Theorem lvecring 42983

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

Step	Hyp	Ref	Expression
1		lveclmod 21101	. 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2		lvecring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20863	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  Scalarcsca 17223  Ringcrg 20214  LModclmod 20855  LVecclvec 21097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-lmod 20857  df-lvec 21098

This theorem is referenced by: (None)