Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cvslvec Structured version   Visualization version   GIF version

Theorem cvslvec 23125
 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 23124 . . . 4 ℂVec = (ℂMod ∩ LVec)
32elin2 3944 . . 3 (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec))
43simprbi 483 . 2 (𝑊 ∈ ℂVec → 𝑊 ∈ LVec)
51, 4syl 17 1 (𝜑𝑊 ∈ LVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139  LVecclvec 19304  ℂModcclm 23062  ℂVecccvs 23123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-cvs 23124 This theorem is referenced by:  cvsunit  23131  cvsdivcl  23133  isncvsngp  23149
 Copyright terms: Public domain W3C validator