Theorem recms

Description: The real numbers form a complete metric space. (Contributed by Thierry Arnoux, 1-Nov-2017)

	Ref	Expression
Assertion	recms	$$⊢{ℝ}_{\mathrm{fld}}\in \mathrm{CMetSp}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)$$
2	1	recld2	$$⊢ℝ\in \mathrm{Clsd}\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
3		cncms	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{CMetSp}$$
4		ax-resscn	$$⊢ℝ\subseteq ℂ$$
5		df-refld	$$⊢{ℝ}_{\mathrm{fld}}={ℂ}_{\mathrm{fld}}{\upharpoonright }_{𝑠}ℝ$$
6		cnfldbas	$$⊢ℂ={\mathrm{Base}}_{{ℂ}_{\mathrm{fld}}}$$
7	5 6 1	cmsss	$$⊢\left({ℂ}_{\mathrm{fld}}\in \mathrm{CMetSp}\wedge ℝ\subseteq ℂ\right)\to \left({ℝ}_{\mathrm{fld}}\in \mathrm{CMetSp}\leftrightarrow ℝ\in \mathrm{Clsd}\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)\right)$$
8	3 4 7	mp2an	$$⊢{ℝ}_{\mathrm{fld}}\in \mathrm{CMetSp}\leftrightarrow ℝ\in \mathrm{Clsd}\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
9	2 8	mpbir	$$⊢{ℝ}_{\mathrm{fld}}\in \mathrm{CMetSp}$$