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Theorem ishl 25214
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 25189 . 2 ℂHil = (Ban ∩ ℂPreHil)
21elin2 4190 1 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098  ℂPreHilccph 25018  Bancbn 25185  ℂHilchl 25186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-hl 25189
This theorem is referenced by:  hlbn  25215  hlcph  25216  ishl2  25222  cphssphl  25223  cmslsschl  25229  chlcsschl  25230
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