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Theorem assasca 21284
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
assasca.f 𝐹 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
assasca (π‘Š ∈ AssAlg β†’ 𝐹 ∈ CRing)

Proof of Theorem assasca
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2 assasca.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
3 eqid 2733 . . . 4 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
4 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
5 eqid 2733 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
61, 2, 3, 4, 5isassa 21278 . . 3 (π‘Š ∈ AssAlg ↔ ((π‘Š ∈ LMod ∧ π‘Š ∈ Ring ∧ 𝐹 ∈ CRing) ∧ βˆ€π‘§ ∈ (Baseβ€˜πΉ)βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)βˆ€π‘¦ ∈ (Baseβ€˜π‘Š)(((𝑧( ·𝑠 β€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑦) = (𝑧( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)) ∧ (π‘₯(.rβ€˜π‘Š)(𝑧( ·𝑠 β€˜π‘Š)𝑦)) = (𝑧( ·𝑠 β€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑦)))))
76simplbi 499 . 2 (π‘Š ∈ AssAlg β†’ (π‘Š ∈ LMod ∧ π‘Š ∈ Ring ∧ 𝐹 ∈ CRing))
87simp3d 1145 1 (π‘Š ∈ AssAlg β†’ 𝐹 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Ringcrg 19969  CRingccrg 19970  LModclmod 20336  AssAlgcasa 21272
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 5264
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 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-assa 21275
This theorem is referenced by:  assa2ass  21285  issubassa3  21287  asclrhm  21309  assamulgscmlem2  21319  asclmulg  32311
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