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Theorem cvslvec 22828
Description: A complex 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 22827 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 3784 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simprbi 480 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
51, 4syl 17 1 (𝜑𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  LVecclvec 19016  ℂModcclm 22765  ℂVecccvs 22826
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
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 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-in 3567  df-cvs 22827
This theorem is referenced by:  cvsunit  22834  cvsdivcl  22836  isncvsngp  22852
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