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Theorem cvsclm 22849
Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
Hypothesis
Ref Expression
cvslvec.1 (𝜑𝑊 ∈ ℂVec)
Assertion
Ref Expression
cvsclm (𝜑𝑊 ∈ ℂMod)

Proof of Theorem cvsclm
StepHypRef Expression
1 cvslvec.1 . 2 (𝜑𝑊 ∈ ℂVec)
2 df-cvs 22847 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 3784 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simplbi 476 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
51, 4syl 17 1 (𝜑𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  LVecclvec 19034  ℂModcclm 22785  ℂVecccvs 22846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-cvs 22847
This theorem is referenced by:  cvsunit  22854  cvsdiv  22855  cvsmuleqdivd  22857  cvsdiveqd  22858  isncvsngp  22872  ncvsprp  22875  ncvsm1  22877  ncvsdif  22878  ncvspi  22879  ncvspds  22884  cnncvsmulassdemo  22887  ttgcontlem1  25682
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