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Theorem cphlmod 23194
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 23192 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
2 nlmlmod 22703 . 2 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  LModclmod 19085  NrmModcnlm 22606  ℂPreHilccph 23186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fv 6057  df-ov 6817  df-nlm 22612  df-cph 23188
This theorem is referenced by:  cphclm  23209  cph2ass  23233  cphtchnm  23249  nmparlem  23258  cphipval2  23260  4cphipval2  23261  cphipval  23262  minveclem1  23415  minveclem2  23417  minveclem4  23423  minveclem6  23425  pjthlem1  23428  pjthlem2  23429
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