vonvolmbl - Mathbox for Glauco Siliprandi

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Theorem vonvolmbl 45022

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 3450	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
2	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑦 ∈ V)
3		reex 11151	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ V)
5		vonvolmbl.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
6	4, 5	ssexd 5286	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ V)
7		snfi 8995	. . . . . . . . . . . . . . 15 ⊢ {𝐴} ∈ Fin
8	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐴} ∈ Fin)
9	8	elexd 3466	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ∈ V)
10	2, 6, 9	inmap 43551	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ∩ 𝐵) ↑_m {𝐴}))
11	10	eqcomd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∩ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
12	11	fveq2d 6851	. . . . . . . . . 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 43554	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ∖ 𝐵) ↑_m {𝐴}))
15	14	eqcomd 2737	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∖ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
16	15	fveq2d 6851	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
17	12, 16	oveq12d 7380	. . . . . . . . 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 724	. . . . . . . 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 7397	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ V)
20	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → ℝ ∈ V)
21		elpwi 4572	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → 𝑦 ⊆ ℝ)
22		mapss 8834	. . . . . . . . . . . . 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 4571	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}))
25	24	adantl 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) → (𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}))
26		simpl 483	. . . . . . . . . 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 4170	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
28	27	fveq2d 6851	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
29		difeq1 4080	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
30	29	fveq2d 6851	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
31	28, 30	oveq12d 7380	. . . . . . . . . . . 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 6847	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘𝑥) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
33	31, 32	eqeq12d 2747	. . . . . . . . . . 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 3580	. . . . . . . . . 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 712	. . . . . . . 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 2732	. . . . . . . 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 2737	. . . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝐴 ∈ 𝑉)
41	21	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝑦 ⊆ ℝ)
42	40, 41	ovnovol 45020	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (vol‘𝑦))
43	42	adantlr 713	. . . . . 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 43378	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∩ 𝐵) ⊆ ℝ)
45	40, 44	ovnovol 45020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∩ 𝐵)))
46	41	ssdifssd 4107	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∖ 𝐵) ⊆ ℝ)
47	40, 46	ovnovol 45020	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∖ 𝐵)))
48	45, 47	oveq12d 7380	. . . . . . 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 713	. . . . . 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 3139	. . . 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 413	. . 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 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐴 ∈ 𝑉)
54	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐵 ⊆ ℝ)
55		simplr 767	. . . . . 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 4572	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴}) → 𝑥 ⊆ (ℝ ↑_m {𝐴}))
57	56	adantl 482	. . . . . 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 5896	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
59	58	cbviunv 5005	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑥 ran 𝑔 = ∪ 𝑓 ∈ 𝑥 ran 𝑓
60	53, 54, 55, 57, 59	vonvolmbllem 45021	. . . . 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 3139	. . . 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 413	. . 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 211	. 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 8834	. . . 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 44999	. . 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 705	. 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 44354	. . . 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 705	. 2 ⊢ (𝜑 → (𝐵 ∈ dom vol ↔ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))))
71	63, 67, 70	3bitr4d 310	1 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3060 Vcvv 3446 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 𝒫 cpw 4565 {csn 4591 ∪ ciun 4959 dom cdm 5638 ran crn 5639 ‘cfv 6501 (class class class)co 7362 ↑_m cmap 8772 Fincfn 8890 ℝcr 11059 +_𝑒 cxad 13040 vol*covol 24863 volcvol 24864 voln*covoln 44897 volncvoln 44899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9586 ax-cc 10380 ax-ac2 10408 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137 ax-pre-sup 11138 ax-addf 11139 ax-mulf 11140

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9356 df-sup 9387 df-inf 9388 df-oi 9455 df-dju 9846 df-card 9884 df-acn 9887 df-ac 10061 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12423 df-z 12509 df-dec 12628 df-uz 12773 df-q 12883 df-rp 12925 df-xneg 13042 df-xadd 13043 df-xmul 13044 df-ioo 13278 df-ico 13280 df-icc 13281 df-fz 13435 df-fzo 13578 df-fl 13707 df-seq 13917 df-exp 13978 df-hash 14241 df-cj 14996 df-re 14997 df-im 14998 df-sqrt 15132 df-abs 15133 df-clim 15382 df-rlim 15383 df-sum 15583 df-prod 15800 df-struct 17030 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-starv 17162 df-tset 17166 df-ple 17167 df-ds 17169 df-unif 17170 df-rest 17318 df-0g 17337 df-topgen 17339 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-grp 18765 df-minusg 18766 df-subg 18939 df-cmn 19578 df-abl 19579 df-mgp 19911 df-ur 19928 df-ring 19980 df-cring 19981 df-oppr 20063 df-dvdsr 20084 df-unit 20085 df-invr 20115 df-dvr 20126 df-drng 20227 df-psmet 20825 df-xmet 20826 df-met 20827 df-bl 20828 df-mopn 20829 df-cnfld 20834 df-top 22280 df-topon 22297 df-bases 22333 df-cmp 22775 df-ovol 24865 df-vol 24866 df-sumge0 44724 df-ome 44851 df-caragen 44853 df-ovoln 44898 df-voln 44900

This theorem is referenced by: vonvol 45023 vonvolmbl2 45024 vonvol2 45025

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