Theorem assasca 21844

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 21842	. 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2		assasca.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3	2	lmodring 20865	. 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
4	1, 3	syl 17	1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6492  Scalarcsca 17221  Ringcrg 20212  LModclmod 20857  AssAlgcasa 21832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-lmod 20859  df-assa 21835

This theorem is referenced by:  assa2ass  21845  assa2ass2  21846  asclrhm  21872  assamulgscmlem2  21882  asclmulg  21884