Theorem ishl 24870

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 24845	. 2 ⊢ ℂHil = (Ban ∩ ℂPreHil)
2	1	elin2 4196	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ℂPreHilccph 24674  Bancbn 24841  ℂHilchl 24842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-hl 24845

This theorem is referenced by:  hlbn  24871  hlcph  24872  ishl2  24878  cphssphl  24879  cmslsschl  24885  chlcsschl  24886