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Theorem lvecring 41563
Description: The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
Hypothesis
Ref Expression
lvecring.1 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
lvecring (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Proof of Theorem lvecring
StepHypRef Expression
1 lveclmod 20943 . 2 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2 lvecring.1 . . 3 𝐹 = (Scalar‘𝑊)
32lmodring 20703 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
41, 3syl 17 1 (𝑊 ∈ LVec → 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6533  Scalarcsca 17198  Ringcrg 20127  LModclmod 20695  LVecclvec 20939
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 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-lmod 20697  df-lvec 20940
This theorem is referenced by: (None)
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