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Theorem cphlmod 25190
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 25188 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmlmod 24683 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  LModclmod 20832  NrmModcnlm 24577  ℂPreHilccph 25182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5303
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-xp 5680  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fv 6554  df-ov 7419  df-nlm 24583  df-cph 25184
This theorem is referenced by:  cphclm  25205  cph2ass  25229  cphtcphnm  25246  nmparlem  25255  cphipval2  25257  4cphipval2  25258  cphipval  25259  cphsscph  25267  cmscsscms  25389  minveclem1  25440  minveclem2  25442  minveclem4  25448  minveclem6  25450  pjthlem1  25453  pjthlem2  25454
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