Metamath Proof Explorer


Theorem ngpmet

Description: The (induced) metric of a normed group is a metric. Part of Definition 2.2-1 of Kreyszig p. 58. (Contributed by NM, 4-Dec-2006) (Revised by AV, 14-Oct-2021)

Ref Expression
Hypotheses ngpmet.x X=BaseG
ngpmet.d D=distGX×X
Assertion ngpmet GNrmGrpDMetX

Proof

Step Hyp Ref Expression
1 ngpmet.x X=BaseG
2 ngpmet.d D=distGX×X
3 ngpms GNrmGrpGMetSp
4 1 2 msmet GMetSpDMetX
5 3 4 syl GNrmGrpDMetX