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Theorem assaring 19260
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 2621 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2621 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
4 eqid 2621 . . . 4 ( ·𝑠𝑊) = ( ·𝑠𝑊)
5 eqid 2621 . . . 4 (.r𝑊) = (.r𝑊)
61, 2, 3, 4, 5isassa 19255 . . 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 476 . 2 (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing))
87simp2d 1072 1 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  cfv 5857  (class class class)co 6615  Basecbs 15800  .rcmulr 15882  Scalarcsca 15884   ·𝑠 cvsca 15885  Ringcrg 18487  CRingccrg 18488  LModclmod 18803  AssAlgcasa 19249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-assa 19252
This theorem is referenced by:  issubassa  19264  assapropd  19267  aspval  19268  asclmul1  19279  asclmul2  19280  asclrhm  19282  rnascl  19283  aspval2  19287  assamulgscmlem1  19288  assamulgscmlem2  19289  mplind  19442  evlseu  19456  pf1subrg  19652  zlmassa  19812  matinv  20423  assaascl0  41485  assaascl1  41486
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