vonioo - Mathbox for Glauco Siliprandi

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Theorem vonioo 46638

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 6718	. . . . . . 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 6718	. . . . . . 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 46539	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0)
13	12	eqcomd 2741	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = (𝐴(𝐿‘∅)𝐵))
14		fveq2 6907	. . . . . 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 8947	. . . . . . 7 ⊢ (𝑋 = ∅ → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
18	16, 17	eqtrd 2775	. . . . . 6 ⊢ (𝑋 = ∅ → 𝐼 = X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)))
19	14, 18	fveq12d 6914	. . . . 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 9081	. . . . . . 7 ⊢ ∅ ∈ Fin
22	21	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ∅ ∈ Fin)
23		eqid 2735	. . . . . 6 ⊢ dom (voln‘∅) = dom (voln‘∅)
24		ressxr 11303	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ℝ ⊆ ℝ^*)
26	6, 25	fssd 6754	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ^*)
27	11, 25	fssd 6754	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ^*)
28	22, 23, 26, 27	ioovonmbl 46633	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘∅))
29	28	von0val 46627	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘∅)‘X𝑘 ∈ ∅ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) = 0)
30	20, 29	eqtrd 2775	. . 3 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = 0)
31		fveq2 6907	. . . . 5 ⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅))
32	31	oveqd 7448	. . . 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 2785	. 2 ⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
35		neqne 2946	. . . 4 ⊢ (¬ 𝑋 = ∅ → 𝑋 ≠ ∅)
36	35	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
37		nfv 1912	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑋 ≠ ∅)
38		nfra1 3282	. . . . . . . . 9 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)
39	37, 38	nfan 1897	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘))
40	2	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ)
41	7	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ)
42		volico 45939	. . . . . . . . . . . 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 3246	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘))
46	45	iftrued 4539	. . . . . . . . . . 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 2775	. . . . . . . . 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 3255	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∀𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘)))
51	50	prodeq2d 15954	. . . . . 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 2741	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
53		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐴‘𝑘) = (𝐴‘𝑗))
54		fveq2 6907	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐵‘𝑘) = (𝐵‘𝑗))
55	53, 54	breq12d 5161	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ (𝐴‘𝑗) < (𝐵‘𝑗)))
56	55	cbvralvw 3235	. . . . . . . 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 6907	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘))
71	70	oveq1d 7446	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((𝐴‘𝑗) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑚)))
72	71	cbvmptv 5261	. . . . . . . . . 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 7439	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (1 / 𝑚) = (1 / 𝑛))
75	74	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / 𝑛)))
76	75	mpteq2dv 5250	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
77	73, 76	eqtrd 2775	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
78	77	cbvmptv 5261	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚)))) = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))
79		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑛X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚)‘𝑘)[,)(𝐵‘𝑘))
80		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑋
81		nffvmpt1 6918	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)
82		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑘
83	81, 82	nffv 6917	. . . . . . . . . 10 ⊢ Ⅎ𝑚(((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)
84		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑚[,)
85		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐵‘𝑘)
86	83, 84, 85	nfov 7461	. . . . . . . . 9 ⊢ Ⅎ𝑚((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
87	80, 86	nfixpw 8955	. . . . . . . 8 ⊢ Ⅎ𝑚X𝑘 ∈ 𝑋 ((((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛)‘𝑘)[,)(𝐵‘𝑘))
88		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑚) = ((𝑚 ∈ ℕ ↦ (𝑗 ∈ 𝑋 ↦ ((𝐴‘𝑗) + (1 / 𝑚))))‘𝑛))
89	88	fveq1d 6909	. . . . . . . . . 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 7446	. . . . . . . . 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 8952	. . . . . . . 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 5259	. . . . . . 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 46637	. . . . . 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 46540	. . . . . 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 2785	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ≠ ∅) ∧ ∀𝑘 ∈ 𝑋 (𝐴‘𝑘) < (𝐵‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
98		rexnal 3098	. . . . . . . . . 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 11405	. . . . . . . . . . 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 3166	. . . . . . . 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 2903	. . . . . . . . 9 ⊢ Ⅎ𝑘(voln‘𝑋)
113		nfixp1 8957	. . . . . . . . . 10 ⊢ Ⅎ𝑘X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))
114	15, 113	nfcxfr 2901	. . . . . . . . 9 ⊢ Ⅎ𝑘𝐼
115	112, 114	nffv 6917	. . . . . . . 8 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼)
116		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐴(𝐿‘𝑋)𝐵)
117	115, 116	nfeq 2917	. . . . . . 7 ⊢ Ⅎ𝑘((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)
118	59	vonmea 46530	. . . . . . . . . . . 12 ⊢ (𝜑 → (voln‘𝑋) ∈ Meas)
119	118	mea0 46410	. . . . . . . . . . 11 ⊢ (𝜑 → ((voln‘𝑋)‘∅) = 0)
120	119	3ad2ant1 1132	. . . . . . . . . 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 1136	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝑘 ∈ 𝑋)
123		simp3 1137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ≤ (𝐴‘𝑘))
124	24, 40	sselid 3993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ^*)
125	124	3adant3 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴‘𝑘) ∈ ℝ^*)
126	24, 41	sselid 3993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ^*)
127	126	3adant3 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐵‘𝑘) ∈ ℝ^*)
128		ioo0 13409	. . . . . . . . . . . . . . . 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 3247	. . . . . . . . . . . . . 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 8970	. . . . . . . . . . . . 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 2775	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → 𝐼 = ∅)
136	135	fveq2d 6911	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∅))
137		ne0i 4347	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))))
141		eleq1w 2822	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝑋 ↔ 𝑘 ∈ 𝑋))
142		fveq2 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘))
143	142, 70	breq12d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ (𝐵‘𝑘) ≤ (𝐴‘𝑘)))
144	141, 143	3anbi23d 1438	. . . . . . . . . . . . 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 1912	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗))
147	59	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑋 ∈ Fin)
148		volicore 46537	. . . . . . . . . . . . . . . 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 11287	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
151	150	3ad2antl1 1184	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℂ)
152		simp2 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → 𝑗 ∈ 𝑋)
153	53, 54	oveq12d 7449	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑗 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑗)[,)(𝐵‘𝑗)))
154	153	fveq2d 6911	. . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐴‘𝑗) ∈ ℝ)
157	7	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐵‘𝑗) ∈ ℝ)
158		volico 45939	. . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → (𝐵‘𝑗) ≤ (𝐴‘𝑗))
162	157, 156	lenltd 11405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → ((𝐵‘𝑗) ≤ (𝐴‘𝑗) ↔ ¬ (𝐴‘𝑗) < (𝐵‘𝑗)))
163	162	3adant3 1131	. . . . . . . . . . . . . . . . . 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 4542	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → if((𝐴‘𝑗) < (𝐵‘𝑗), ((𝐵‘𝑗) − (𝐴‘𝑗)), 0) = 0)
166	160, 165	eqtrd 2775	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) ∧ 𝑘 = 𝑗) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
169	146, 147, 151, 152, 168	fprodeq0g 16027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋 ∧ (𝐵‘𝑗) ≤ (𝐴‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
170	145, 169	chvarvv 1996	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = 0)
171	140, 170	eqtrd 2775	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) = 0)
172	120, 136, 171	3eqtr4d 2785	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋 ∧ (𝐵‘𝑘) ≤ (𝐴‘𝑘)) → ((voln‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵))
173	172	3exp 1118	. . . . . . . 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 3264	. . . . . 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 813	. . 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 813	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 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8865 Xcixp 8936 Fincfn 8984 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 (,)cioo 13384 [,)cico 13386 ∏cprod 15936 volcvol 25512 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 ax-mulf 11233

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-tpos 8250 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-pws 17496 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-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-field 20749 df-abv 20827 df-staf 20857 df-srng 20858 df-lmod 20877 df-lss 20948 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-refld 21641 df-phl 21662 df-dsmm 21770 df-frlm 21785 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-nm 24611 df-ngp 24612 df-tng 24613 df-nrg 24614 df-nlm 24615 df-cncf 24918 df-clm 25110 df-cph 25216 df-tcph 25217 df-rrx 25433 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: vonn0ioo 46643

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