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 ∈ ℂHil ↔ W ∈ Ban ∧ W ∈ CPreHil

Proof

Step Hyp Ref Expression
1 df-hl ⊢ ℂHil = Ban ∩ CPreHil
2 1 elin2 ⊢ W ∈ ℂHil ↔ W ∈ Ban ∧ W ∈ CPreHil