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Theorem cvslvec 25158
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 25157 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 4203 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simprbi 496 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
51, 4syl 17 1 (𝜑𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  LVecclvec 21101  ℂModcclm 25095  ℂVecccvs 25156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-cvs 25157
This theorem is referenced by:  cvsunit  25164  cvsdivcl  25166  isncvsngp  25183
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