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Theorem ishl 24726
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
StepHypRef Expression
1 df-hl 24701 . 2 ℂHil = (Ban ∩ ℂPreHil)
21elin2 4157 1 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  wcel 2106  ℂPreHilccph 24530  Bancbn 24697  ℂHilchl 24698
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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-in 3917  df-hl 24701
This theorem is referenced by:  hlbn  24727  hlcph  24728  ishl2  24734  cphssphl  24735  cmslsschl  24741  chlcsschl  24742
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