vonioolem1 - Mathbox for Glauco Siliprandi

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Theorem vonioolem1 47321

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 7223	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
7	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
8	4, 7	fvmpt2d 7004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
9		ovexd 7446	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
10	8, 9	fvmpt2d 7004	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
11	10	oveq2d 7427	. . . . . . 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 7080	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
14	13	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
15	14	recnd 11237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ)
16		vonioolem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
18	17	ffvelcdmda 7080	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
19	18	recnd 11237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ)
20		nnrecre 12278	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
21	20	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
22	21	recnd 11237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ)
23	15, 19, 22	subsub4d 11600	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛))))
24	11, 23	eqtr4d 2807	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
25	24	prodeq2dv 15976	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
26	25	mpteq2dva 5206	. . . 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 2804	. . 3 ⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))))
28		nfv 1941	. . . 4 ⊢ Ⅎ𝑘𝜑
29		rpssre 13024	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
30		vonioolem1.t	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
31	16	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
32		difrp 13056	. . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
33	31, 13, 32	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
34	30, 33	mpbid 235	. . . . . 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 11237	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ)
37		eqid 2769	. . . 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 46549	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
39	27, 38	eqbrtrd 5135	. 2 ⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
40		vonioolem1.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
41		nnex 12239	. . . . . 6 ⊢ ℕ ∈ V
42	41	mptex 7222	. . . . 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 2873	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
45		vonioolem1.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
46	41	mptex 7222	. . . . 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 2873	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
49		vonioolem1.n	. . . 4 ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)
50		1rp 13020	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
51	50	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
52		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
53	28, 52, 34	rnmptssd 7120	. . . . . . . . . 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 11290	. . . . . . . . . . . . 13 ⊢ < Or ℝ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ)
57	52	rnmptfi 45816	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ Fin → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
58	5, 57	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
59		vonioolem1.u	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ≠ ∅)
60	28, 34, 52, 59	rnmptn0 6246	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅)
61	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
62	53, 61	sstrd 3953	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
63		fiinfcl 9463	. . . . . . . . . . . 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 1397	. . . . . . . . . . 11 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
65	54, 64	eqeltrid 2873	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
66	53, 65	sseldd 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
67	51, 66	rpdivcld 13077	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ⁺)
68	67	rpred 13060	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ)
69	67	rpge0d 13064	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (1 / 𝐸))
70		flge0nn0 13853	. . . . . . 7 ⊢ (((1 / 𝐸) ∈ ℝ ∧ 0 ≤ (1 / 𝐸)) → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
71	68, 69, 70	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
72		nn0p1nn 12543	. . . . . 6 ⊢ ((⌊‘(1 / 𝐸)) ∈ ℕ₀ → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
73	71, 72	syl 18	. . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
74	73	nnzd 12617	. . . 4 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℤ)
75	49, 74	eqeltrid 2873	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76	49	recnnltrp 46019	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℝ⁺ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
77	66, 76	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
78	77	simpld 499	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
79		uznnssnn 12919	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (ℤ_≥‘𝑁) ⊆ ℕ)
80	78, 79	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ ℕ)
81	40, 80	eqsstrid 3981	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ⊆ ℕ)
82	81	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ)
83		simpr 489	. . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
88		eqid 2769	. . . . . . . . . 10 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
89	18, 21	readdcld 11238	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
90	89	fmpttd 7111	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
91	8	feq1d 6688	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
92	90, 91	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
93	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
94	87, 88, 92, 93	hoimbl 47272	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
95	94	elexd 3484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
96	86, 95	fvmpt2d 7004	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
97	84, 96	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
98	97	fveq2d 6886	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
99	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin)
100	59	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅)
101	84, 92	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ)
102	12	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ)
103		eqid 2769	. . . . . 6 ⊢ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
104	99, 100, 101, 102, 103	vonn0hoi 47311	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
105	101	ffvelcdmda 7080	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
106	84, 14	syldanl 613	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
107		volico 46624	. . . . . . . 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 595	. . . . . . 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 613	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
110	84, 21	syldanl 613	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
111	78	nnrecred 12287	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
112	111	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
113	35	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
114	40	eleq2i 2861	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑁))
115	114	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
116		eluzle 12875	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑁 ≤ 𝑛)
117	115, 116	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛)
118	117	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛)
119	78	nnrpd 13058	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
120	119	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈ ℝ⁺)
121		nnrp 13028	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
122	84, 121	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ⁺)
123	120, 122	lerecd 13079	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁 ≤ 𝑛 ↔ (1 / 𝑛) ≤ (1 / 𝑁)))
124	118, 123	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (1 / 𝑛) ≤ (1 / 𝑁))
125	124	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁))
126	111	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
127	29, 66	sselid 3941	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ)
128	127	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ)
129	77	simprd 500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 / 𝑁) < 𝐸)
130	129	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸)
131	62	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
132	58	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
133		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
134		ovexd 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V)
135	52	elrnmpt1 5951	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
136	133, 134, 135	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
137	136	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
138		infrefilb 12201	. . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
140	54, 139	eqbrtrid 5148	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
141	126, 128, 35, 130, 140	ltletrd 11370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
142	141	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
143	110, 112, 113, 125, 142	lelttrd 11368	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	84, 18	syldanl 613	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
145	144, 110, 106	ltaddsub2d 11815	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘) ↔ (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘))))
146	143, 145	mpbird 260	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘))
147	109, 146	eqbrtrd 5135	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘))
148	147	iftrued 4498	. . . . . . 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 2804	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
150	149	prodeq2dv 15976	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
151	98, 104, 150	3eqtrd 2808	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
152		fvexd 6897	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V)
153	45	fvmpt2 7002	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
154	84, 152, 153	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
155		prodex 15959	. . . . . 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 7002	. . . . 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 595	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
159	151, 154, 158	3eqtr4rd 2815	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛))
160	40, 44, 48, 75, 159	climeq 15618	. 2 ⊢ (𝜑 → (𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))))
161	39, 160	mpbid 235	1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 Vcvv 3461 ⊆ wss 3911 ∅c0 4292 ifcif 4490 class class class wbr 5111 ↦ cmpt 5194 Or wor 5569 dom cdm 5662 ran crn 5663 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Xcixp 8895 Fincfn 8943 infcinf 9401 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 < clt 11243 ≤ cle 11244 − cmin 11441 / cdiv 11871 ℕcn 12233 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ℝ⁺crp 13016 [,)cico 13374 ⌊cfl 13823 ⇝ cli 15535 ∏cprod 15957 volcvol 25591 volncvoln 47179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cc 10419 ax-ac2 10447 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-acn 9928 df-ac 10100 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-prod 15958 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cn 23353 df-cnp 23354 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-ovol 25592 df-vol 25593 df-salg 46950 df-sumge0 47004 df-mea 47091 df-ome 47131 df-caragen 47133 df-ovoln 47178 df-voln 47180

This theorem is referenced by: vonioolem2 47322

W3C validator