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Theorem assasca 21417
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
assasca (π‘Š ∈ AssAlg β†’ 𝐹 ∈ Ring)

Proof of Theorem assasca
StepHypRef Expression
1 assalmod 21415 . 2 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
2 assasca.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
32lmodring 20479 . 2 (π‘Š ∈ LMod β†’ 𝐹 ∈ Ring)
41, 3syl 17 1 (π‘Š ∈ AssAlg β†’ 𝐹 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  Scalarcsca 17200  Ringcrg 20056  LModclmod 20471  AssAlgcasa 21405
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 5307
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 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-lmod 20473  df-assa 21408
This theorem is referenced by:  assa2ass  21418  asclrhm  21444  assamulgscmlem2  21454  asclmulg  32635
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