Theorem msmet

Description: The distance function, suitably truncated, is a metric on X . (Contributed by Mario Carneiro, 12-Nov-2013)

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

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	isms2	$$⊢M\in \mathrm{MetSp}\leftrightarrow \left(D\in \mathrm{Met}\left(X\right)\wedge \mathrm{TopOpen}\left(M\right)=\mathrm{MetOpen}\left(D\right)\right)$$
5	4	simplbi	$$⊢M\in \mathrm{MetSp}\to D\in \mathrm{Met}\left(X\right)$$