Theorem vonicclem2 47139

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 1922	. . . 4 ⊢ Ⅎ𝑛𝜑
2		vonicclem2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
3	2	vonmea 47029	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
4		1zzd 12553	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
5		nnuz 12822	. . . 4 ⊢ ℕ = (ℤ≥‘1)
6	2	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7		eqid 2741	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
8		vonicclem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
9	8	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10		vonicclem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
11	10	ffvelcdmda 7028	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
12	11	adantlr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
13		nnrecre 12214	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
14	13	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
15	12, 14	readdcld 11170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ ℝ)
16	15	fmpttd 7059	. . . . . . 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 7171	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
20	19	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
21	18, 20	fvmpt2d 6952	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
22	21	feq1d 6640	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ))
23	16, 22	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ)
24	6, 7, 9, 23	hoimbl 47086	. . . . 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 7058	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
27		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
28		ressxr 11185	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
29	8	ffvelcdmda 7028	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
30	28, 29	sselid 3914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
31	30	adantlr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
32		ovexd 7394	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ V)
33	21, 32	fvmpt2d 6952	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐵‘𝑘) + (1 / 𝑛)))
34	33, 15	eqeltrd 2841	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
35	34	rexrd 11191	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ*)
36	9	ffvelcdmda 7028	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
37	36	leidd 11712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38		1red 11141	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnre 12176	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
40	39, 38	readdcld 11170	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41		peano2nn 12181	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42		nnne0 12206	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
44	38, 40, 43	redivcld 11978	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
45	44	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
46	39	ltp1d 12081	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47		nnrp 12949	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
48	41	nnrpd 12979	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
49	47, 48	ltrecd 12999	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛)))
50	46, 49	mpbid 234	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) < (1 / 𝑛))
51	44, 13, 50	ltled 11290	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
52	51	ad2antlr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
53	45, 14, 12, 52	leadd2dd 11761	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛)))
54		oveq2 7367	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
55	54	oveq2d 7375	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐵‘𝑘) + (1 / 𝑛)) = ((𝐵‘𝑘) + (1 / 𝑚)))
56	55	mpteq2dv 5168	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
57	56	cbvmptv 5178	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
58	17, 57	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
59		oveq2 7367	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
60	59	oveq2d 7375	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐵‘𝑘) + (1 / 𝑚)) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
61	60	mpteq2dv 5168	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
62		simpr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
63	62	peano2nnd 12186	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
64	6	mptexd 7171	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
65	58, 61, 63, 64	fvmptd3 6962	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
66		ovexd 7394	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
67	65, 66	fvmpt2d 6952	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
68	67, 33	breq12d 5087	. . . . . . . 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 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
70		icossico 13364	. . . . . . 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 845	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
72	27, 71	ixpssixp 45551	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
73		fveq2 6830	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
74	73	fveq1d 6832	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
75	74	oveq2d 7375	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
76	75	ixpeq2dv 8855	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
77	76	cbvmptv 5178	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
78	25, 77	eqtri 2764	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
79		fveq2 6830	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
80	79	fveq1d 6832	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
81	80	oveq2d 7375	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
82	81	ixpeq2dv 8855	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83		ovex 7392	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
84	83	rgenw 3059	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85		ixpexg 8864	. . . . . . . . 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 6962	. . . . . 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 3456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ V)
91	89, 90	fvmpt2d 6952	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
92	88, 91	sseq12d 3949	. . . . 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 259	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛))
94		1nn 12180	. . . . . 6 ⊢ 1 ∈ ℕ
95	94, 5	eleqtri 2839	. . . . 5 ⊢ 1 ∈ (ℤ≥‘1)
96	95	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ (ℤ≥‘1))
97		fveq2 6830	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
98	97	fveq1d 6832	. . . . . . . . 9 ⊢ (𝑛 = 1 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
99	98	oveq2d 7375	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
100	99	ixpeq2dv 8855	. . . . . . 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 7392	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103	102	rgenw 3059	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104		ixpexg 8864	. . . . . . . . 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 6962	. . . . . 6 ⊢ (𝜑 → (𝐷‘1) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
108	107	fveq2d 6834	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))))
109		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑘𝜑
110		simpl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝜑)
111	94	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 1 ∈ ℕ)
112		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
113	94	elexi 3455	. . . . . . . 8 ⊢ 1 ∈ V
114		eleq1 2829	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115	114	anbi2d 637	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116	115	anbi1d 638	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋)))
117	98	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝐶‘𝑛)‘𝑘) ∈ ℝ ↔ ((𝐶‘1)‘𝑘) ∈ ℝ))
118	116, 117	imbi12d 346	. . . . . . . 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 3504	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120	110, 111, 112, 119	syl21anc 844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121	109, 2, 29, 120	vonhoire 47127	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122	108, 121	eqeltrd 2841	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123		eqid 2741	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
124	1, 3, 4, 5, 26, 93, 96, 122, 123	meaiininc 46942	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
125	109, 29, 11	iinhoiicc 47129	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)))
126	33	oveq2d 7375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
127	126	ixpeq2dva 8854	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
128	91, 127	eqtrd 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
129	128	iineq2dv 4949	. . . . . . 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 2786	. . . . . 6 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
133	132	eqcomd 2747	. . . . 5 ⊢ (𝜑 → 𝐼 = ∩ 𝑛 ∈ ℕ (𝐷‘𝑛))
134	133	fveq2d 6834	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
135	134	eqcomd 2747	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
136	124, 135	breqtrd 5100	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137		2fveq3 6835	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
138	137	cbvmptv 5178	. . . 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 2750	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
143	2, 8, 10, 140, 141, 17, 25, 142	vonicclem1 47138	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	139, 143	eqbrtrd 5096	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
145		climuni 15509	. 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 591	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055  Vcvv 3433   ⊆ wss 3884  ∅c0 4263  ∩ ciin 4924   class class class wbr 5074   ↦ cmpt 5155  dom cdm 5620  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  Xcixp 8839  Fincfn 8887  ℝcr 11033  0cc0 11034  1c1 11035   + caddc 11037  ℝ^*cxr 11174   < clt 11175   ≤ cle 11176   − cmin 11373   / cdiv 11803  ℕcn 12169  ℤ_≥cuz 12783  [,)cico 13295  [,]cicc 13296   ⇝ cli 15441  ∏cprod 15863  volncvoln 46993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681  ax-inf2 9557  ax-cc 10353  ax-ac2 10381  ax-cnex 11090  ax-resscn 11091  ax-1cn 11092  ax-icn 11093  ax-addcl 11094  ax-addrcl 11095  ax-mulcl 11096  ax-mulrcl 11097  ax-mulcom 11098  ax-addass 11099  ax-mulass 11100  ax-distr 11101  ax-i2m1 11102  ax-1ne0 11103  ax-1rid 11104  ax-rnegex 11105  ax-rrecex 11106  ax-cnre 11107  ax-pre-lttri 11108  ax-pre-lttrn 11109  ax-pre-ltadd 11110  ax-pre-mulgt0 11111  ax-pre-sup 11112  ax-addf 11113

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-iin 4926  df-disj 5042  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623  df-om 7810  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-er 8637  df-map 8769  df-pm 8770  df-ixp 8840  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-fi 9318  df-sup 9349  df-inf 9350  df-oi 9419  df-dju 9820  df-card 9858  df-acn 9861  df-ac 10033  df-pnf 11177  df-mnf 11178  df-xr 11179  df-ltxr 11180  df-le 11181  df-sub 11375  df-neg 11376  df-div 11804  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ioo 13297  df-ico 13299  df-icc 13300  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-prod 15864  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-hom 17239  df-cco 17240  df-rest 17380  df-topn 17381  df-0g 17399  df-gsum 17400  df-topgen 17401  df-pt 17402  df-prds 17405  df-xrs 17461  df-qtop 17466  df-imas 17467  df-xps 17469  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18747  df-mulg 19039  df-cntz 19286  df-cmn 19751  df-psmet 21342  df-xmet 21343  df-met 21344  df-bl 21345  df-mopn 21346  df-cnfld 21351  df-top 22880  df-topon 22897  df-topsp 22919  df-bases 22932  df-cn 23213  df-cnp 23214  df-cmp 23373  df-tx 23548  df-hmeo 23741  df-xms 24306  df-ms 24307  df-tms 24308  df-cncf 24866  df-ovol 25452  df-vol 25453  df-salg 46764  df-sumge0 46818  df-mea 46905  df-ome 46945  df-caragen 46947  df-ovoln 46992  df-voln 46994

This theorem is referenced by:  vonicc  47140