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

Theorem lveclmodd 20981
Description: A vector space is a left module. (Contributed by SN, 16-May-2024.)
Hypothesis
Ref Expression
lveclmodd.1 (𝜑𝑊 ∈ LVec)
Assertion
Ref Expression
lveclmodd (𝜑𝑊 ∈ LMod)

Proof of Theorem lveclmodd
StepHypRef Expression
1 lveclmodd.1 . 2 (𝜑𝑊 ∈ LVec)
2 lveclmod 20980 . 2 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝜑𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  LModclmod 20732  LVecclvec 20976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-lvec 20977
This theorem is referenced by:  lvecgrpd  20982  quslvec  33012  ply1degltdimlem  33252  algextdeglem8  33328  prjspner1  41972
  Copyright terms: Public domain W3C validator