vonioolem1 - Mathbox for Glauco Siliprandi

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Theorem vonioolem1 43744

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 6991	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
7	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V)
8	4, 7	fvmpt2d 6782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
9		ovexd 7199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V)
10	8, 9	fvmpt2d 6782	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
11	10	oveq2d 7180	. . . . . . 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	ffvelrnda 6855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
14	13	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
15	14	recnd 10740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ)
16		vonioolem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
17	16	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
18	17	ffvelrnda 6855	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
19	18	recnd 10740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ)
20		nnrecre 11751	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
21	20	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
22	21	recnd 10740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ)
23	15, 19, 22	subsub4d 11099	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛))))
24	11, 23	eqtr4d 2776	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
25	24	prodeq2dv 15362	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))
26	25	mpteq2dva 5122	. . . 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 2773	. . 3 ⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))))
28		nfv 1920	. . . 4 ⊢ Ⅎ𝑘𝜑
29		rpssre 12472	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
30		vonioolem1.t	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
31	16	ffvelrnda 6855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
32		difrp 12503	. . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ⁺))
33	31, 13, 32	syl2anc 587	. . . . . . 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	sseldi 3873	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ)
36	35	recnd 10740	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ)
37		eqid 2738	. . . 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 42975	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
39	27, 38	eqbrtrd 5049	. 2 ⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
40		vonioolem1.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑁)
41		nnex 11715	. . . . . 6 ⊢ ℕ ∈ V
42	41	mptex 6990	. . . . 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 2837	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
45		vonioolem1.s	. . . 4 ⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
46	41	mptex 6990	. . . . 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 2837	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
49		vonioolem1.n	. . . 4 ⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1)
50		1rp 12469	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
51	50	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ⁺)
52		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))
53	28, 52, 34	rnmptssd 42257	. . . . . . . . . 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 10792	. . . . . . . . . . . . 13 ⊢ < Or ℝ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → < Or ℝ)
57	52	rnmptfi 42229	. . . . . . . . . . . . 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 6070	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅)
61	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
62	53, 61	sstrd 3885	. . . . . . . . . . . 12 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
63		fiinfcl 9031	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
65	54, 64	eqeltrid 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
66	53, 65	sseldd 3876	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
67	51, 66	rpdivcld 12524	. . . . . . . 8 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ⁺)
68	67	rpred 12507	. . . . . . 7 ⊢ (𝜑 → (1 / 𝐸) ∈ ℝ)
69	67	rpge0d 12511	. . . . . . 7 ⊢ (𝜑 → 0 ≤ (1 / 𝐸))
70		flge0nn0 13274	. . . . . . 7 ⊢ (((1 / 𝐸) ∈ ℝ ∧ 0 ≤ (1 / 𝐸)) → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
71	68, 69, 70	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈ ℕ₀)
72		nn0p1nn 12008	. . . . . 6 ⊢ ((⌊‘(1 / 𝐸)) ∈ ℕ₀ → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
73	71, 72	syl 17	. . . . 5 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℕ)
74	73	nnzd 12160	. . . 4 ⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈ ℤ)
75	49, 74	eqeltrid 2837	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
76	49	recnnltrp 42438	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ ℝ⁺ → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
77	66, 76	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸))
78	77	simpld 498	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
79		uznnssnn 12370	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (ℤ_≥‘𝑁) ⊆ ℕ)
80	78, 79	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ ℕ)
81	40, 80	eqsstrid 3923	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ⊆ ℕ)
82	81	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ)
83		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
84	82, 83	sseldd 3876	. . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
88		eqid 2738	. . . . . . . . . 10 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
89	18, 21	readdcld 10741	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ)
90	89	fmpttd 6883	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
91	8	feq1d 6483	. . . . . . . . . . 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 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ)
94	87, 88, 92, 93	hoimbl 43695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋))
95	94	elexd 3417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V)
96	86, 95	fvmpt2d 6782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
97	84, 96	syldan 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
98	97	fveq2d 6672	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
99	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin)
100	59	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅)
101	84, 92	syldan 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ)
102	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ)
103		eqid 2738	. . . . . 6 ⊢ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
104	99, 100, 101, 102, 103	vonn0hoi 43734	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))))
105	101	ffvelrnda 6855	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
106	84, 14	syldanl 605	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
107		volico 43050	. . . . . . . 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 587	. . . . . . 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 605	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛)))
110	84, 21	syldanl 605	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
111	78	nnrecred 11760	. . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑁))
115	114	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑁))
116		eluzle 12330	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑁 ≤ 𝑛)
117	115, 116	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛)
118	117	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛)
119	78	nnrpd 12505	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
120	119	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈ ℝ⁺)
121		nnrp 12476	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
122	84, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ⁺)
123	120, 122	lerecd 12526	. . . . . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁))
126	111	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ)
127	29, 66	sseldi 3873	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ)
128	127	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ)
129	77	simprd 499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 / 𝑁) < 𝐸)
130	129	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸)
131	62	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)
132	58	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin)
133		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
134		ovexd 7199	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V)
135	52	elrnmpt1 5795	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
136	133, 134, 135	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
137	136	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))))
138		infrefilb 11697	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
140	54, 139	eqbrtrid 5062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘)))
141	126, 128, 35, 130, 140	ltletrd 10871	. . . . . . . . . . . 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 10869	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	84, 18	syldanl 605	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
145	144, 110, 106	ltaddsub2d 11312	. . . . . . . . . 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 5049	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘))
148	147	iftrued 4419	. . . . . . 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 2773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
150	149	prodeq2dv 15362	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
151	98, 104, 150	3eqtrd 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
152		fvexd 6683	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V)
153	45	fvmpt2 6780	. . . . 5 ⊢ ((𝑛 ∈ ℕ ∧ ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
154	84, 152, 153	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛)))
155		prodex 15346	. . . . . 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 6780	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))
159	151, 154, 158	3eqtr4rd 2784	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛))
160	40, 44, 48, 75, 159	climeq 15007	. 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 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 Vcvv 3397 ⊆ wss 3841 ∅c0 4209 ifcif 4411 class class class wbr 5027 ↦ cmpt 5107 Or wor 5437 dom cdm 5519 ran crn 5520 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 Xcixp 8500 Fincfn 8548 infcinf 8971 ℝcr 10607 0cc0 10608 1c1 10609 + caddc 10611 < clt 10746 ≤ cle 10747 − cmin 10941 / cdiv 11368 ℕcn 11709 ℕ₀cn0 11969 ℤcz 12055 ℤ_≥cuz 12317 ℝ⁺crp 12465 [,)cico 12816 ⌊cfl 13244 ⇝ cli 14924 ∏cprod 15344 volcvol 24208 volncvoln 43602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cc 9928 ax-ac2 9956 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-disj 4993 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-oadd 8128 df-omul 8129 df-er 8313 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-dju 9396 df-card 9434 df-acn 9437 df-ac 9609 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-fl 13246 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-rlim 14929 df-sum 15129 df-prod 15345 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-mulg 18336 df-subg 18387 df-cntz 18558 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cn 21971 df-cnp 21972 df-cmp 22131 df-tx 22306 df-hmeo 22499 df-xms 23066 df-ms 23067 df-tms 23068 df-cncf 23623 df-ovol 24209 df-vol 24210 df-salg 43376 df-sumge0 43427 df-mea 43514 df-ome 43554 df-caragen 43556 df-ovoln 43601 df-voln 43603

This theorem is referenced by: vonioolem2 43745

W3C validator