Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvecring Structured version   Visualization version   GIF version

Theorem lvecring 41799
Description: The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
Hypothesis
Ref Expression
lvecring.1 𝐹 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
lvecring (π‘Š ∈ LVec β†’ 𝐹 ∈ Ring)

Proof of Theorem lvecring
StepHypRef Expression
1 lveclmod 20998 . 2 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
2 lvecring.1 . . 3 𝐹 = (Scalarβ€˜π‘Š)
32lmodring 20758 . 2 (π‘Š ∈ LMod β†’ 𝐹 ∈ Ring)
41, 3syl 17 1 (π‘Š ∈ LVec β†’ 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6553  Scalarcsca 17243  Ringcrg 20180  LModclmod 20750  LVecclvec 20994
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 2699  ax-nul 5310
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 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-lmod 20752  df-lvec 20995
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator