Theorem assasca 21408

Description: The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.)

Hypothesis

Ref	Expression
assasca.f	⊢ 𝐹 = (Scalar‘𝑊)

Assertion

Ref	Expression
assasca	⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Proof of Theorem assasca

Step	Hyp	Ref	Expression
1		assalmod 21406	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2		assasca.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20471	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  Scalarcsca 17196  Ringcrg 20049  LModclmod 20463  AssAlgcasa 21396

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7408  df-lmod 20465  df-assa 21399

This theorem is referenced by:  assa2ass  21409  asclrhm  21435  assamulgscmlem2  21445  asclmulg  32623