Theorem mblvon

Description: The n-dimensional Lebesgue measure of a measurable set is the same as its n-dimensional Lebesgue outer measure. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	mblvon.1	$$⊢\phi \to X\in \mathrm{Fin}$$
	mblvon.2	$$⊢\phi \to A\in \mathrm{dom}\mathrm{voln}\left(X\right)$$
Assertion	mblvon	$$⊢\phi \to \mathrm{voln}\left(X\right)\left(A\right)=\mathrm{voln*}\left(X\right)\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		mblvon.1	$$⊢\phi \to X\in \mathrm{Fin}$$
2		mblvon.2	$$⊢\phi \to A\in \mathrm{dom}\mathrm{voln}\left(X\right)$$
3	1	vonval	$$⊢\phi \to \mathrm{voln}\left(X\right)={\mathrm{voln*}\left(X\right)\upharpoonright }_{\mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)}$$
4	3	fveq1d	$$⊢\phi \to \mathrm{voln}\left(X\right)\left(A\right)=\left({\mathrm{voln*}\left(X\right)\upharpoonright }_{\mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)}\right)\left(A\right)$$
5	1	dmvon	$$⊢\phi \to \mathrm{dom}\mathrm{voln}\left(X\right)=\mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)$$
6	2 5	eleqtrd	$$⊢\phi \to A\in \mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)$$
7		fvres	$$⊢A\in \mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)\to \left({\mathrm{voln*}\left(X\right)\upharpoonright }_{\mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)}\right)\left(A\right)=\mathrm{voln*}\left(X\right)\left(A\right)$$
8	6 7	syl	$$⊢\phi \to \left({\mathrm{voln*}\left(X\right)\upharpoonright }_{\mathrm{CaraGen}\left(\mathrm{voln*}\left(X\right)\right)}\right)\left(A\right)=\mathrm{voln*}\left(X\right)\left(A\right)$$
9	4 8	eqtrd	$$⊢\phi \to \mathrm{voln}\left(X\right)\left(A\right)=\mathrm{voln*}\left(X\right)\left(A\right)$$