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Theorem assasca 21636
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 21634 . 2 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
2 assasca.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
32lmodring 20622 . 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 1539   ∈ wcel 2104  β€˜cfv 6542  Scalarcsca 17204  Ringcrg 20127  LModclmod 20614  AssAlgcasa 21624
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-lmod 20616  df-assa 21627
This theorem is referenced by:  assa2ass  21637  asclrhm  21663  assamulgscmlem2  21673  asclmulg  32909
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