vonvolmbl - Mathbox for Glauco Siliprandi

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Theorem vonvolmbl 43874

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 3412	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
2	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑦 ∈ V)
3		reex 10820	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ V)
5		vonvolmbl.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
6	4, 5	ssexd 5217	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ V)
7		snfi 8721	. . . . . . . . . . . . . . 15 ⊢ {𝐴} ∈ Fin
8	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐴} ∈ Fin)
9	8	elexd 3428	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ∈ V)
10	2, 6, 9	inmap 42422	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ∩ 𝐵) ↑_m {𝐴}))
11	10	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∩ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
12	11	fveq2d 6721	. . . . . . . . . 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 42425	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ∖ 𝐵) ↑_m {𝐴}))
15	14	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∖ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
16	15	fveq2d 6721	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
17	12, 16	oveq12d 7231	. . . . . . . . 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 7248	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ V)
20	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → ℝ ∈ V)
21		elpwi 4522	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → 𝑦 ⊆ ℝ)
22		mapss 8570	. . . . . . . . . . . . 13 ⊢ ((ℝ ∈ V ∧ 𝑦 ⊆ ℝ) → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
23	20, 21, 22	syl2anc 587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
24	19, 23	elpwd 4521	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ 𝒫 (ℝ ↑_m {𝐴}))
25	24	adantl 485	. . . . . . . . . 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 486	. . . . . . . . . 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 4120	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
28	27	fveq2d 6721	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
29		difeq1 4030	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
30	29	fveq2d 6721	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
31	28, 30	oveq12d 7231	. . . . . . . . . . . 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 6717	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘𝑥) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
33	31, 32	eqeq12d 2753	. . . . . . . . . . 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 3535	. . . . . . . . . 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 587	. . . . . . . . 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 2738	. . . . . . . 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 2781	. . . . . . 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 2743	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝐴 ∈ 𝑉)
41	21	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝑦 ⊆ ℝ)
42	40, 41	ovnovol 43872	. . . . . . 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 42269	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∩ 𝐵) ⊆ ℝ)
45	40, 44	ovnovol 43872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∩ 𝐵)))
46	41	ssdifssd 4057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∖ 𝐵) ⊆ ℝ)
47	40, 46	ovnovol 43872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∖ 𝐵)))
48	45, 47	oveq12d 7231	. . . . . . 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 2785	. . . . 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 3105	. . . 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 416	. . 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 769	. . . . . 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 4522	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴}) → 𝑥 ⊆ (ℝ ↑_m {𝐴}))
57	56	adantl 485	. . . . . 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 5805	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
59	58	cbviunv 4949	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑥 ran 𝑔 = ∪ 𝑓 ∈ 𝑥 ran 𝑓
60	53, 54, 55, 57, 59	vonvolmbllem 43873	. . . . 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 3105	. . . 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 416	. . 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 215	. 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 8570	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
65	4, 5, 64	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
66	8	isvonmbl 43851	. . 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 43209	. . . 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 314	1 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∖ cdif 3863 ∩ cin 3865 ⊆ wss 3866 𝒫 cpw 4513 {csn 4541 ∪ ciun 4904 dom cdm 5551 ran crn 5552 ‘cfv 6380 (class class class)co 7213 ↑_m cmap 8508 Fincfn 8626 ℝcr 10728 +_𝑒 cxad 12702 vol*covol 24359 volcvol 24360 voln*covoln 43749 volncvoln 43751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-ac2 10077 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-ac 9730 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-prod 15468 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-rest 16927 df-0g 16946 df-topgen 16948 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-minusg 18369 df-subg 18540 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-cnfld 20364 df-top 21791 df-topon 21808 df-bases 21843 df-cmp 22284 df-ovol 24361 df-vol 24362 df-sumge0 43576 df-ome 43703 df-caragen 43705 df-ovoln 43750 df-voln 43752

This theorem is referenced by: vonvol 43875 vonvolmbl2 43876 vonvol2 43877

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