Theorem ishl 24431

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.)

Assertion

Ref	Expression
ishl	⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Proof of Theorem ishl

Step	Hyp	Ref	Expression
1		df-hl 24406	. 2 ⊢ ℂHil = (Ban ∩ ℂPreHil)
2	1	elin2 4127	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ℂPreHilccph 24235  Bancbn 24402  ℂHilchl 24403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-hl 24406

This theorem is referenced by:  hlbn  24432  hlcph  24433  ishl2  24439  cphssphl  24440  cmslsschl  24446  chlcsschl  24447