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 = ( Base ` G ) |
|
| ngpmet.d | |- D = ( ( dist ` G ) |` ( X X. X ) ) |
||
| Assertion | ngpmet | |- ( G e. NrmGrp -> D e. ( Met ` X ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ngpmet.x | |- X = ( Base ` G ) |
|
| 2 | ngpmet.d | |- D = ( ( dist ` G ) |` ( X X. X ) ) |
|
| 3 | ngpms | |- ( G e. NrmGrp -> G e. MetSp ) |
|
| 4 | 1 2 | msmet | |- ( G e. MetSp -> D e. ( Met ` X ) ) |
| 5 | 3 4 | syl | |- ( G e. NrmGrp -> D e. ( Met ` X ) ) |