Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | ngpms | |- ( G e. NrmGrp -> G e. MetSp ) |
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 e. NrmGrp <-> ( G e. Grp /\ G e. MetSp /\ ( ( norm ` G ) o. ( -g ` G ) ) C_ ( dist ` G ) ) ) |
5 | 4 | simp2bi | |- ( G e. NrmGrp -> G e. MetSp ) |