 13284 ⌊cfl 13728 ⇝ cli 15426 ∏cprod 15845 Meascmea 46440 volncvoln 46529

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cc 10364 ax-ac2 10392 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-acn 9871 df-ac 10045 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-rlim 15431 df-sum 15629 df-prod 15846 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-pws 17388 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-subrng 20466 df-subrg 20490 df-drng 20651 df-field 20652 df-abv 20729 df-staf 20759 df-srng 20760 df-lmod 20800 df-lss 20870 df-lmhm 20961 df-lvec 21042 df-sra 21112 df-rgmod 21113 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-refld 21547 df-phl 21568 df-dsmm 21674 df-frlm 21689 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cn 23147 df-cnp 23148 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-xms 24241 df-ms 24242 df-tms 24243 df-nm 24503 df-ngp 24504 df-tng 24505 df-nrg 24506 df-nlm 24507 df-cncf 24804 df-clm 24996 df-cph 25101 df-tcph 25102 df-rrx 25318 df-ovol 25398 df-vol 25399 df-salg 46300 df-sumge0 46354 df-mea 46441 df-ome 46481 df-caragen 46483 df-ovoln 46528 df-voln 46530

This theorem is referenced by: vonioo 46673

W3C validator