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Theorem assasca 21409
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
StepHypRef Expression
1 assalmod 21407 . 2 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2 assasca.f . . 3 𝐹 = (Scalar‘𝑊)
32lmodring 20472 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
41, 3syl 17 1 (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6541  Scalarcsca 17197  Ringcrg 20050  LModclmod 20464  AssAlgcasa 21397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6493  df-fv 6549  df-ov 7409  df-lmod 20466  df-assa 21400
This theorem is referenced by:  assa2ass  21410  asclrhm  21436  assamulgscmlem2  21446  asclmulg  32624
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