Theorem ishl 25284

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 25259	. 2 ⊢ ℂHil = (Ban ∩ ℂPreHil)
2	1	elin2 4194	1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2099  ℂPreHilccph 25088  Bancbn 25255  ℂHilchl 25256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-hl 25259

This theorem is referenced by:  hlbn  25285  hlcph  25286  ishl2  25292  cphssphl  25293  cmslsschl  25299  chlcsschl  25300