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Theorem cphlmod 25222
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 25220 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmlmod 24715 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  LModclmod 20875  NrmModcnlm 24609  ℂPreHilccph 25214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-nlm 24615  df-cph 25216
This theorem is referenced by:  cphclm  25237  cph2ass  25261  cphtcphnm  25278  nmparlem  25287  cphipval2  25289  4cphipval2  25290  cphipval  25291  cphsscph  25299  cmscsscms  25421  minveclem1  25472  minveclem2  25474  minveclem4  25480  minveclem6  25482  pjthlem1  25485  pjthlem2  25486
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