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Theorem cvsclm 23729
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 23727 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 4173 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simplbi 500 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod)
51, 4syl 17 1 (𝜑𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  LVecclvec 19873  ℂModcclm 23665  ℂVecccvs 23726
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-cvs 23727
This theorem is referenced by:  cvsunit  23734  cvsdiv  23735  cvsmuleqdivd  23737  cvsdiveqd  23738  isncvsngp  23752  ncvsprp  23755  ncvsm1  23757  ncvsdif  23758  ncvspi  23759  ncvspds  23764  cnncvsmulassdemo  23767  ttgcontlem1  26670
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