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

Theorem lveclmodd 20996
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 20995 . 2 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
31, 2syl 17 1 (𝜑𝑊 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  LModclmod 20747  LVecclvec 20991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-iota 6495  df-fv 6551  df-lvec 20992
This theorem is referenced by:  lvecgrpd  20997  quslvec  33120  ply1degltdimlem  33377  algextdeglem8  33449  prjspner1  42115
  Copyright terms: Public domain W3C validator