Theorem assasca 21636

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 21634	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2		assasca.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20622	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ‘cfv 6542  Scalarcsca 17204  Ringcrg 20127  LModclmod 20614  AssAlgcasa 21624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  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 6494  df-fv 6550  df-ov 7414  df-lmod 20616  df-assa 21627

This theorem is referenced by:  assa2ass  21637  asclrhm  21663  assamulgscmlem2  21673  asclmulg  32909