Theorem vonicclem2 46640

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 1912	. . . 4 ⊢ Ⅎ𝑛𝜑
2		vonicclem2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
3	2	vonmea 46530	. . . 4 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
4		1zzd 12646	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
5		nnuz 12919	. . . 4 ⊢ ℕ = (ℤ≥‘1)
6	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin)
7		eqid 2735	. . . . . 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 7104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
12	11	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
13		nnrecre 12306	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
14	13	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
15	12, 14	readdcld 11288	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ ℝ)
16	15	fmpttd 7135	. . . . . . 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 7244	. . . . . . . . . 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 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))))
22	21	feq1d 6721	. . . . . . 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 46587	. . . . 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 7134	. . . 4 ⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋))
27		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
28		ressxr 11303	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
29	8	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
30	28, 29	sselid 3993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
31	30	adantlr 715	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ*)
32		ovexd 7466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / 𝑛)) ∈ V)
33	21, 32	fvmpt2d 7029	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐵‘𝑘) + (1 / 𝑛)))
34	33, 15	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ)
35	34	rexrd 11309	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ*)
36	9	ffvelcdmda 7104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
37	36	leidd 11827	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ≤ (𝐴‘𝑘))
38		1red 11260	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ)
39		nnre 12271	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
40	39, 38	readdcld 11288	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ)
41		peano2nn 12276	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
42		nnne0 12298	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ≠ 0)
43	41, 42	syl 17	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0)
44	38, 40, 43	redivcld 12093	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ∈ ℝ)
45	44	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ)
46	39	ltp1d 12196	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1))
47		nnrp 13044	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ+)
48	41	nnrpd 13073	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
49	47, 48	ltrecd 13093	. . . . . . . . . . . 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 11407	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
52	51	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛))
53	45, 14, 12, 52	leadd2dd 11876	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐵‘𝑘) + (1 / 𝑛)))
54		oveq2 7439	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚))
55	54	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((𝐵‘𝑘) + (1 / 𝑛)) = ((𝐵‘𝑘) + (1 / 𝑚)))
56	55	mpteq2dv 5250	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
57	56	cbvmptv 5261	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
58	17, 57	eqtri 2763	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / 𝑚))))
59		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1)))
60	59	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → ((𝐵‘𝑘) + (1 / 𝑚)) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
61	60	mpteq2dv 5250	. . . . . . . . . . 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 12281	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ)
64	6	mptexd 7244	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))) ∈ V)
65	58, 61, 63, 64	fvmptd3 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) + (1 / (𝑛 + 1)))))
66		ovexd 7466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) + (1 / (𝑛 + 1))) ∈ V)
67	65, 66	fvmpt2d 7029	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐵‘𝑘) + (1 / (𝑛 + 1))))
68	67, 33	breq12d 5161	. . . . . . . 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 13454	. . . . . . 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 45032	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
73		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚))
74	73	fveq1d 6909	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))
75	74	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
76	75	ixpeq2dv 8952	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
77	76	cbvmptv 5261	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
78	25, 77	eqtri 2763	. . . . . . 7 ⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)))
79		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1)))
80	79	fveq1d 6909	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘))
81	80	oveq2d 7447	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
82	81	ixpeq2dv 8952	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑚)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)))
83		ovex 7464	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
84	83	rgenw 3063	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘(𝑛 + 1))‘𝑘)) ∈ V
85		ixpexg 8961	. . . . . . . . 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 7039	. . . . . 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 3502	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) ∈ V)
91	89, 90	fvmpt2d 7029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)))
92	88, 91	sseq12d 4029	. . . . 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 12275	. . . . . 6 ⊢ 1 ∈ ℕ
95	94, 5	eleqtri 2837	. . . . 5 ⊢ 1 ∈ (ℤ≥‘1)
96	95	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ (ℤ≥‘1))
97		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (𝐶‘𝑛) = (𝐶‘1))
98	97	fveq1d 6909	. . . . . . . . 9 ⊢ (𝑛 = 1 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘1)‘𝑘))
99	98	oveq2d 7447	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
100	99	ixpeq2dv 8952	. . . . . . 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 7464	. . . . . . . . . 10 ⊢ ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
103	102	rgenw 3063	. . . . . . . . 9 ⊢ ∀𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)) ∈ V
104		ixpexg 8961	. . . . . . . . 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 7039	. . . . . 6 ⊢ (𝜑 → (𝐷‘1) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘)))
108	107	fveq2d 6911	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))))
109		nfv 1912	. . . . . 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 3501	. . . . . . . 8 ⊢ 1 ∈ V
114		eleq1 2827	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝑛 ∈ ℕ ↔ 1 ∈ ℕ))
115	114	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑛 = 1 → ((𝜑 ∧ 𝑛 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ)))
116	115	anbi1d 631	. . . . . . . . 9 ⊢ (𝑛 = 1 → (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 1 ∈ ℕ) ∧ 𝑘 ∈ 𝑋)))
117	98	eleq1d 2824	. . . . . . . . 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 3558	. . . . . . 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 46628	. . . . 5 ⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘1)‘𝑘))) ∈ ℝ)
122	108, 121	eqeltrd 2839	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘(𝐷‘1)) ∈ ℝ)
123		eqid 2735	. . . 4 ⊢ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
124	1, 3, 4, 5, 26, 93, 96, 122, 123	meaiininc 46443	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
125	109, 29, 11	iinhoiicc 46630	. . . . . . 7 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,](𝐵‘𝑘)))
126	33	oveq2d 7447	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
127	126	ixpeq2dva 8951	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐶‘𝑛)‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
128	91, 127	eqtrd 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)((𝐵‘𝑘) + (1 / 𝑛))))
129	128	iineq2dv 5022	. . . . . . 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 2785	. . . . . 6 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼)
133	132	eqcomd 2741	. . . . 5 ⊢ (𝜑 → 𝐼 = ∩ 𝑛 ∈ ℕ (𝐷‘𝑛))
134	133	fveq2d 6911	. . . 4 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)))
135	134	eqcomd 2741	. . 3 ⊢ (𝜑 → ((voln‘𝑋)‘∩ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼))
136	124, 135	breqtrd 5174	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼))
137		2fveq3 6912	. . . . 5 ⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚)))
138	137	cbvmptv 5261	. . . 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 2744	. . . 4 ⊢ (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛)))
143	2, 8, 10, 140, 141, 17, 25, 142	vonicclem1 46639	. . 3 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
144	139, 143	eqbrtrd 5170	. 2 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
145		climuni 15585	. 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 584	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  ∩ ciin 4997   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Xcixp 8936  Fincfn 8984  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   − cmin 11490   / cdiv 11918  ℕcn 12264  ℤ_≥cuz 12876  [,)cico 13386  [,]cicc 13387   ⇝ cli 15517  ∏cprod 15936  volncvoln 46494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-sum 15720  df-prod 15937  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-ovol 25513  df-vol 25514  df-salg 46265  df-sumge0 46319  df-mea 46406  df-ome 46446  df-caragen 46448  df-ovoln 46493  df-voln 46495

This theorem is referenced by:  vonicc  46641