Theorem xmsxmet

Description: The distance function, suitably truncated, is an extended metric on X . (Contributed by Mario Carneiro, 2-Sep-2015)

	Ref	Expression
Hypotheses	msf.x	$$⊢X={\mathrm{Base}}_M$$
	msf.d	$$⊢D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
Assertion	xmsxmet	$$⊢M\in \mathrm{\infty MetSp}\to D\in \mathrm{\infty Met}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		msf.x	$$⊢X={\mathrm{Base}}_M$$
2		msf.d	$$⊢D={\mathrm{dist}\left(M\right)\upharpoonright }_{\left(X\times X\right)}$$
3		eqid	$$⊢\mathrm{TopOpen}\left(M\right)=\mathrm{TopOpen}\left(M\right)$$
4	3 1 2	isxms2	$$⊢M\in \mathrm{\infty MetSp}\leftrightarrow \left(D\in \mathrm{\infty Met}\left(X\right)\wedge \mathrm{TopOpen}\left(M\right)=\mathrm{MetOpen}\left(D\right)\right)$$
5	4	simplbi	$$⊢M\in \mathrm{\infty MetSp}\to D\in \mathrm{\infty Met}\left(X\right)$$