vonvolmbl - Mathbox for Glauco Siliprandi

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Theorem vonvolmbl 47104

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 3435	. . . . . . . . . . . . . 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 5252	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ V)
7		snfi 8980	. . . . . . . . . . . . . . 15 ⊢ {𝐴} ∈ Fin
8	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝐴} ∈ Fin)
9	8	elexd 3454	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝐴} ∈ V)
10	2, 6, 9	inmap 45654	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ∩ 𝐵) ↑_m {𝐴}))
11	10	eqcomd 2745	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∩ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
12	11	fveq2d 6831	. . . . . . . . . 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 45657	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ∖ 𝐵) ↑_m {𝐴}))
15	14	eqcomd 2745	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∖ 𝐵) ↑_m {𝐴}) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
16	15	fveq2d 6831	. . . . . . . . . 10 ⊢ (𝜑 → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
17	12, 16	oveq12d 7374	. . . . . . . . 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 732	. . . . . . . 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 7391	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ∈ V)
20	3	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → ℝ ∈ V)
21		elpwi 4536	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝒫 ℝ → 𝑦 ⊆ ℝ)
22		mapss 8827	. . . . . . . . . . . . 13 ⊢ ((ℝ ∈ V ∧ 𝑦 ⊆ ℝ) → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
23	20, 21, 22	syl2anc 590	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 ℝ → (𝑦 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
24	19, 23	elpwd 4535	. . . . . . . . . . 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 4142	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∩ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴})))
28	27	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∩ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∩ (𝐵 ↑_m {𝐴}))))
29		difeq1 4050	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → (𝑥 ∖ (𝐵 ↑_m {𝐴})) = ((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴})))
30	29	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘(𝑥 ∖ (𝐵 ↑_m {𝐴}))) = ((voln‘{𝐴})‘((𝑦 ↑_m {𝐴}) ∖ (𝐵 ↑_m {𝐴}))))
31	28, 30	oveq12d 7374	. . . . . . . . . . . 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 6827	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ↑_m {𝐴}) → ((voln‘{𝐴})‘𝑥) = ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})))
33	31, 32	eqeq12d 2755	. . . . . . . . . . 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 3558	. . . . . . . . . 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 590	. . . . . . . . 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 720	. . . . . . . 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 2740	. . . . . . . 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 2778	. . . . . . 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 2745	. . . . . 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 47102	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘(𝑦 ↑_m {𝐴})) = (vol‘𝑦))
43	42	adantlr 721	. . . . . 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 4175	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∩ 𝐵) ⊆ ℝ)
45	40, 44	ovnovol 47102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∩ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∩ 𝐵)))
46	41	ssdifssd 4077	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → (𝑦 ∖ 𝐵) ⊆ ℝ)
47	40, 46	ovnovol 47102	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 ℝ) → ((voln‘{𝐴})‘((𝑦 ∖ 𝐵) ↑_m {𝐴})) = (vol‘(𝑦 ∖ 𝐵)))
48	45, 47	oveq12d 7374	. . . . . . 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 721	. . . . . 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 2782	. . . . 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 3131	. . . 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 732	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐴 ∈ 𝑉)
54	5	ad2antrr 732	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))) ∧ 𝑥 ∈ 𝒫 (ℝ ↑_m {𝐴})) → 𝐵 ⊆ ℝ)
55		simplr 774	. . . . . 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 4536	. . . . . . 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 5878	. . . . . . 7 ⊢ (𝑔 = 𝑓 → ran 𝑔 = ran 𝑓)
59	58	cbviunv 4968	. . . . . 6 ⊢ ∪ 𝑔 ∈ 𝑥 ran 𝑔 = ∪ 𝑓 ∈ 𝑥 ran 𝑓
60	53, 54, 55, 57, 59	vonvolmbllem 47103	. . . . 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 3131	. . . 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 213	. 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 8827	. . . 4 ⊢ ((ℝ ∈ V ∧ 𝐵 ⊆ ℝ) → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
65	4, 5, 64	syl2anc 590	. . 3 ⊢ (𝜑 → (𝐵 ↑_m {𝐴}) ⊆ (ℝ ↑_m {𝐴}))
66	8	isvonmbl 47081	. . 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 713	. 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 46436	. . . 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 713	. 2 ⊢ (𝜑 → (𝐵 ∈ dom vol ↔ ∀𝑦 ∈ 𝒫 ℝ(vol‘𝑦) = ((vol‘(𝑦 ∩ 𝐵)) +_𝑒 (vol*‘(𝑦 ∖ 𝐵)))))
71	63, 67, 70	3bitr4d 312	1 ⊢ (𝜑 → ((𝐵 ↑_m {𝐴}) ∈ dom (voln‘{𝐴}) ↔ 𝐵 ∈ dom vol))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3053 Vcvv 3431 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4529 {csn 4555 ∪ ciun 4921 dom cdm 5618 ran crn 5619 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Fincfn 8883 ℝcr 11028 +_𝑒 cxad 13052 vol*covol 25447 volcvol 25448 voln*covoln 46979 volncvoln 46981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 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 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-top 22877 df-topon 22894 df-bases 22929 df-cmp 23370 df-ovol 25449 df-vol 25450 df-sumge0 46806 df-ome 46933 df-caragen 46935 df-ovoln 46980 df-voln 46982

This theorem is referenced by: vonvol 47105 vonvolmbl2 47106 vonvol2 47107

W3C validator