Theorem nrgngp

Description: A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Assertion	nrgngp	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( norm ‘ 𝑅 ) = ( norm ‘ 𝑅 )
2		eqid	⊢ ( AbsVal ‘ 𝑅 ) = ( AbsVal ‘ 𝑅 )
3	1 2	isnrg	⊢ ( 𝑅 ∈ NrmRing ↔ ( 𝑅 ∈ NrmGrp ∧ ( norm ‘ 𝑅 ) ∈ ( AbsVal ‘ 𝑅 ) ) )
4	3	simplbi	⊢ ( 𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp )