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Theorem cvslvec 23143
Description: A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
Hypothesis
Ref Expression
cvslvec.1 (𝜑𝑊 ∈ ℂVec)
Assertion
Ref Expression
cvslvec (𝜑𝑊 ∈ LVec)

Proof of Theorem cvslvec
StepHypRef Expression
1 cvslvec.1 . 2 (𝜑𝑊 ∈ ℂVec)
2 df-cvs 23142 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 3950 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simprbi 478 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
51, 4syl 17 1 (𝜑𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  LVecclvec 19314  ℂModcclm 23080  ℂVecccvs 23141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-in 3728  df-cvs 23142
This theorem is referenced by:  cvsunit  23149  cvsdivcl  23151  isncvsngp  23167
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