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Theorem assaring 20554
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assaring (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)

Proof of Theorem assaring
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2801 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2801 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4 eqid 2801 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2801 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 20549 . . 3 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀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 ∧ (Scalar‘𝑊) ∈ CRing))
87simp2d 1140 1 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cfv 6328  (class class class)co 7139  Basecbs 16479  .rcmulr 16562  Scalarcsca 16564   ·𝑠 cvsca 16565  Ringcrg 19294  CRingccrg 19295  LModclmod 19631  AssAlgcasa 20543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-assa 20546
This theorem is referenced by:  issubassa  20559  assapropd  20562  aspval  20563  asclmul1  20575  asclmul2  20576  ascldimul  20577  ascldimulOLD  20578  asclrhm  20580  rnascl  20581  aspval2  20588  assamulgscmlem1  20589  assamulgscmlem2  20590  zlmassa  20592  mplind  20745  evlseu  20759  pf1subrg  20976  matinv  21286  selvval2lem4  39428  assaascl0  44784  assaascl1  44785
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