Theorem lvecring 42570

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 21038	. 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2		lvecring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20799	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  Scalarcsca 17161  Ringcrg 20149  LModclmod 20791  LVecclvec 21034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349  df-lmod 20793  df-lvec 21035

This theorem is referenced by: (None)