vonioo - Mathbox for Glauco Siliprandi

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Theorem vonioo 46687

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

**Hypotheses**
Ref	Expression
vonioo.x	⊢ (𝜑 → 𝑋 ∈ Fin)
vonioo.a	⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
vonioo.b	⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
vonioo.i	⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
vonioo.l	⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))

**Assertion**
Ref	Expression
vonioo	⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Distinct variable groups: 𝐴,𝑎,𝑏,𝑘 𝐵,𝑎,𝑏,𝑘 𝑘,𝐿 𝑋,𝑎,𝑏,𝑘,𝑥 𝜑,𝑎,𝑏,𝑘,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐼(𝑥,𝑘,𝑎,𝑏) 𝐿(𝑥,𝑎,𝑏)

Proof of Theorem vonioo Dummy variables 𝑗 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vonioo.l	. . . . 5 ⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑_m 𝑥), 𝑏 ∈ (ℝ ↑_m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))))
2		vonioo.a	. . . . . . 7 ⊢ (𝜑 → 𝐴:𝑋⟶ℝ)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
4		feq2 6670	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
5	4	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
6	3, 5	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
7		vonioo.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
9		feq2 6670	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
11	8, 10	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
12	1, 6, 11	hoidmv0val 46588	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
13	12	eqcomd 2736	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = (𝐴(𝐿‘∅)𝐵))
14		fveq2 6861	. . . . . 6 ⊢ (𝑋 = ∅ → (voln‘𝑋) = (voln‘∅))
15		vonioo.i	. . . . . . . 8 ⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
16	15	a1i 11	. . . . . . 7 ⊢ (𝑋 = ∅ → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
17		ixpeq1 8884	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
18	16, 17	eqtrd 2765	. . . . . 6 ⊢ (𝑋 = ∅ → 𝐼 = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
19	14, 18	fveq12d 6868	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
20	19	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
21		0fi 9016	. . . . . . 7 ⊢ ∅ ∈ Fin
22	21	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
23		eqid 2730	. . . . . 6 ⊢ dom (voln‘∅) = dom (voln‘∅)
24		ressxr 11225	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ℝ ⊆ ℝ^*)
26	6, 25	fssd 6708	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ^*)
27	11, 25	fssd 6708	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ^*)
28	22, 23, 26, 27	ioovonmbl 46682	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘∅))
29	28	von0val 46676	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) = 0)
30	20, 29	eqtrd 2765	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = 0)
31		fveq2 6861	. . . . 5 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
32	31	oveqd 7407	. . . 4 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
33	32	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
34	13, 30, 33	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
35		neqne 2934	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
36	35	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
37		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑋 ≠ ∅)
38		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)
39	37, 38	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
40	2	ffvelcdmda 7059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
41	7	ffvelcdmda 7059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
42		volico 45988	. . . . . . . . . . . 12 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
43	40, 41, 42	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
44	43	ad4ant14 752	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
45		rspa 3227	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
46	45	iftrued 4499	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑋) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
47	46	adantll 714	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑋) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
48	44, 47	eqtrd 2765	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
49	48	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝑘 ∈ 𝑋 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘))))
50	39, 49	ralrimi 3236	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
51	50	prodeq2d 15894	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
52	51	eqcomd 2736	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
53		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
54		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
55	53, 54	breq12d 5123	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑗) < (𝐵‘𝑗)))
56	55	cbvralvw 3216	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
57	56	biimpi 216	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) → ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
58	57	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
59		vonioo.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ Fin)
60	59	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
61	60	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ∈ Fin)
62	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ)
63	62	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐴:𝑋⟶ℝ)
64	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐵:𝑋⟶ℝ)
65	64	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐵:𝑋⟶ℝ)
66		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
67	66	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ≠ ∅)
68	56, 45	sylanbr 582	. . . . . . . 8 ⊢ ((∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
69	68	adantll 714	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
70		fveq2 6861	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
71	70	oveq1d 7405	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐴‘𝑗) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑚)))
72	71	cbvmptv 5214	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))
73	72	a1i 11	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
74		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
75	74	oveq2d 7406	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑛)))
76	75	mpteq2dv 5204	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
77	73, 76	eqtrd 2765	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
78	77	cbvmptv 5214	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚)))) = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
79		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑛X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘))
80		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑋
81		nffvmpt1 6872	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)
82		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑘
83	81, 82	nffv 6871	. . . . . . . . . 10 ⊢ Ⅎ𝑚(((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)
84		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑚[,)
85		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐵‘𝑘)
86	83, 84, 85	nfov 7420	. . . . . . . . 9 ⊢ Ⅎ𝑚((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
87	80, 86	nfixpw 8892	. . . . . . . 8 ⊢ Ⅎ𝑚X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
88		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚) = ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛))
89	88	fveq1d 6863	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘) = (((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘))
90	89	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
91	90	ixpeq2dv 8889	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
92	79, 87, 91	cbvmpt 5212	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) = (𝑛 ∈ ℕ ↦ X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))
93	61, 63, 65, 67, 69, 15, 78, 92	vonioolem2 46686	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
94	58, 93	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
95	1, 60, 66, 62, 64	hoidmvn0val 46589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
96	95	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
97	52, 94, 96	3eqtr4d 2775	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
98		rexnal 3083	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
99	98	bicomi 224	. . . . . . . . 9 ⊢ (¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
100	99	biimpi 216	. . . . . . . 8 ⊢ (¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
101	100	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
102		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
103	41	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ∈ ℝ)
104	40	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐴‘𝑘) ∈ ℝ)
105	103, 104	lenltd 11327	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) ↔ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)))
106	102, 105	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
107	106	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (¬ (𝐴‘𝑘) < (𝐵‘𝑘) → (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
108	107	reximdva 3147	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
109	108	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
110	101, 109	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
111	110	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
112		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑘(voln‘𝑋)
113		nfixp1 8894	. . . . . . . . . 10 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
114	15, 113	nfcxfr 2890	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐼
115	112, 114	nffv 6871	. . . . . . . 8 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼)
116		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐴(𝐿‘𝑋)𝐵)
117	115, 116	nfeq 2906	. . . . . . 7 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)
118	59	vonmea 46579	. . . . . . . . . . . 12 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
119	118	mea0 46459	. . . . . . . . . . 11 ⊢ (𝜑 → ((voln‘𝑋)‘∅) = 0)
120	119	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘∅) = 0)
121	15	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
122		simp2 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝑘 ∈ 𝑋)
123		simp3 1138	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
124	24, 40	sselid 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
125	124	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴‘𝑘) ∈ ℝ^*)
126	24, 41	sselid 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
127	126	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ∈ ℝ^*)
128		ioo0 13338	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
129	125, 127, 128	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
130	123, 129	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
131		rspe 3228	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
132	122, 130, 131	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
133		ixp0 8907	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
134	132, 133	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
135	121, 134	eqtrd 2765	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = ∅)
136	135	fveq2d 6865	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∅))
137		ne0i 4307	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑋 → 𝑋 ≠ ∅)
138	137	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 ≠ ∅)
139	138, 95	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
140	139	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
141		eleq1w 2812	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑋 ↔ 𝑘 ∈ 𝑋))
142		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
143	142, 70	breq12d 5123	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
144	141, 143	3anbi23d 1441	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ↔ (𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘))))
145	144	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0) ↔ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)))
146		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗))
147	59	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑋 ∈ Fin)
148		volicore 46586	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
149	40, 41, 148	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
150	149	recnd 11209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
151	150	3ad2antl1 1186	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
152		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑗 ∈ 𝑋)
153	53, 54	oveq12d 7408	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑗)[,)(𝐵‘𝑗)))
154	153	fveq2d 6865	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
155	154	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
156	2	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
157	7	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
158		volico 45988	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑗) ∈ ℝ ∧ (𝐵‘𝑗) ∈ ℝ) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
159	156, 157, 158	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
160	159	3adant3 1132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
161		simp3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (𝐵‘𝑗) ≤ (𝐴‘𝑗))
162	157, 156	lenltd 11327	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
163	162	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
164	161, 163	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ¬ (𝐴‘𝑗) < (𝐵‘𝑗))
165	164	iffalsed 4502	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0) = 0)
166	160, 165	eqtrd 2765	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
167	166	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
168	155, 167	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
169	146, 147, 151, 152, 168	fprodeq0g 15967	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
170	145, 169	chvarvv 1989	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
171	140, 170	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = 0)
172	120, 136, 171	3eqtr4d 2775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
173	172	3exp 1119	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
174	173	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
175	37, 117, 174	rexlimd 3245	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)))
176	175	imp 406	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
177	111, 176	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
178	97, 177	pm2.61dan 812	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
179	36, 178	syldan 591	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
180	34, 179	pm2.61dan 812	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 ⊆ wss 3917 ∅c0 4299 ifcif 4491 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ∈ cmpo 7392 ↑_m cmap 8802 Xcixp 8873 Fincfn 8921 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 / cdiv 11842 ℕcn 12193 (,)cioo 13313 [,)cico 13315 ∏cprod 15876 volcvol 25371 volncvoln 46543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-ac2 10423 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-acn 9902 df-ac 10076 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-prod 15877 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-pws 17419 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-field 20648 df-abv 20725 df-staf 20755 df-srng 20756 df-lmod 20775 df-lss 20845 df-lmhm 20936 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-refld 21521 df-phl 21542 df-dsmm 21648 df-frlm 21663 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cn 23121 df-cnp 23122 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-nm 24477 df-ngp 24478 df-tng 24479 df-nrg 24480 df-nlm 24481 df-cncf 24778 df-clm 24970 df-cph 25075 df-tcph 25076 df-rrx 25292 df-ovol 25372 df-vol 25373 df-salg 46314 df-sumge0 46368 df-mea 46455 df-ome 46495 df-caragen 46497 df-ovoln 46542 df-voln 46544

This theorem is referenced by: vonn0ioo 46692

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