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Theorem ishl 24729
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 24704 . 2 ℂHil = (Ban ∩ ℂPreHil)
21elin2 4158 1 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wcel 2107  ℂPreHilccph 24533  Bancbn 24700  ℂHilchl 24701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-in 3918  df-hl 24704
This theorem is referenced by:  hlbn  24730  hlcph  24731  ishl2  24737  cphssphl  24738  cmslsschl  24744  chlcsschl  24745
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