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Theorem assasca 19369
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
assasca.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
assasca (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)

Proof of Theorem assasca
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 assasca.f . . . 4 𝐹 = (Scalar‘𝑊)
3 eqid 2651 . . . 4 (Base‘𝐹) = (Base‘𝐹)
4 eqid 2651 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2651 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 19363 . . 3 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑧 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠𝑊)𝑥)(.r𝑊)𝑦) = (𝑧( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)) ∧ (𝑥(.r𝑊)(𝑧( ·𝑠𝑊)𝑦)) = (𝑧( ·𝑠𝑊)(𝑥(.r𝑊)𝑦)))))
76simplbi 475 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
87simp3d 1095 1 (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  Ringcrg 18593  CRingccrg 18594  LModclmod 18911  AssAlgcasa 19357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-assa 19360
This theorem is referenced by:  assa2ass  19370  issubassa  19372  asclrhm  19390  assamulgscmlem2  19397
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