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Theorem assasca 21900
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 21898 . 2 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2 assasca.f . . 3 𝐹 = (Scalar‘𝑊)
32lmodring 20883 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
41, 3syl 17 1 (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Scalarcsca 17301  Ringcrg 20251  LModclmod 20875  AssAlgcasa 21888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-lmod 20877  df-assa 21891
This theorem is referenced by:  assa2ass  21901  assa2ass2  21902  asclrhm  21928  assamulgscmlem2  21938  asclmulg  21940
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