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Theorem assasca 21097
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 2733 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 assasca.f . . . 4 𝐹 = (Scalar‘𝑊)
3 eqid 2733 . . . 4 (Base‘𝐹) = (Base‘𝐹)
4 eqid 2733 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2733 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 21091 . . 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 497 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing))
87simp3d 1142 1 (𝑊 ∈ AssAlg → 𝐹 ∈ CRing)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1537  wcel 2101  wral 3059  cfv 6447  (class class class)co 7295  Basecbs 16940  .rcmulr 16991  Scalarcsca 16993   ·𝑠 cvsca 16994  Ringcrg 19811  CRingccrg 19812  LModclmod 20151  AssAlgcasa 21085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-nul 5233
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2939  df-ral 3060  df-rab 3224  df-v 3436  df-sbc 3719  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-iota 6399  df-fv 6455  df-ov 7298  df-assa 21088
This theorem is referenced by:  assa2ass  21098  issubassa3  21100  asclrhm  21122  assamulgscmlem2  21132  asclmulg  31694
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