Theorem lvecring 42493

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 21078	. 2 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2		lvecring.1	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20839	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ‘cfv 6542  Scalarcsca 17280  Ringcrg 20203  LModclmod 20831  LVecclvec 21074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5288

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-iota 6495  df-fv 6550  df-ov 7417  df-lmod 20833  df-lvec 21075

This theorem is referenced by: (None)