vonioo - Mathbox for Glauco Siliprandi

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Theorem vonioo 47125

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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
4		feq2 6634	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
5	4	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
6	3, 5	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
7		vonioo.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
9		feq2 6634	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
11	8, 10	mpbid 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
12	1, 6, 11	hoidmv0val 47026	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
13	12	eqcomd 2745	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = (𝐴(𝐿‘∅)𝐵))
14		fveq2 6827	. . . . . 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 8846	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
18	16, 17	eqtrd 2774	. . . . . 6 ⊢ (𝑋 = ∅ → 𝐼 = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
19	14, 18	fveq12d 6834	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
20	19	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
21		0fi 8979	. . . . . . 7 ⊢ ∅ ∈ Fin
22	21	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
23		eqid 2739	. . . . . 6 ⊢ dom (voln‘∅) = dom (voln‘∅)
24		ressxr 11180	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ℝ ⊆ ℝ^*)
26	6, 25	fssd 6672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ^*)
27	11, 25	fssd 6672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ^*)
28	22, 23, 26, 27	ioovonmbl 47120	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘∅))
29	28	von0val 47114	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) = 0)
30	20, 29	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = 0)
31		fveq2 6827	. . . . 5 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
32	31	oveqd 7373	. . . 4 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
33	32	adantl 482	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
34	13, 30, 33	3eqtr4d 2784	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
35		neqne 2942	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
36	35	adantl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
37		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑋 ≠ ∅)
38		nfra1 3263	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)
39	37, 38	nfan 1906	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
40	2	ffvelcdmda 7025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
41	7	ffvelcdmda 7025	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
42		volico 46426	. . . . . . . . . . . 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 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
44	43	ad4ant14 758	. . . . . . . . . 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 3228	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
46	45	iftrued 4462	. . . . . . . . . . 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 720	. . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
49	48	ex 413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝑘 ∈ 𝑋 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘))))
50	39, 49	ralrimi 3237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
51	50	prodeq2d 15877	. . . . . 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 2745	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
53		fveq2 6827	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
54		fveq2 6827	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
55	53, 54	breq12d 5085	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑗) < (𝐵‘𝑗)))
56	55	cbvralvw 3217	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
57	56	bilani 505	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
58		vonioo.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ Fin)
59	58	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
60	59	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ∈ Fin)
61	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ)
62	61	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐴:𝑋⟶ℝ)
63	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐵:𝑋⟶ℝ)
64	63	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐵:𝑋⟶ℝ)
65		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
66	65	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ≠ ∅)
67	56, 45	sylanbr 588	. . . . . . . 8 ⊢ ((∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
68	67	adantll 720	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
69		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
70	69	oveq1d 7371	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐴‘𝑗) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑚)))
71	70	cbvmptv 5176	. . . . . . . . . 10 ⊢ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))
72	71	a1i 11	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))
73		oveq2 7364	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
74	73	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑛)))
75	74	mpteq2dv 5166	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
76	72, 75	eqtrd 2774	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
77	76	cbvmptv 5176	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚)))) = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
78		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑛X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘))
79		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑋
80		nffvmpt1 6838	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)
81		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑘
82	80, 81	nffv 6837	. . . . . . . . . 10 ⊢ Ⅎ𝑚(((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)
83		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑚[,)
84		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐵‘𝑘)
85	82, 83, 84	nfov 7386	. . . . . . . . 9 ⊢ Ⅎ𝑚((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
86	79, 85	nfixpw 8854	. . . . . . . 8 ⊢ Ⅎ𝑚X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
87		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚) = ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛))
88	87	fveq1d 6829	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘) = (((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘))
89	88	oveq1d 7371	. . . . . . . . 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)))
90	89	ixpeq2dv 8851	. . . . . . . 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)))
91	78, 86, 90	cbvmpt 5174	. . . . . . 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)))
92	60, 62, 64, 66, 68, 15, 77, 91	vonioolem2 47124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
93	57, 92	syldan 597	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
94	1, 59, 65, 61, 63	hoidmvn0val 47027	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
95	94	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
96	52, 93, 95	3eqtr4d 2784	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
97		rexnal 3091	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
98	97	bilanri 507	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
99		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
100	41	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ∈ ℝ)
101	40	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐴‘𝑘) ∈ ℝ)
102	100, 101	lenltd 11283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) ↔ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)))
103	99, 102	mpbird 258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
104	103	ex 413	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (¬ (𝐴‘𝑘) < (𝐵‘𝑘) → (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
105	104	reximdva 3152	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
106	105	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
107	98, 106	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
108	107	adantlr 721	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
109		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑘(voln‘𝑋)
110		nfixp1 8856	. . . . . . . . . 10 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
111	15, 110	nfcxfr 2899	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐼
112	109, 111	nffv 6837	. . . . . . . 8 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼)
113		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐴(𝐿‘𝑋)𝐵)
114	112, 113	nfeq 2914	. . . . . . 7 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)
115	58	vonmea 47017	. . . . . . . . . . . 12 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
116	115	mea0 46897	. . . . . . . . . . 11 ⊢ (𝜑 → ((voln‘𝑋)‘∅) = 0)
117	116	3ad2ant1 1139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘∅) = 0)
118	15	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
119		simp2 1143	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝑘 ∈ 𝑋)
120		simp3 1144	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
121	24, 40	sselid 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
122	121	3adant3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴‘𝑘) ∈ ℝ^*)
123	24, 41	sselid 3913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
124	123	3adant3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ∈ ℝ^*)
125		ioo0 13314	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
126	122, 124, 125	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
127	120, 126	mpbird 258	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
128		rspe 3229	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
129	119, 127, 128	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
130		ixp0 8869	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
131	129, 130	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
132	118, 131	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = ∅)
133	132	fveq2d 6831	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∅))
134		ne0i 4269	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑋 → 𝑋 ≠ ∅)
135	134	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 ≠ ∅)
136	135, 94	syldan 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
137	136	3adant3 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
138		eleq1w 2822	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑋 ↔ 𝑘 ∈ 𝑋))
139		fveq2 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
140	139, 69	breq12d 5085	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
141	138, 140	3anbi23d 1447	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ↔ (𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘))))
142	141	imbi1d 342	. . . . . . . . . . . 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)))
143		nfv 1921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗))
144	58	3ad2ant1 1139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑋 ∈ Fin)
145		volicore 47024	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
146	40, 41, 145	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
147	146	recnd 11164	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
148	147	3ad2antl1 1192	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
149		simp2 1143	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑗 ∈ 𝑋)
150	53, 54	oveq12d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑗)[,)(𝐵‘𝑗)))
151	150	fveq2d 6831	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
152	151	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
153	2	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
154	7	ffvelcdmda 7025	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
155		volico 46426	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴‘𝑗) ∈ ℝ ∧ (𝐵‘𝑗) ∈ ℝ) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
156	153, 154, 155	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
157	156	3adant3 1138	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
158		simp3 1144	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (𝐵‘𝑗) ≤ (𝐴‘𝑗))
159	154, 153	lenltd 11283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
160	159	3adant3 1138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
161	158, 160	mpbid 233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ¬ (𝐴‘𝑗) < (𝐵‘𝑗))
162	161	iffalsed 4465	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0) = 0)
163	157, 162	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
164	163	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
165	152, 164	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
166	143, 144, 148, 149, 165	fprodeq0g 15950	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
167	142, 166	chvarvv 1996	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
168	137, 167	eqtrd 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = 0)
169	117, 133, 168	3eqtr4d 2784	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
170	169	3exp 1125	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
171	170	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
172	37, 114, 171	rexlimd 3246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)))
173	172	imp 407	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
174	108, 173	syldan 597	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
175	96, 174	pm2.61dan 818	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
176	36, 175	syldan 597	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
177	34, 176	pm2.61dan 818	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 ⊆ wss 3883 ∅c0 4261 ifcif 4454 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 ↑_m cmap 8763 Xcixp 8835 Fincfn 8883 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 (,)cioo 13289 [,)cico 13291 ∏cprod 15859 volcvol 25448 volncvoln 46981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-pws 17403 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-drng 20703 df-field 20704 df-abv 20781 df-staf 20811 df-srng 20812 df-lmod 20852 df-lss 20922 df-lmhm 21012 df-lvec 21093 df-sra 21163 df-rgmod 21164 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-refld 21580 df-phl 21601 df-dsmm 21707 df-frlm 21722 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cn 23210 df-cnp 23211 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-xms 24303 df-ms 24304 df-tms 24305 df-nm 24565 df-ngp 24566 df-tng 24567 df-nrg 24568 df-nlm 24569 df-cncf 24863 df-clm 25048 df-cph 25153 df-tcph 25154 df-rrx 25370 df-ovol 25449 df-vol 25450 df-salg 46752 df-sumge0 46806 df-mea 46893 df-ome 46933 df-caragen 46935 df-ovoln 46980 df-voln 46982

This theorem is referenced by: vonn0ioo 47130

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