Theorem assasca 21829

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 21827	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2		assasca.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20831	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  Scalarcsca 17192  Ringcrg 20180  LModclmod 20823  AssAlgcasa 21817

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 2709  ax-nul 5253

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 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-lmod 20825  df-assa 21820

This theorem is referenced by:  assa2ass  21830  assa2ass2  21831  asclrhm  21858  assamulgscmlem2  21868  asclmulg  21870