vonvolmbl - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > vonvolmbl		Structured version Visualization version GIF version

Theorem vonvolmbl 47107

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 3434	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
2	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑦 ∈ V)
3		reex 11120	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
4	3	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ℝ ∈ V)
5		vonvolmbl.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
6	4, 5	ssexd 5261	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ V)
7		snfi 8983	. . . . . . . . . . . . . . 15 ⊢ {𝐴} ∈ Fin
8	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐴} ∈ Fin)
9	8	elexd 3454	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ∈ V)
10	2, 6, 9	inmap 45656	. . . . . . . . . . . 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 6838	. . . . . . . . . 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 45659	. . . . . . . . . . . 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 6838	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
17	12, 16	oveq12d 7378	. . . . . . . . 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 727	. . . . . . . 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 7395	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ V)
20	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → ℝ ∈ V)
21		elpwi 4549	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → 𝑦 ⊆ ℝ)
22		mapss 8830	. . . . . . . . . . . . 13 ⊢ ((ℝ ∈ V ∧ 𝑦 ⊆ ℝ) → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
23	20, 21, 22	syl2anc 585	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
24	19, 23	elpwd 4548	. . . . . . . . . . 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 4154	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
28	27	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
29		difeq1 4060	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
30	29	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
31	28, 30	oveq12d 7378	. . . . . . . . . . . 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 6834	. . . . . . . . . . . 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 3563	. . . . . . . . . 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 585	. . . . . . . . 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 715	. . . . . . . 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 2776	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝐴 ∈ 𝑉)
41	21	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → 𝑦 ⊆ ℝ)
42	40, 41	ovnovol 47105	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (vol‘𝑦))
43	42	adantlr 716	. . . . . 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 4188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∩ 𝐵) ⊆ ℝ)
45	40, 44	ovnovol 47105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∩ 𝐵)))
46	41	ssdifssd 4088	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∖ 𝐵) ⊆ ℝ)
47	40, 46	ovnovol 47105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∖ 𝐵)))
48	45, 47	oveq12d 7378	. . . . . . 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 716	. . . . . 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 2780	. . . . 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 3130	. . . 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 727	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐴 ∈ 𝑉)
54	5	ad2antrr 727	. . . . . 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 4549	. . . . . . 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 5885	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
59	58	cbviunv 4982	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑥 ran 𝑔 = ∪ 𝑓 ∈ 𝑥 ran 𝑓
60	53, 54, 55, 57, 59	vonvolmbllem 47106	. . . . 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 3130	. . . 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 8830	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
65	4, 5, 64	syl2anc 585	. . 3 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
66	8	isvonmbl 47084	. . 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 708	. 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 46439	. . . 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 708	. 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 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4542 {csn 4568 ∪ ciun 4934 dom cdm 5624 ran crn 5625 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 Fincfn 8886 ℝcr 11028 +_𝑒 cxad 13052 vol*covol 25439 volcvol 25440 voln*covoln 46982 volncvoln 46984

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cc 10348 ax-ac2 10376 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-rlim 15442 df-sum 15640 df-prod 15860 df-rest 17376 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-top 22869 df-topon 22886 df-bases 22921 df-cmp 23362 df-ovol 25441 df-vol 25442 df-sumge0 46809 df-ome 46936 df-caragen 46938 df-ovoln 46983 df-voln 46985

This theorem is referenced by: vonvol 47108 vonvolmbl2 47109 vonvol2 47110

W3C validator