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Theorem ishl 25410
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 25385 . 2 ℂHil = (Ban ∩ ℂPreHil)
21elin2 4213 1 (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  ℂPreHilccph 25214  Bancbn 25381  ℂHilchl 25382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-hl 25385
This theorem is referenced by:  hlbn  25411  hlcph  25412  ishl2  25418  cphssphl  25419  cmslsschl  25425  chlcsschl  25426
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