Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007) Use bncmet (or preferably bncms ) instead. (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | iscbn.x | |- X = ( BaseSet ` U ) |
|
| iscbn.8 | |- D = ( IndMet ` U ) |
||
| Assertion | cbncms | |- ( U e. CBan -> D e. ( CMet ` X ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iscbn.x | |- X = ( BaseSet ` U ) |
|
| 2 | iscbn.8 | |- D = ( IndMet ` U ) |
|
| 3 | 1 2 | iscbn | |- ( U e. CBan <-> ( U e. NrmCVec /\ D e. ( CMet ` X ) ) ) |
| 4 | 3 | simprbi | |- ( U e. CBan -> D e. ( CMet ` X ) ) |