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

Theorem nvclvec 22311
 Description: A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Assertion
Ref Expression
nvclvec (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Proof of Theorem nvclvec
StepHypRef Expression
1 isnvc 22309 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
21simprbi 479 1 (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  LVecclvec 18923  NrmModcnlm 22195  NrmVeccnvc 22196 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-nvc 22202 This theorem is referenced by:  nvctvc  22314  lssnvc  22316
 Copyright terms: Public domain W3C validator