vonioo - Mathbox for Glauco Siliprandi

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Theorem vonioo 47287

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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ)
4		feq2 6685	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
5	4	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ))
6	3, 5	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ)
7		vonioo.b	. . . . . . 7 ⊢ (𝜑 → 𝐵:𝑋⟶ℝ)
8	7	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ)
9		feq2 6685	. . . . . . 7 ⊢ (𝑋 = ∅ → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
10	9	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ))
11	8, 10	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ)
12	1, 6, 11	hoidmv0val 47188	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
13	12	eqcomd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = (𝐴(𝐿‘∅)𝐵))
14		fveq2 6882	. . . . . 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 8905	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
18	16, 17	eqtrd 2804	. . . . . 6 ⊢ (𝑋 = ∅ → 𝐼 = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
19	14, 18	fveq12d 6889	. . . . 5 ⊢ (𝑋 = ∅ → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
20	19	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))))
21		0fi 9038	. . . . . . 7 ⊢ ∅ ∈ Fin
22	21	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
23		eqid 2769	. . . . . 6 ⊢ dom (voln‘∅) = dom (voln‘∅)
24		ressxr 11252	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ℝ ⊆ ℝ^*)
26	6, 25	fssd 6724	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ^*)
27	11, 25	fssd 6724	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ^*)
28	22, 23, 26, 27	ioovonmbl 47282	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘∅))
29	28	von0val 47276	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) = 0)
30	20, 29	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = 0)
31		fveq2 6882	. . . . 5 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
32	31	oveqd 7428	. . . 4 ⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
33	32	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵))
34	13, 30, 33	3eqtr4d 2814	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
35		neqne 2972	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
36	35	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
37		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑋 ≠ ∅)
38		nfra1 3295	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)
39	37, 38	nfan 1926	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
40	2	ffvelcdmda 7080	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
41	7	ffvelcdmda 7080	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
42		volico 46588	. . . . . . . . . . . 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 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0))
44	43	ad4ant14 764	. . . . . . . . . 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 3260	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
46	45	iftrued 4500	. . . . . . . . . . 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 726	. . . . . . . . . 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 2804	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
49	48	ex 417	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝑘 ∈ 𝑋 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘))))
50	39, 49	ralrimi 3269	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
51	50	prodeq2d 15974	. . . . . 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 2775	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
53		fveq2 6882	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
54		fveq2 6882	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
55	53, 54	breq12d 5126	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑗) < (𝐵‘𝑗)))
56	55	cbvralvw 3249	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗))
57	56	bilani 509	. . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ∈ Fin)
60	59	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ∈ Fin)
61	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐴:𝑋⟶ℝ)
62	61	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐴:𝑋⟶ℝ)
63	7	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝐵:𝑋⟶ℝ)
64	63	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝐵:𝑋⟶ℝ)
65		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅)
66	65	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → 𝑋 ≠ ∅)
67	56, 45	sylanbr 593	. . . . . . . 8 ⊢ ((∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
68	67	adantll 726	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
69		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
70	69	oveq1d 7426	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐴‘𝑗) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑚)))
71	70	cbvmptv 5219	. . . . . . . . . 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 7419	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
74	73	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑛)))
75	74	mpteq2dv 5209	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
76	72, 75	eqtrd 2804	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
77	76	cbvmptv 5219	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚)))) = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
78		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑛X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘))
79		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑋
80		nffvmpt1 6893	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)
81		nfcv 2931	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑘
82	80, 81	nffv 6892	. . . . . . . . . 10 ⊢ Ⅎ𝑚(((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)
83		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑚[,)
84		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐵‘𝑘)
85	82, 83, 84	nfov 7441	. . . . . . . . 9 ⊢ Ⅎ𝑚((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
86	79, 85	nfixpw 8913	. . . . . . . 8 ⊢ Ⅎ𝑚X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
87		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚) = ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛))
88	87	fveq1d 6884	. . . . . . . . . 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 7426	. . . . . . . . 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 8910	. . . . . . . 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 5217	. . . . . . 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 47286	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑗 ∈ 𝑋 (𝐴‘𝑗) < (𝐵‘𝑗)) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))
93	57, 92	syldan 602	. . . . 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 47189	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
95	94	adantr 485	. . . . 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 2814	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
97		rexnal 3123	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) ↔ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
98	97	bilanri 511	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
99		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ¬ (𝐴‘𝑘) < (𝐵‘𝑘))
100	41	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ∈ ℝ)
101	40	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐴‘𝑘) ∈ ℝ)
102	100, 101	lenltd 11355	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) ↔ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)))
103	99, 102	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ¬ (𝐴‘𝑘) < (𝐵‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
104	103	ex 417	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (¬ (𝐴‘𝑘) < (𝐵‘𝑘) → (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
105	104	reximdva 3184	. . . . . . . 8 ⊢ (𝜑 → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
106	105	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → (∃𝑘 ∈ 𝑋 ¬ (𝐴‘𝑘) < (𝐵‘𝑘) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
107	98, 106	mpd 16	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
108	107	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘))
109		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑘(voln‘𝑋)
110		nfixp1 8915	. . . . . . . . . 10 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
111	15, 110	nfcxfr 2929	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐼
112	109, 111	nffv 6892	. . . . . . . 8 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼)
113		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐴(𝐿‘𝑋)𝐵)
114	112, 113	nfeq 2944	. . . . . . 7 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)
115	58	vonmea 47179	. . . . . . . . . . . 12 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
116	115	mea0 47059	. . . . . . . . . . 11 ⊢ (𝜑 → ((voln‘𝑋)‘∅) = 0)
117	116	3ad2ant1 1149	. . . . . . . . . 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 1153	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝑘 ∈ 𝑋)
120		simp3 1154	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
121	24, 40	sselid 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
122	121	3adant3 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴‘𝑘) ∈ ℝ^*)
123	24, 41	sselid 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
124	123	3adant3 1148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ∈ ℝ^*)
125		ioo0 13396	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ^* ∧ (𝐵‘𝑘) ∈ ℝ^*) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
126	122, 124, 125	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
127	120, 126	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
128		rspe 3261	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
129	119, 127, 128	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
130		ixp0 8928	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
131	129, 130	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = ∅)
132	118, 131	eqtrd 2804	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = ∅)
133	132	fveq2d 6886	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∅))
134		ne0i 4302	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑋 → 𝑋 ≠ ∅)
135	134	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑋 ≠ ∅)
136	135, 94	syldan 602	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
137	136	3adant3 1148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
138		eleq1w 2852	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑋 ↔ 𝑘 ∈ 𝑋))
139		fveq2 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
140	139, 69	breq12d 5126	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
141	138, 140	3anbi23d 1465	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ↔ (𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘))))
142	141	imbi1d 344	. . . . . . . . . . . 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 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗))
144	58	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑋 ∈ Fin)
145		volicore 47186	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
146	40, 41, 145	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ)
147	146	recnd 11236	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
148	147	3ad2antl1 1202	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
149		simp2 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑗 ∈ 𝑋)
150	53, 54	oveq12d 7429	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑗)[,)(𝐵‘𝑗)))
151	150	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑗 → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
152	151	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))))
153	2	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
154	7	ffvelcdmda 7080	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
155		volico 46588	. . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0))
157	156	3adant3 1148	. . . . . . . . . . . . . . . 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 1154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (𝐵‘𝑗) ≤ (𝐴‘𝑗))
159	154, 153	lenltd 11355	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
160	159	3adant3 1148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
161	158, 160	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ¬ (𝐴‘𝑗) < (𝐵‘𝑗))
162	161	iffalsed 4503	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0) = 0)
163	157, 162	eqtrd 2804	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
164	163	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑗)[,)(𝐵‘𝑗))) = 0)
165	152, 164	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
166	143, 144, 148, 149, 165	fprodeq0g 16047	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
167	142, 166	chvarvv 2016	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
168	137, 167	eqtrd 2804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = 0)
169	117, 133, 168	3eqtr4d 2814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
170	169	3exp 1135	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
171	170	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))))
172	37, 114, 171	rexlimd 3278	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → (∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)))
173	172	imp 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∃𝑘 ∈ 𝑋 (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
174	108, 173	syldan 602	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ¬ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
175	96, 174	pm2.61dan 824	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
176	36, 175	syldan 602	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
177	34, 176	pm2.61dan 824	1 ⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 ⊆ wss 3913 ∅c0 4294 ifcif 4492 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ↑_m cmap 8823 Xcixp 8894 Fincfn 8942 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 ℝ^*cxr 11241 < clt 11242 ≤ cle 11243 − cmin 11440 / cdiv 11870 ℕcn 12232 (,)cioo 13371 [,)cico 13373 ∏cprod 15956 volcvol 25590 volncvoln 47143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cc 10418 ax-ac2 10446 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-omul 8457 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-acn 9927 df-ac 10099 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-rlim 15539 df-sum 15737 df-prod 15957 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-pws 17501 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-rhm 20553 df-subrng 20630 df-subrg 20654 df-drng 20814 df-field 20815 df-abv 20889 df-staf 20919 df-srng 20920 df-lmod 20960 df-lss 21030 df-lmhm 21120 df-lvec 21201 df-sra 21271 df-rgmod 21272 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-cnfld 21491 df-refld 21723 df-phl 21744 df-dsmm 21850 df-frlm 21865 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cn 23352 df-cnp 23353 df-cmp 23512 df-tx 23687 df-hmeo 23880 df-xms 24445 df-ms 24446 df-tms 24447 df-nm 24707 df-ngp 24708 df-tng 24709 df-nrg 24710 df-nlm 24711 df-cncf 25005 df-clm 25190 df-cph 25295 df-tcph 25296 df-rrx 25512 df-ovol 25591 df-vol 25592 df-salg 46914 df-sumge0 46968 df-mea 47055 df-ome 47095 df-caragen 47097 df-ovoln 47142 df-voln 47144

This theorem is referenced by: vonn0ioo 47292

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