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Theorem cvsclm 24505
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 24503 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 4158 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simplbi 499 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
51, 4syl 17 1 (𝜑𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  LVecclvec 20578  ℂModcclm 24441  ℂVecccvs 24502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-cvs 24503
This theorem is referenced by:  cvsunit  24510  cvsdiv  24511  cvsmuleqdivd  24513  cvsdiveqd  24514  isncvsngp  24529  ncvsprp  24532  ncvsm1  24534  ncvsdif  24535  ncvspi  24536  ncvspds  24541  cnncvsmulassdemo  24544  ttgcontlem1  27875
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