Metamath Proof Explorer


Theorem ishl

Description: The predicate "is a subcomplex Hilbert space". A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007) (Revised by Mario Carneiro, 15-Oct-2015)

Ref Expression
Assertion ishl
|- ( W e. CHil <-> ( W e. Ban /\ W e. CPreHil ) )

Proof

Step Hyp Ref Expression
1 df-hl
 |-  CHil = ( Ban i^i CPreHil )
2 1 elin2
 |-  ( W e. CHil <-> ( W e. Ban /\ W e. CPreHil ) )