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

Theorem assaring 20021
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 2818 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2818 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2818 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4 eqid 2818 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2818 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 20016 . . 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 498 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing))
87simp2d 1135 1 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  Ringcrg 19226  CRingccrg 19227  LModclmod 19563  AssAlgcasa 20010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-assa 20013
This theorem is referenced by:  issubassa  20026  assapropd  20029  aspval  20030  asclmul1  20042  asclmul2  20043  ascldimul  20044  ascldimulOLD  20045  asclrhm  20047  rnascl  20048  aspval2  20055  assamulgscmlem1  20056  assamulgscmlem2  20057  mplind  20210  evlseu  20224  pf1subrg  20439  zlmassa  20599  matinv  21214  selvval2lem4  39014  assaascl0  44361  assaascl1  44362
  Copyright terms: Public domain W3C validator