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Theorem hlcph 25236
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 25234 . 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 2098  โ„‚PreHilccph 25038  Bancbn 25205  โ„‚Hilchl 25206
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 25209
This theorem is referenced by:  hlphl  25237  hlprlem  25239  cmslsschl  25249  chlcsschl  25250  pjthlem1  25309  pjthlem2  25310  cldcss  25313
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