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Theorem hlcph 24731
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 24729 . 2 (๐‘Š โˆˆ โ„‚Hil โ†” (๐‘Š โˆˆ Ban โˆง ๐‘Š โˆˆ โ„‚PreHil))
21simprbi 498 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 2107  โ„‚PreHilccph 24533  Bancbn 24700  โ„‚Hilchl 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:  hlphl  24732  hlprlem  24734  cmslsschl  24744  chlcsschl  24745  pjthlem1  24804  pjthlem2  24805  cldcss  24808
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