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Theorem assasca 21882
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 21880 . 2 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
2 assasca.f . . 3 𝐹 = (Scalar‘𝑊)
32lmodring 20866 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Ring)
41, 3syl 17 1 (𝑊 ∈ AssAlg → 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  Scalarcsca 17300  Ringcrg 20230  LModclmod 20858  AssAlgcasa 21870
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-lmod 20860  df-assa 21873
This theorem is referenced by:  assa2ass  21883  assa2ass2  21884  asclrhm  21910  assamulgscmlem2  21920  asclmulg  21922
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