Metamath Proof Explorer

Theorem bncms

Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015)

Ref Expression
Assertion bncms 
|- ( W e. Ban -> W e. CMetSp )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
2 1 isbn 
 |-  ( W e. Ban <-> ( W e. NrmVec /\ W e. CMetSp /\ ( Scalar ` W ) e. CMetSp ) )
3 2 simp2bi 
 |-  ( W e. Ban -> W e. CMetSp )