vonioolem1 - Mathbox for Glauco Siliprandi

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Theorem vonioolem1 46671

Description: The sequence of the measures of the half-open intervals converges to the measure of their union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
vonioolem1.x	⊢ (𝜑 → 𝑋 ∈ Fin)
vonioolem1.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
vonioolem1.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
vonioolem1.u	⊢ (𝜑 → 𝑋 ≠ ∅)
vonioolem1.t	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
vonioolem1.c	⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
vonioolem1.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
vonioolem1.s	⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
vonioolem1.r	⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
vonioolem1.e	⊢ 𝐸 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
vonioolem1.n	⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)
vonioolem1.z	⊢ 𝑍 = (ℤ_≥‘𝑁)

**Assertion**
Ref	Expression
vonioolem1	⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑘,𝑛 𝐶,𝑘 𝑆,𝑛 𝑇,𝑛 𝑘,𝑋,𝑛 𝑘,𝑍,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝐴(𝑘) 𝐶(𝑛) 𝐷(𝑘,𝑛) 𝑆(𝑘) 𝑇(𝑘) 𝐸(𝑘,𝑛) 𝑁(𝑘,𝑛)

**Proof of Theorem vonioolem1**
Step	Hyp	Ref	Expression
1		vonioolem1.r	. . . . 5 ⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))))
3		vonioolem1.c	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
4	3	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))))
5		vonioolem1.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ Fin)
6	5	mptexd 7180	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
8	4, 7	fvmpt2d 6963	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
9		ovexd 7404	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
10	8, 9	fvmpt2d 6963	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
11	10	oveq2d 7385	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛))))
12		vonioolem1.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
13	12	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
14	13	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
15	14	recnd 11178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ)
16		vonioolem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
18	17	ffvelcdmda 7038	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
19	18	recnd 11178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ)
20		nnrecre 12204	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
21	20	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
22	21	recnd 11178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ)
23	15, 19, 22	subsub4d 11540	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛))))
24	11, 23	eqtr4d 2767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
25	24	prodeq2dv 15864	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
26	25	mpteq2dva 5195	. . . 4 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))))
27	2, 26	eqtrd 2764	. . 3 ⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))))
28		nfv 1914	. . . 4 ⊢ Ⅎ𝑘𝜑
29		rpssre 12935	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
30		vonioolem1.t	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
31	16	ffvelcdmda 7038	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
32		difrp 12967	. . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
33	31, 13, 32	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
34	30, 33	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
35	29, 34	sselid 3941	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
36	35	recnd 11178	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ)
37		eqid 2729	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
38	28, 5, 36, 37	fprodsubrecnncnv 45899	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
39	27, 38	eqbrtrd 5124	. 2 ⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
40		vonioolem1.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
41		nnex 12168	. . . . . 6 ⊢ ℕ ∈ V
42	41	mptex 7179	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) ∈ V
43	42	a1i 11	. . . 4 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) ∈ V)
44	1, 43	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
45		vonioolem1.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
46	41	mptex 7179	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V
47	46	a1i 11	. . . 4 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V)
48	45, 47	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
49		vonioolem1.n	. . . 4 ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)
50		1rp 12931	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
51	50	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
52		eqid 2729	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
53	28, 52, 34	rnmptssd 45183	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ⁺)
54		vonioolem1.e	. . . . . . . . . . 11 ⊢ 𝐸 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )
55		ltso 11230	. . . . . . . . . . . . 13 ⊢ < Or ℝ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ)
57	52	rnmptfi 45158	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
58	5, 57	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
59		vonioolem1.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ ∅)
60	28, 34, 52, 59	rnmptn0 6205	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅)
61	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
62	53, 61	sstrd 3954	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
63		fiinfcl 9430	. . . . . . . . . . . 12 ⊢ (( < Or ℝ ∧ (ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
64	56, 58, 60, 62, 63	syl13anc 1374	. . . . . . . . . . 11 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
65	54, 64	eqeltrid 2832	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
66	53, 65	sseldd 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
67	51, 66	rpdivcld 12988	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ⁺)
68	67	rpred 12971	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ)
69	67	rpge0d 12975	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (1 / 𝐸))
70		flge0nn0 13758	. . . . . . 7 ⊢ (((1 / 𝐸) ∈ ℝ ∧ 0 ≤ (1 / 𝐸)) → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
71	68, 69, 70	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
72		nn0p1nn 12457	. . . . . 6 ⊢ ((⌊‘(1 / 𝐸)) ∈ ℕ₀ → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
73	71, 72	syl 17	. . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
74	73	nnzd 12532	. . . 4 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℤ)
75	49, 74	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76	49	recnnltrp 45366	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℝ⁺ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
77	66, 76	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
78	77	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
79		uznnssnn 12830	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (ℤ_≥‘𝑁) ⊆ ℕ)
80	78, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ ℕ)
81	40, 80	eqsstrid 3982	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ⊆ ℕ)
82	81	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ)
83		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
84	82, 83	sseldd 3944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℕ)
85		vonioolem1.d	. . . . . . . . 9 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
86	85	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
87	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
88		eqid 2729	. . . . . . . . . 10 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
89	18, 21	readdcld 11179	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
90	89	fmpttd 7069	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
91	8	feq1d 6652	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
92	90, 91	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
93	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
94	87, 88, 92, 93	hoimbl 46622	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
95	94	elexd 3468	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
96	86, 95	fvmpt2d 6963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
97	84, 96	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
98	97	fveq2d 6844	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
99	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin)
100	59	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅)
101	84, 92	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ)
102	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ)
103		eqid 2729	. . . . . 6 ⊢ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
104	99, 100, 101, 102, 103	vonn0hoi 46661	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
105	101	ffvelcdmda 7038	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
106	84, 14	syldanl 602	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
107		volico 45974	. . . . . . . 8 ⊢ ((((𝐶‘𝑛)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0))
108	105, 106, 107	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0))
109	84, 10	syldanl 602	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
110	84, 21	syldanl 602	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
111	78	nnrecred 12213	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
112	111	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
113	35	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
114	40	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑁))
115	114	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
116		eluzle 12782	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑁 ≤ 𝑛)
117	115, 116	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛)
118	117	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛)
119	78	nnrpd 12969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
120	119	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈ ℝ⁺)
121		nnrp 12939	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
122	84, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ⁺)
123	120, 122	lerecd 12990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁 ≤ 𝑛 ↔ (1 / 𝑛) ≤ (1 / 𝑁)))
124	118, 123	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (1 / 𝑛) ≤ (1 / 𝑁))
125	124	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁))
126	111	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
127	29, 66	sselid 3941	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ)
128	127	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ)
129	77	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 / 𝑁) < 𝐸)
130	129	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸)
131	62	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
132	58	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
133		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
134		ovexd 7404	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V)
135	52	elrnmpt1 5913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
136	133, 134, 135	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
137	136	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
138		infrefilb 12145	. . . . . . . . . . . . . . 15 ⊢ ((ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
139	131, 132, 137, 138	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
140	54, 139	eqbrtrid 5137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
141	126, 128, 35, 130, 140	ltletrd 11310	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
142	141	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
143	110, 112, 113, 125, 142	lelttrd 11308	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	84, 18	syldanl 602	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
145	144, 110, 106	ltaddsub2d 11755	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘) ↔ (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘))))
146	143, 145	mpbird 257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘))
147	109, 146	eqbrtrd 5124	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘))
148	147	iftrued 4492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
149	108, 148	eqtrd 2764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
150	149	prodeq2dv 15864	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
151	98, 104, 150	3eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
152		fvexd 6855	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V)
153	45	fvmpt2 6961	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
154	84, 152, 153	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
155		prodex 15847	. . . . . 6 ⊢ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V
156	155	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V)
157	1	fvmpt2 6961	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
158	84, 156, 157	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
159	151, 154, 158	3eqtr4rd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛))
160	40, 44, 48, 75, 159	climeq 15509	. 2 ⊢ (𝜑 → (𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))))
161	39, 160	mpbid 232	1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 ⊆ wss 3911 ∅c0 4292 ifcif 4484 class class class wbr 5102 ↦ cmpt 5183 Or wor 5538 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 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 [,)cico 13284 ⌊cfl 13728 ⇝ cli 15426 ∏cprod 15845 volcvol 25397 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

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-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-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-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 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-cncf 24804 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: vonioolem2 46672

W3C validator