Theorem xmetutop

Description: The topology induced by a uniform structure generated by an extended metric D is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018)

	Ref	Expression
Assertion	xmetutop	$$⊢\left(X\ne \varnothing \wedge D\in \mathrm{\infty Met}\left(X\right)\right)\to \mathrm{unifTop}\left(\mathrm{metUnif}\left(D\right)\right)=\mathrm{MetOpen}\left(D\right)$$

Proof

Step	Hyp	Ref	Expression
1		xmetpsmet	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to D\in \mathrm{PsMet}\left(X\right)$$
2		psmetutop	$$⊢\left(X\ne \varnothing \wedge D\in \mathrm{PsMet}\left(X\right)\right)\to \mathrm{unifTop}\left(\mathrm{metUnif}\left(D\right)\right)=\mathrm{topGen}\left(\mathrm{ran}\mathrm{ball}\left(D\right)\right)$$
3	1 2	sylan2	$$⊢\left(X\ne \varnothing \wedge D\in \mathrm{\infty Met}\left(X\right)\right)\to \mathrm{unifTop}\left(\mathrm{metUnif}\left(D\right)\right)=\mathrm{topGen}\left(\mathrm{ran}\mathrm{ball}\left(D\right)\right)$$
4		eqid	$$⊢\mathrm{MetOpen}\left(D\right)=\mathrm{MetOpen}\left(D\right)$$
5	4	mopnval	$$⊢D\in \mathrm{\infty Met}\left(X\right)\to \mathrm{MetOpen}\left(D\right)=\mathrm{topGen}\left(\mathrm{ran}\mathrm{ball}\left(D\right)\right)$$
6	5	adantl	$$⊢\left(X\ne \varnothing \wedge D\in \mathrm{\infty Met}\left(X\right)\right)\to \mathrm{MetOpen}\left(D\right)=\mathrm{topGen}\left(\mathrm{ran}\mathrm{ball}\left(D\right)\right)$$
7	3 6	eqtr4d	$$⊢\left(X\ne \varnothing \wedge D\in \mathrm{\infty Met}\left(X\right)\right)\to \mathrm{unifTop}\left(\mathrm{metUnif}\left(D\right)\right)=\mathrm{MetOpen}\left(D\right)$$