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Theorem cphlmod 24915
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 24913 . 2 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmMod)
2 nlmlmod 24415 . 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 2106  LModclmod 20614  NrmModcnlm 24309  β„‚PreHilccph 24907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-ov 7414  df-nlm 24315  df-cph 24909
This theorem is referenced by:  cphclm  24930  cph2ass  24954  cphtcphnm  24971  nmparlem  24980  cphipval2  24982  4cphipval2  24983  cphipval  24984  cphsscph  24992  cmscsscms  25114  minveclem1  25165  minveclem2  25167  minveclem4  25173  minveclem6  25175  pjthlem1  25178  pjthlem2  25179
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