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Theorem hlcph 25291
Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
hlcph (๐‘Š โˆˆ โ„‚Hil โ†’ ๐‘Š โˆˆ โ„‚PreHil)

Proof of Theorem hlcph
StepHypRef Expression
1 ishl 25289 . 2 (๐‘Š โˆˆ โ„‚Hil โ†” (๐‘Š โˆˆ Ban โˆง ๐‘Š โˆˆ โ„‚PreHil))
21simprbi 496 1 (๐‘Š โˆˆ โ„‚Hil โ†’ ๐‘Š โˆˆ โ„‚PreHil)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆˆ wcel 2099  โ„‚PreHilccph 25093  Bancbn 25260  โ„‚Hilchl 25261
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 3473  df-in 3954  df-hl 25264
This theorem is referenced by:  hlphl  25292  hlprlem  25294  cmslsschl  25304  chlcsschl  25305  pjthlem1  25364  pjthlem2  25365  cldcss  25368
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