Theorem ngpms

Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Assertion	ngpms	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( norm ‘ 𝐺 ) = ( norm ‘ 𝐺 )
2		eqid	⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 )
3		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
4	1 2 3	isngp	⊢ ( 𝐺 ∈ NrmGrp ↔ ( 𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ( ( norm ‘ 𝐺 ) ∘ ( -g ‘ 𝐺 ) ) ⊆ ( dist ‘ 𝐺 ) ) )
5	4	simp2bi	⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp )