vonioolem1 - Mathbox for Glauco Siliprandi

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Theorem vonioolem1 45694

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 7227	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
7	6	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
8	4, 7	fvmpt2d 7010	. . . . . . . . 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 7010	. . . . . . . 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 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
14	13	adantlr 711	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
15	14	recnd 11246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ)
16		vonioolem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
18	17	ffvelcdmda 7085	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
19	18	recnd 11246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ)
20		nnrecre 12258	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
21	20	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
22	21	recnd 11246	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ)
23	15, 19, 22	subsub4d 11606	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛))))
24	11, 23	eqtr4d 2773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
25	24	prodeq2dv 15871	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
26	25	mpteq2dva 5247	. . . 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 2770	. . 3 ⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))))
28		nfv 1915	. . . 4 ⊢ Ⅎ𝑘𝜑
29		rpssre 12985	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
30		vonioolem1.t	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
31	16	ffvelcdmda 7085	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
32		difrp 13016	. . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
33	31, 13, 32	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
34	30, 33	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺)
35	29, 34	sselid 3979	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
36	35	recnd 11246	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ)
37		eqid 2730	. . . 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 44922	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
39	27, 38	eqbrtrd 5169	. 2 ⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
40		vonioolem1.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
41		nnex 12222	. . . . . 6 ⊢ ℕ ∈ V
42	41	mptex 7226	. . . . 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 2835	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
45		vonioolem1.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
46	41	mptex 7226	. . . . 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 2835	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
49		vonioolem1.n	. . . 4 ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)
50		1rp 12982	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
51	50	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
52		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
53	28, 52, 34	rnmptssd 44193	. . . . . . . . . 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 11298	. . . . . . . . . . . . 13 ⊢ < Or ℝ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ)
57	52	rnmptfi 44168	. . . . . . . . . . . . 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 6242	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅)
61	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
62	53, 61	sstrd 3991	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
63		fiinfcl 9498	. . . . . . . . . . . 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 1370	. . . . . . . . . . 11 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
65	54, 64	eqeltrid 2835	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
66	53, 65	sseldd 3982	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
67	51, 66	rpdivcld 13037	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ⁺)
68	67	rpred 13020	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ)
69	67	rpge0d 13024	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (1 / 𝐸))
70		flge0nn0 13789	. . . . . . 7 ⊢ (((1 / 𝐸) ∈ ℝ ∧ 0 ≤ (1 / 𝐸)) → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
71	68, 69, 70	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
72		nn0p1nn 12515	. . . . . 6 ⊢ ((⌊‘(1 / 𝐸)) ∈ ℕ₀ → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
73	71, 72	syl 17	. . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
74	73	nnzd 12589	. . . 4 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℤ)
75	49, 74	eqeltrid 2835	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76	49	recnnltrp 44385	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℝ⁺ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
77	66, 76	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
78	77	simpld 493	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
79		uznnssnn 12883	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (ℤ_≥‘𝑁) ⊆ ℕ)
80	78, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ ℕ)
81	40, 80	eqsstrid 4029	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ⊆ ℕ)
82	81	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ)
83		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
84	82, 83	sseldd 3982	. . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
88		eqid 2730	. . . . . . . . . 10 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
89	18, 21	readdcld 11247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
90	89	fmpttd 7115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
91	8	feq1d 6701	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
92	90, 91	mpbird 256	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
93	12	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
94	87, 88, 92, 93	hoimbl 45645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
95	94	elexd 3493	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
96	86, 95	fvmpt2d 7010	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
97	84, 96	syldan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
98	97	fveq2d 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
99	5	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin)
100	59	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅)
101	84, 92	syldan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ)
102	12	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ)
103		eqid 2730	. . . . . 6 ⊢ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
104	99, 100, 101, 102, 103	vonn0hoi 45684	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
105	101	ffvelcdmda 7085	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
106	84, 14	syldanl 600	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
107		volico 44997	. . . . . . . 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 582	. . . . . . 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 600	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
110	84, 21	syldanl 600	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
111	78	nnrecred 12267	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ)
112	111	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
113	35	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
114	40	eleq2i 2823	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑁))
115	114	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
116		eluzle 12839	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑁 ≤ 𝑛)
117	115, 116	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛)
118	117	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛)
119	78	nnrpd 13018	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
120	119	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈ ℝ⁺)
121		nnrp 12989	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
122	84, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ⁺)
123	120, 122	lerecd 13039	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁 ≤ 𝑛 ↔ (1 / 𝑛) ≤ (1 / 𝑁)))
124	118, 123	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (1 / 𝑛) ≤ (1 / 𝑁))
125	124	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁))
126	111	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
127	29, 66	sselid 3979	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ)
128	127	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ)
129	77	simprd 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 / 𝑁) < 𝐸)
130	129	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸)
131	62	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
132	58	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
133		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
134		ovexd 7446	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V)
135	52	elrnmpt1 5956	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
136	133, 134, 135	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
137	136	adantl 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
138		infrefilb 12204	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
140	54, 139	eqbrtrid 5182	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
141	126, 128, 35, 130, 140	ltletrd 11378	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
142	141	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
143	110, 112, 113, 125, 142	lelttrd 11376	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	84, 18	syldanl 600	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
145	144, 110, 106	ltaddsub2d 11819	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘) ↔ (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘))))
146	143, 145	mpbird 256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘))
147	109, 146	eqbrtrd 5169	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘))
148	147	iftrued 4535	. . . . . . 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 2770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
150	149	prodeq2dv 15871	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
151	98, 104, 150	3eqtrd 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
152		fvexd 6905	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V)
153	45	fvmpt2 7008	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
154	84, 152, 153	syl2anc 582	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
155		prodex 15855	. . . . . 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 7008	. . . . 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 582	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
159	151, 154, 158	3eqtr4rd 2781	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛))
160	40, 44, 48, 75, 159	climeq 15515	. 2 ⊢ (𝜑 → (𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))))
161	39, 160	mpbid 231	1 ⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ⊆ wss 3947 ∅c0 4321 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 Or wor 5586 dom cdm 5675 ran crn 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 Xcixp 8893 Fincfn 8941 infcinf 9438 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 / cdiv 11875 ℕcn 12216 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 [,)cico 13330 ⌊cfl 13759 ⇝ cli 15432 ∏cprod 15853 volcvol 25212 volncvoln 45552

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12979 df-xneg 13096 df-xadd 13097 df-xmul 13098 df-ioo 13332 df-ico 13334 df-icc 13335 df-fz 13489 df-fzo 13632 df-fl 13761 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-prod 15854 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-rest 17372 df-topn 17373 df-0g 17391 df-gsum 17392 df-topgen 17393 df-pt 17394 df-prds 17397 df-xrs 17452 df-qtop 17457 df-imas 17458 df-xps 17460 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-mulg 18987 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-mopn 21140 df-cnfld 21145 df-top 22616 df-topon 22633 df-topsp 22655 df-bases 22669 df-cn 22951 df-cnp 22952 df-cmp 23111 df-tx 23286 df-hmeo 23479 df-xms 24046 df-ms 24047 df-tms 24048 df-cncf 24618 df-ovol 25213 df-vol 25214 df-salg 45323 df-sumge0 45377 df-mea 45464 df-ome 45504 df-caragen 45506 df-ovoln 45551 df-voln 45553

This theorem is referenced by: vonioolem2 45695

W3C validator