vonioolem2 - Mathbox for Glauco Siliprandi

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Theorem vonioolem2 47517

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 47410	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
3		1zzd 12653	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
4		nnuz 12930	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
5	1	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
6		eqid 2762	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
7		vonioolem2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
8	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
9	8	ffvelcdmda 7081	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
10		nnrecre 12306	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
11	10	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
12	9, 11	readdcld 11266	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
13	12	fmpttd 7112	. . . . . . 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 7227	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
17	16	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
18	15, 17	fvmpt2d 7004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
19	18	feq1d 6688	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
20	13, 19	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
21		vonioolem2.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
22	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
23	5, 6, 20, 22	hoimbl 47467	. . . . 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 7111	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
26		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
27		oveq2 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
28	27	oveq2d 7433	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) + (1 / 𝑛)) = ((𝐴‘𝑘) + (1 / 𝑚)))
29	28	mpteq2dv 5203	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
30	29	cbvmptv 5213	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
31	14, 30	eqtri 2785	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
32		oveq2 7425	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
33	32	oveq2d 7433	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
34	33	mpteq2dv 5203	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
35		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
36	35	peano2nnd 12278	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
37	5	mptexd 7227	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
38	31, 34, 36, 37	fvmptd3 7014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))))
39		ovexd 7452	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
40	38, 39	fvmpt2d 7004	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴‘𝑘) + (1 / (𝑛 + 1))))
41		1red 11237	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
42		nnre 12268	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
43	42, 41	readdcld 11266	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
44		peano2nn 12273	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
45		nnne0 12298	. . . . . . . . . . . . 13 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
46	44, 45	syl 18	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
47	41, 43, 46	redivcld 12071	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
48	47	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
49	9, 48	readdcld 11266	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ ℝ)
50	40, 49	eqeltrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ)
51	50	rexrd 11287	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ^*)
52		ressxr 11281	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
53	21	ffvelcdmda 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
54	52, 53	sselid 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
55	54	adantlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
56	42	ltp1d 12173	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
57		nnrp 13058	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
58	44	nnrpd 13088	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ⁺)
59	57, 58	ltrecd 13108	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
60	56, 59	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
61	47, 10, 60	ltled 11386	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
62	61	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
63	48, 11, 9, 62	leadd2dd 11857	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛)))
64		ovexd 7452	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
65	18, 64	fvmpt2d 7004	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
66	40, 65	breq12d 5120	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
68	53	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
69		eqidd 2763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (𝐵‘𝑘))
70	68, 69	eqled 11341	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘))
71		icossico 13473	. . . . . . 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 852	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
73	26, 72	ixpssixp 45932	. . . . 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 3476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
76	74, 75	fvmpt2d 7004	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
77		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
78	77	fveq1d 6884	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
79	78	oveq1d 7432	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
80	79	ixpeq2dv 8924	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
81	80	cbvmptv 5213	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
82	24, 81	eqtri 2785	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))
83		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
84	83	fveq1d 6884	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
85	84	oveq1d 7432	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
86	85	ixpeq2dv 8924	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
87		ovex 7450	. . . . . . . . . 10 ⊢ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
88	87	rgenw 3082	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V
89		ixpexg 8933	. . . . . . . . 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 7014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))
93	76, 92	sseq12d 3967	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)))
95	1, 6, 7, 21	hoimbl 47467	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
96		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑘𝜑
97	7	ffvelcdmda 7081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
98	96, 1, 97, 53	vonhoire 47508	. . . . 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 1837	. . . . . . . . 9 ⊢ Ⅎ𝑘⊤
102		ioossico 13495	. . . . . . . . . 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 45932	. . . . . . . 8 ⊢ (⊤ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
105	104	mptru 1577	. . . . . . 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 3968	. . . . 5 ⊢ (𝜑 → 𝐼 ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)))
108	52	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
109	7, 108	fssd 6724	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ^*)
110	21, 108	fssd 6724	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ^*)
111	1, 6, 109, 110	ioovonmbl 47513	. . . . . 6 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
112	99, 111	eqeltrid 2866	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ dom (voln‘𝑋))
113	2, 95, 98, 107, 112	meassre 47313	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ)
114	2	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas)
115	76, 23	eqeltrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ dom (voln‘𝑋))
116	112	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋))
117	52, 97	sselid 3932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
118	117	adantlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
119	57	rpreccld 13100	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
120	119	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ⁺)
121	9, 120	ltaddrpd 13123	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)))
122		icossioo 13497	. . . . . . . 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 852	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
124	26, 123	ixpssixp 45932	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
125	65	oveq1d 7432	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
126	125	ixpeq2dva 8923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
127	76, 126	eqtrd 2797	. . . . . . 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 3967	. . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ 𝐼)
131	114, 6, 115, 116, 130	meassle 47299	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷‘𝑛)) ≤ ((voln‘𝑋)‘𝐼))
132		eqid 2762	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
133	2, 3, 4, 25, 94, 113, 131, 132	meaiuninc2 47318	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
134	96, 1, 97, 54	iunhoiioo 47512	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
135	127	iuneq2dv 4979	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)))
136	134, 135, 100	3eqtr4d 2807	. . . . . 6 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
137	136	eqcomd 2768	. . . . 5 ⊢ (𝜑 → 𝐼 = ∪ 𝑛 ∈ ℕ (𝐷‘𝑛))
138	137	fveq2d 6886	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)))
139	138	eqcomd 2768	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
140	133, 139	breqtrd 5135	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
141		2fveq3 6887	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
142	141	cbvmptv 5213	. . . 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 2771	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
147		eqcom 2769	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 ↔ 𝑚 = 𝑛)
148	147	imbi1i 352	. . . . . . . . 9 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)))
149		eqcom 2769	. . . . . . . . . 10 ⊢ (((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘) ↔ ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
150	149	imbi2i 339	. . . . . . . . 9 ⊢ ((𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
151	148, 150	bitri 278	. . . . . . . 8 ⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)))
152	78, 151	mpbi 233	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))
153	152	oveq2d 7433	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
154	153	prodeq2ad 46430	. . . . 5 ⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
155	154	cbvmptv 5213	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
156		eqid 2762	. . . 4 ⊢ inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
157		eqid 2762	. . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
159		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
160	158, 159	oveq12d 7435	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) − (𝐴‘𝑗)) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
161	160	cbvmptv 5213	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
162	161	rneqi 5925	. . . . . . . . 9 ⊢ ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
163	162	infeq1i 9453	. . . . . . . 8 ⊢ inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
164	163	oveq2i 7428	. . . . . . 7 ⊢ (1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < )) = (1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))
165	164	fveq2i 6885	. . . . . 6 ⊢ (⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) = (⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )))
166	165	oveq1i 7427	. . . . 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 6885	. . . 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 47516	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
169	143, 168	eqbrtrd 5131	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
170		climuni 15643	. 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 596	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2957 ∀wral 3078 Vcvv 3453 ⊆ wss 3902 ∅c0 4282 ∪ ciun 4954 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ran crn 5660 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 Xcixp 8908 Fincfn 8956 infcinf 9415 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 − cmin 11469 / cdiv 11899 ℕcn 12261 ℤ_≥cuz 12891 ℝ⁺crp 13046 (,)cioo 13402 [,)cico 13404 ⌊cfl 13855 ⇝ cli 15575 ∏cprod 15996 Meascmea 47285 volncvoln 47374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cc 10441 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207 ax-mulf 11208

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5075 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-omul 8464 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-rlim 15580 df-sum 15778 df-prod 15997 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-pws 17540 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-ghm 19347 df-cntz 19450 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-rhm 20619 df-subrng 20714 df-subrg 20738 df-drng 20898 df-field 20899 df-abv 20981 df-staf 21011 df-srng 21012 df-lmod 21052 df-lss 21122 df-lmhm 21212 df-lvec 21293 df-sra 21363 df-rgmod 21364 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-cnfld 21592 df-refld 21824 df-phl 21845 df-dsmm 21951 df-frlm 21966 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cn 23458 df-cnp 23459 df-cmp 23618 df-tx 23794 df-hmeo 23987 df-xms 24552 df-ms 24553 df-tms 24554 df-nm 24814 df-ngp 24815 df-tng 24816 df-nrg 24817 df-nlm 24818 df-cncf 25112 df-clm 25297 df-cph 25402 df-tcph 25403 df-rrx 25619 df-ovol 25698 df-vol 25699 df-salg 47145 df-sumge0 47199 df-mea 47286 df-ome 47326 df-caragen 47328 df-ovoln 47373 df-voln 47375

This theorem is used by: vonioo 47518

W3C validator