Theorem cnnrg

Description: The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Assertion	cnnrg	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{NrmRing}$$

Proof

Step	Hyp	Ref	Expression
1		cnngp	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{NrmGrp}$$
2		absabv	$$⊢\mathrm{abs}\in \mathrm{AbsVal}\left({ℂ}_{\mathrm{fld}}\right)$$
3		cnfldnm	$$⊢\mathrm{abs}=\mathrm{norm}\left({ℂ}_{\mathrm{fld}}\right)$$
4		eqid	$$⊢\mathrm{AbsVal}\left({ℂ}_{\mathrm{fld}}\right)=\mathrm{AbsVal}\left({ℂ}_{\mathrm{fld}}\right)$$
5	3 4	isnrg	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{NrmRing}\leftrightarrow \left({ℂ}_{\mathrm{fld}}\in \mathrm{NrmGrp}\wedge \mathrm{abs}\in \mathrm{AbsVal}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
6	1 2 5	mpbir2an	$$⊢{ℂ}_{\mathrm{fld}}\in \mathrm{NrmRing}$$