MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  assaring Structured version   Visualization version   GIF version

Theorem assaring 21904
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 2740 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2740 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2740 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4 eqid 2740 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2740 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 21899 . . 3 (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) ∧ ∀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 497 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring))
87simprd 495 1 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wral 3067  cfv 6573  (class class class)co 7448  Basecbs 17258  .rcmulr 17312  Scalarcsca 17314   ·𝑠 cvsca 17315  Ringcrg 20260  LModclmod 20880  AssAlgcasa 21893
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 2110  ax-9 2118  ax-ext 2711  ax-nul 5324
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-assa 21896
This theorem is referenced by:  issubassa  21910  assapropd  21915  aspval  21916  asclmul1  21929  asclmul2  21930  ascldimul  21931  asclrhm  21933  rnascl  21934  aspval2  21941  assamulgscmlem1  21942  assamulgscmlem2  21943  asclmulg  21945  zlmassa  21946  mplind  22117  evlseu  22130  pf1subrg  22373  matinv  22704  lactlmhm  33647  assalactf1o  33648  assarrginv  33649  irngnzply1lem  33690  assaascl0  48109  assaascl1  48110
  Copyright terms: Public domain W3C validator