Theorem vonicclem2 47255

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 1934	. . . 4 ⊢ Ⅎ𝑛𝜑
2		vonicclem2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
3	2	vonmea 47145	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
4		1zzd 12602	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
5		nnuz 12878	. . . 4 ⊢ ℕ = (ℤ≥‘1)
6	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7		eqid 2762	. . . . . 6 ⊢ dom (voln‘𝑋) = dom (voln‘𝑋)
8		vonicclem2.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
9	8	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ)
10		vonicclem2.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
11	10	ffvelcdmda 7065	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
12	11	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
13		nnrecre 12255	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
14	13	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
15	12, 14	readdcld 11211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ ℝ)
16	15	fmpttd 7096	. . . . . . 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 7208	. . . . . . . . . 10 ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
20	19	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) ∈ V)
21	18, 20	fvmpt2d 6989	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
22	21	feq1d 6673	. . . . . . 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 47202	. . . . 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 7095	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
27		nfv 1934	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
28		ressxr 11226	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
29	8	ffvelcdmda 7065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
30	28, 29	sselid 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
31	30	adantlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
32		ovexd 7431	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ V)
33	21, 32	fvmpt2d 6989	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐵‘𝑘) + (1 / 𝑛)))
34	33, 15	eqeltrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
35	34	rexrd 11232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ*)
36	9	ffvelcdmda 7065	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
37	36	leidd 11753	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38		1red 11182	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnre 12217	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
40	39, 38	readdcld 11211	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41		peano2nn 12222	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42		nnne0 12247	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
44	38, 40, 43	redivcld 12019	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
45	44	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
46	39	ltp1d 12122	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47		nnrp 13005	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
48	41	nnrpd 13035	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
49	47, 48	ltrecd 13055	. . . . . . . . . . . 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 11331	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
52	51	ad2antlr 737	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
53	45, 14, 12, 52	leadd2dd 11802	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛)))
54		oveq2 7404	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
55	54	oveq2d 7412	. . . . . . . . . . . . . 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 2785	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
59		oveq2 7404	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
60	59	oveq2d 7412	. . . . . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
63	62	peano2nnd 12227	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
64	6	mptexd 7208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
65	58, 61, 63, 64	fvmptd3 6999	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
66		ovexd 7431	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
67	65, 66	fvmpt2d 6989	. . . . . . . . 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 259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘))
70		icossico 13420	. . . . . . 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 849	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
72	27, 71	ixpssixp 45667	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
73		fveq2 6867	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
74	73	fveq1d 6869	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
75	74	oveq2d 7412	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
76	75	ixpeq2dv 8895	. . . . . . . . 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 2785	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
79		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
80	79	fveq1d 6869	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
81	80	oveq2d 7412	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
82	81	ixpeq2dv 8895	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83		ovex 7429	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
84	83	rgenw 3080	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85		ixpexg 8904	. . . . . . . . 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 6999	. . . . . 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 3477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ V)
91	89, 90	fvmpt2d 6989	. . . . . 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 259	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) ⊆ (𝐷‘𝑛))
94		1nn 12221	. . . . . 6 ⊢ 1 ∈ ℕ
95	94, 5	eleqtri 2860	. . . . 5 ⊢ 1 ∈ (ℤ≥‘1)
96	95	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ (ℤ≥‘1))
97		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
98	97	fveq1d 6869	. . . . . . . . 9 ⊢ (𝑛 = 1 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
99	98	oveq2d 7412	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
100	99	ixpeq2dv 8895	. . . . . . 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 7429	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103	102	rgenw 3080	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104		ixpexg 8904	. . . . . . . . 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 6999	. . . . . 6 ⊢ (𝜑 → (𝐷‘1) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
108	107	fveq2d 6871	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))))
109		nfv 1934	. . . . . 6 ⊢ Ⅎ𝑘𝜑
110		simpl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝜑)
111	94	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 1 ∈ ℕ)
112		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
113	94	elexi 3476	. . . . . . . 8 ⊢ 1 ∈ V
114		eleq1 2850	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115	114	anbi2d 639	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116	115	anbi1d 640	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋)))
117	98	eleq1d 2847	. . . . . . . . 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 3525	. . . . . . 7 ⊢ (((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
120	110, 111, 112, 119	syl21anc 848	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐶‘1)‘𝑘) ∈ ℝ)
121	109, 2, 29, 120	vonhoire 47243	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122	108, 121	eqeltrd 2862	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123		eqid 2762	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
124	1, 3, 4, 5, 26, 93, 96, 122, 123	meaiininc 47058	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
125	109, 29, 11	iinhoiicc 47245	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)))
126	33	oveq2d 7412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
127	126	ixpeq2dva 8894	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
128	91, 127	eqtrd 2797	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
129	128	iineq2dv 4975	. . . . . . 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 2807	. . . . . 6 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
133	132	eqcomd 2768	. . . . 5 ⊢ (𝜑 → 𝐼 = ∩ 𝑛 ∈ ℕ (𝐷‘𝑛))
134	133	fveq2d 6871	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
135	134	eqcomd 2768	. . 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 6872	. . . . 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 2771	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
143	2, 8, 10, 140, 141, 17, 25, 142	vonicclem1 47254	. . 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 15579	. 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 593	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  ∩ ciin 4950   class class class wbr 5100   ↦ cmpt 5181  dom cdm 5647  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396  Xcixp 8879  Fincfn 8927  ℝcr 11072  0cc0 11073  1c1 11074   + caddc 11076  ℝ^*cxr 11215   < clt 11216   ≤ cle 11217   − cmin 11414   / cdiv 11844  ℕcn 12210  ℤ_≥cuz 12839  [,)cico 13351  [,]cicc 13352   ⇝ cli 15511  ∏cprod 15933  volncvoln 47109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-cc 10392  ax-ac2 10420  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151  ax-addf 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-fi 9357  df-sup 9388  df-inf 9389  df-oi 9458  df-dju 9859  df-card 9897  df-acn 9900  df-ac 10072  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-q 12950  df-rp 12994  df-xneg 13114  df-xadd 13115  df-xmul 13116  df-ioo 13353  df-ico 13355  df-icc 13356  df-fz 13513  df-fzo 13660  df-fl 13802  df-seq 14015  df-exp 14075  df-hash 14344  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-clim 15515  df-rlim 15516  df-sum 15714  df-prod 15934  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-starv 17301  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-unif 17309  df-hom 17310  df-cco 17311  df-rest 17451  df-topn 17452  df-0g 17470  df-gsum 17471  df-topgen 17472  df-pt 17473  df-prds 17476  df-xrs 17532  df-qtop 17537  df-imas 17538  df-xps 17540  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-submnd 18818  df-mulg 19110  df-cntz 19357  df-cmn 19822  df-psmet 21413  df-xmet 21414  df-met 21415  df-bl 21416  df-mopn 21417  df-cnfld 21422  df-top 22951  df-topon 22968  df-topsp 22990  df-bases 23003  df-cn 23284  df-cnp 23285  df-cmp 23444  df-tx 23619  df-hmeo 23812  df-xms 24377  df-ms 24378  df-tms 24379  df-cncf 24937  df-ovol 25523  df-vol 25524  df-salg 46880  df-sumge0 46934  df-mea 47021  df-ome 47061  df-caragen 47063  df-ovoln 47108  df-voln 47110

This theorem is referenced by:  vonicc  47256