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Theorem assasca 20691
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 2739 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 assasca.f . . . 4 𝐹 = (Scalar‘𝑊)
3 eqid 2739 . . . 4 (Base‘𝐹) = (Base‘𝐹)
4 eqid 2739 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2739 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 20685 . . 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 501 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
87simp3d 1145 1 (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  cfv 6350  (class class class)co 7183  Basecbs 16599  .rcmulr 16682  Scalarcsca 16684   ·𝑠 cvsca 16685  Ringcrg 19429  CRingccrg 19430  LModclmod 19766  AssAlgcasa 20679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6308  df-fv 6358  df-ov 7186  df-assa 20682
This theorem is referenced by:  assa2ass  20692  issubassa3  20694  ascldimulOLD  20715  asclrhm  20717  assamulgscmlem2  20727  asclmulg  31251
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