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Theorem cphlmod 24691
Description: A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphlmod (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ LMod)

Proof of Theorem cphlmod
StepHypRef Expression
1 cphnlm 24689 . 2 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmMod)
2 nlmlmod 24195 . 2 (π‘Š ∈ NrmMod β†’ π‘Š ∈ LMod)
31, 2syl 17 1 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  LModclmod 20471  NrmModcnlm 24089  β„‚PreHilccph 24683
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 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-nlm 24095  df-cph 24685
This theorem is referenced by:  cphclm  24706  cph2ass  24730  cphtcphnm  24747  nmparlem  24756  cphipval2  24758  4cphipval2  24759  cphipval  24760  cphsscph  24768  cmscsscms  24890  minveclem1  24941  minveclem2  24943  minveclem4  24949  minveclem6  24951  pjthlem1  24954  pjthlem2  24955
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