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Theorem cphlmod 25208
Description: A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphlmod (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)

Proof of Theorem cphlmod
StepHypRef Expression
1 cphnlm 25206 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmlmod 24699 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  LModclmod 20858  NrmModcnlm 24593  ℂPreHilccph 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-nlm 24599  df-cph 25202
This theorem is referenced by:  cphclm  25223  cph2ass  25247  cphtcphnm  25264  nmparlem  25273  cphipval2  25275  4cphipval2  25276  cphipval  25277  cphsscph  25285  cmscsscms  25407  minveclem1  25458  minveclem2  25460  minveclem4  25466  minveclem6  25468  pjthlem1  25471  pjthlem2  25472
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