Theorem vonicclem2 47042

Description: The n-dimensional Lebesgue measure of closed intervals. This is the second statement in Proposition 115G (d) of [Fremlin1] p. 32. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
vonicclem2.x	⊢ (𝜑 → 𝑋 ∈ Fin)
vonicclem2.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
vonicclem2.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
vonicclem2.n	⊢ (𝜑 → 𝑋 ≠ ∅)
vonicclem2.t	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))
vonicclem2.i	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))
vonicclem2.c	⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
vonicclem2.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))

Assertion

Ref	Expression
vonicclem2	⊢ (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 vonicclem2

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . 4 ⊢ Ⅎ𝑛𝜑
2		vonicclem2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
3	2	vonmea 46932	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
4		1zzd 12534	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
5		nnuz 12802	. . . 4 ⊢ ℕ = (ℤ≥‘1)
6	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7		eqid 2737	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
8		vonicclem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10		vonicclem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
11	10	ffvelcdmda 7038	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
12	11	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
13		nnrecre 12199	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
14	13	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
15	12, 14	readdcld 11173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ ℝ)
16	15	fmpttd 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)
17		vonicclem2.c	. . . . . . . . . 10 ⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))))
19	2	mptexd 7180	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
21	18, 20	fvmpt2d 6963	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
22	21	feq1d 6652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
23	16, 22	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
24	6, 7, 9, 23	hoimbl 46989	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ dom (voln‘𝑋))
25		vonicclem2.d	. . . . 5 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
26	24, 25	fmptd 7068	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
27		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
28		ressxr 11188	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
29	8	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
30	28, 29	sselid 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
31	30	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
32		ovexd 7403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ V)
33	21, 32	fvmpt2d 6963	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐵‘𝑘) + (1 / 𝑛)))
34	33, 15	eqeltrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
35	34	rexrd 11194	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ*)
36	9	ffvelcdmda 7038	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
37	36	leidd 11715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38		1red 11145	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnre 12164	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
40	39, 38	readdcld 11173	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41		peano2nn 12169	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42		nnne0 12191	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
44	38, 40, 43	redivcld 11981	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
45	44	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
46	39	ltp1d 12084	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47		nnrp 12929	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
48	41	nnrpd 12959	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
49	47, 48	ltrecd 12979	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
50	46, 49	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
51	44, 13, 50	ltled 11293	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
52	51	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
53	45, 14, 12, 52	leadd2dd 11764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛)))
54		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
55	54	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐵‘𝑘) + (1 / 𝑛)) = ((𝐵‘𝑘) + (1 / 𝑚)))
56	55	mpteq2dv 5194	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
57	56	cbvmptv 5204	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
58	17, 57	eqtri 2760	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
59		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
60	59	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐵‘𝑘) + (1 / 𝑚)) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
61	60	mpteq2dv 5194	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
62		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
63	62	peano2nnd 12174	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
64	6	mptexd 7180	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
65	58, 61, 63, 64	fvmptd3 6973	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
66		ovexd 7403	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
67	65, 66	fvmpt2d 6963	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
68	67, 33	breq12d 5113	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ↔ ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛))))
69	53, 68	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
70		icossico 13344	. . . . . . 7 ⊢ ((((𝐴‘𝑘) ∈ ℝ* ∧ ((𝐶‘𝑛)‘𝑘) ∈ ℝ*) ∧ ((𝐴‘𝑘) ≤ (𝐴‘𝑘) ∧ ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
71	31, 35, 37, 69, 70	syl22anc 839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
72	27, 71	ixpssixp 45451	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
73		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
74	73	fveq1d 6844	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
75	74	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
76	75	ixpeq2dv 8863	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
77	76	cbvmptv 5204	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
78	25, 77	eqtri 2760	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
79		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
80	79	fveq1d 6844	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
81	80	oveq2d 7384	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
82	81	ixpeq2dv 8863	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83		ovex 7401	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
84	83	rgenw 3056	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85		ixpexg 8872	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
86	84, 85	ax-mp 5	. . . . . . . 8 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
87	86	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V)
88	78, 82, 63, 87	fvmptd3 6973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
89	25	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))))
90	24	elexd 3466	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ V)
91	89, 90	fvmpt2d 6963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
92	88, 91	sseq12d 3969	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))))
93	72, 92	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛))
94		1nn 12168	. . . . . 6 ⊢ 1 ∈ ℕ
95	94, 5	eleqtri 2835	. . . . 5 ⊢ 1 ∈ (ℤ≥‘1)
96	95	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ (ℤ≥‘1))
97		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
98	97	fveq1d 6844	. . . . . . . . 9 ⊢ (𝑛 = 1 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
99	98	oveq2d 7384	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
100	99	ixpeq2dv 8863	. . . . . . 7 ⊢ (𝑛 = 1 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
101	94	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℕ)
102		ovex 7401	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103	102	rgenw 3056	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104		ixpexg 8872	. . . . . . . . 9 ⊢ (∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
105	103, 104	ax-mp 5	. . . . . . . 8 ⊢ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
106	105	a1i 11	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V)
107	25, 100, 101, 106	fvmptd3 6973	. . . . . 6 ⊢ (𝜑 → (𝐷‘1) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
108	107	fveq2d 6846	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))))
109		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑘𝜑
110		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝜑)
111	94	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 1 ∈ ℕ)
112		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
113	94	elexi 3465	. . . . . . . 8 ⊢ 1 ∈ V
114		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115	114	anbi2d 631	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116	115	anbi1d 632	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋)))
117	98	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝐶‘𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
118	116, 117	imbi12d 344	. . . . . . . 8 ⊢ (𝑛 = 1 → ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ) ↔ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)))
119	113, 118, 34	vtocl 3517	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120	110, 111, 112, 119	syl21anc 838	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121	109, 2, 29, 120	vonhoire 47030	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122	108, 121	eqeltrd 2837	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123		eqid 2737	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
124	1, 3, 4, 5, 26, 93, 96, 122, 123	meaiininc 46845	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
125	109, 29, 11	iinhoiicc 47032	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)))
126	33	oveq2d 7384	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
127	126	ixpeq2dva 8862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
128	91, 127	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
129	128	iineq2dv 4974	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
130		vonicclem2.i	. . . . . . . 8 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘))
131	130	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)))
132	125, 129, 131	3eqtr4d 2782	. . . . . 6 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
133	132	eqcomd 2743	. . . . 5 ⊢ (𝜑 → 𝐼 = ∩ 𝑛 ∈ ℕ (𝐷‘𝑛))
134	133	fveq2d 6846	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
135	134	eqcomd 2743	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
136	124, 135	breqtrd 5126	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137		2fveq3 6847	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
138	137	cbvmptv 5204	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚)))
139	138	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))))
140		vonicclem2.n	. . . 4 ⊢ (𝜑 → 𝑋 ≠ ∅)
141		vonicclem2.t	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐵‘𝑘))
142	138	eqcomi 2746	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
143	2, 8, 10, 140, 141, 17, 25, 142	vonicclem1 47041	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	139, 143	eqbrtrd 5122	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
145		climuni 15487	. 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)))
146	136, 144, 145	syl2anc 585	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ∅c0 4287  ∩ ciin 4949   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5632  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Xcixp 8847  Fincfn 8895  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041  ℝ^*cxr 11177   < clt 11178   ≤ cle 11179   − cmin 11376   / cdiv 11806  ℕcn 12157  ℤ_≥cuz 12763  [,)cico 13275  [,]cicc 13276   ⇝ cli 15419  ∏cprod 15838  volncvoln 46896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cc 10357  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-er 8645  df-map 8777  df-pm 8778  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-fi 9326  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-q 12874  df-rp 12918  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-fl 13724  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-rlim 15424  df-sum 15622  df-prod 15839  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-starv 17204  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-unif 17212  df-hom 17213  df-cco 17214  df-rest 17354  df-topn 17355  df-0g 17373  df-gsum 17374  df-topgen 17375  df-pt 17376  df-prds 17379  df-xrs 17435  df-qtop 17440  df-imas 17441  df-xps 17443  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-mulg 19010  df-cntz 19258  df-cmn 19723  df-psmet 21313  df-xmet 21314  df-met 21315  df-bl 21316  df-mopn 21317  df-cnfld 21322  df-top 22850  df-topon 22867  df-topsp 22889  df-bases 22902  df-cn 23183  df-cnp 23184  df-cmp 23343  df-tx 23518  df-hmeo 23711  df-xms 24276  df-ms 24277  df-tms 24278  df-cncf 24839  df-ovol 25433  df-vol 25434  df-salg 46667  df-sumge0 46721  df-mea 46808  df-ome 46848  df-caragen 46850  df-ovoln 46895  df-voln 46897

This theorem is referenced by:  vonicc  47043