vonvolmbl - Mathbox for Glauco Siliprandi

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Theorem vonvolmbl 46657

Description: A subset of Real numbers is Lebesgue measurable if and only if its corresponding 1-dimensional set is measurable w.r.t. the 1-dimensional Lebesgue measure. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
vonvolmbl.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
vonvolmbl.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)

**Assertion**
Ref	Expression
vonvolmbl	⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Proof of Theorem vonvolmbl Dummy variables 𝑓 𝑥 𝑦 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3468	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
2	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑦 ∈ V)
3		reex 11225	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ V)
5		vonvolmbl.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
6	4, 5	ssexd 5299	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ V)
7		snfi 9062	. . . . . . . . . . . . . . 15 ⊢ {𝐴} ∈ Fin
8	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐴} ∈ Fin)
9	8	elexd 3488	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ∈ V)
10	2, 6, 9	inmap 45200	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ∩ 𝐵) ↑_m {𝐴}))
11	10	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∩ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
12	11	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
13		vonvolmbl.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
14	2, 6, 13	difmapsn 45203	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ∖ 𝐵) ↑_m {𝐴}))
15	14	eqcomd 2742	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∖ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
16	15	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
17	12, 16	oveq12d 7428	. . . . . . . . 9 ⊢ (𝜑 → (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))) = (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))))
18	17	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))) = (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln*‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))))
19		ovexd 7445	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ V)
20	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → ℝ ∈ V)
21		elpwi 4587	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → 𝑦 ⊆ ℝ)
22		mapss 8908	. . . . . . . . . . . . 13 ⊢ ((ℝ ∈ V ∧ 𝑦 ⊆ ℝ) → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
23	20, 21, 22	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
24	19, 23	elpwd 4586	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}))
25	24	adantl 481	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln*‘{𝐴})‘𝑥) ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}))
26		simpl 482	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥) ∧ 𝑦 ∈ 𝒫 ℝ) → ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥))
27		ineq1 4193	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
28	27	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
29		difeq1 4099	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
30	29	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
31	28, 30	oveq12d 7428	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))))
32		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘𝑥) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
33	31, 32	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥) ↔ (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴}))))
34	33	rspcva 3604	. . . . . . . . . 10 ⊢ (((𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}) ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) → (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
35	25, 26, 34	syl2anc 584	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥) ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
36	35	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
37		eqidd 2737	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = ((voln*‘{𝐴})‘(𝑦 ↑_m {𝐴})))
38	18, 36, 37	3eqtrd 2775	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
39	38	eqcomd 2742	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))))
40	13	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝐴 ∈ 𝑉)
41	21	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝑦 ⊆ ℝ)
42	40, 41	ovnovol 46655	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (vol‘𝑦))
43	42	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (vol*‘𝑦))
44	41	ssinss1d 4227	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∩ 𝐵) ⊆ ℝ)
45	40, 44	ovnovol 46655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∩ 𝐵)))
46	41	ssdifssd 4127	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∖ 𝐵) ⊆ ℝ)
47	40, 46	ovnovol 46655	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∖ 𝐵)))
48	45, 47	oveq12d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))))
49	48	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → (((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) +_𝑒 ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴}))) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))
50	39, 43, 49	3eqtr3d 2779	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) ∧ 𝑦 ∈ 𝒫 ℝ) → (vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))))
51	50	ralrimiva 3133	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)) → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))))
52	51	ex 412	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥) → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))))
53	13	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐴 ∈ 𝑉)
54	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐵 ⊆ ℝ)
55		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))))
56		elpwi 4587	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴}) → 𝑥 ⊆ (ℝ ↑_m {𝐴}))
57	56	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝑥 ⊆ (ℝ ↑_m {𝐴}))
58		rneq 5921	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
59	58	cbviunv 5021	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑥 ran 𝑔 = ∪ 𝑓 ∈ 𝑥 ran 𝑓
60	53, 54, 55, 57, 59	vonvolmbllem 46656	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → (((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥))
61	60	ralrimiva 3133	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))) → ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥))
62	61	ex 412	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵))) → ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥)))
63	52, 62	impbid 212	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln‘{𝐴})‘𝑥) ↔ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol‘(𝑦 ∖ 𝐵)))))
64		mapss 8908	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
65	4, 5, 64	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
66	8	isvonmbl 46634	. . 3 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ ((𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}) ∧ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln*‘{𝐴})‘𝑥))))
67	65, 66	mpbirand 707	. 2 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ ∀𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})(((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) +_𝑒 ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴})))) = ((voln*‘{𝐴})‘𝑥)))
68		ismbl4 45989	. . . 4 ⊢ (𝐵 ∈ dom vol ↔ (𝐵 ⊆ ℝ ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))))
69	68	a1i 11	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom vol ↔ (𝐵 ⊆ ℝ ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵))))))
70	5, 69	mpbirand 707	. 2 ⊢ (𝜑 → (𝐵 ∈ dom vol ↔ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))))
71	63, 67, 70	3bitr4d 311	1 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3052 Vcvv 3464 ∖ cdif 3928 ∩ cin 3930 ⊆ wss 3931 𝒫 cpw 4580 {csn 4606 ∪ ciun 4972 dom cdm 5659 ran crn 5660 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 Fincfn 8964 ℝcr 11133 +_𝑒 cxad 13131 vol*covol 25420 volcvol 25421 voln*covoln 46532 volncvoln 46534

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cc 10454 ax-ac2 10482 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-ac 10135 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-rlim 15510 df-sum 15708 df-prod 15925 df-rest 17441 df-topgen 17462 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-top 22837 df-topon 22854 df-bases 22889 df-cmp 23330 df-ovol 25422 df-vol 25423 df-sumge0 46359 df-ome 46486 df-caragen 46488 df-ovoln 46533 df-voln 46535

This theorem is referenced by: vonvol 46658 vonvolmbl2 46659 vonvol2 46660

W3C validator