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

Theorem assaring 17837
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assaring  |-  ( W  e. AssAlg  ->  W  e.  Ring )

Proof of Theorem assaring
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2441 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2441 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2441 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2441 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2441 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17832 . . 3  |-  ( W  e. AssAlg 
<->  ( ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing )  /\  A. z  e.  ( Base `  (Scalar `  W )
) A. x  e.  ( Base `  W
) A. y  e.  ( Base `  W
) ( ( ( z ( .s `  W ) x ) ( .r `  W
) y )  =  ( z ( .s
`  W ) ( x ( .r `  W ) y ) )  /\  ( x ( .r `  W
) ( z ( .s `  W ) y ) )  =  ( z ( .s
`  W ) ( x ( .r `  W ) y ) ) ) ) )
76simplbi 460 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
87simp2d 1008 1  |-  ( W  e. AssAlg  ->  W  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   ` cfv 5574  (class class class)co 6277   Basecbs 14504   .rcmulr 14570  Scalarcsca 14572   .scvsca 14573   Ringcrg 17066   CRingccrg 17067   LModclmod 17380  AssAlgcasa 17826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280  df-assa 17829
This theorem is referenced by:  issubassa  17841  assapropd  17844  aspval  17845  asclmul1  17856  asclmul2  17857  asclrhm  17859  rnascl  17860  aspval2  17864  assamulgscmlem1  17865  assamulgscmlem2  17866  mplind  18035  evlseu  18053  pf1subrg  18252  zlmassa  18428  matinv  19046  assaascl0  32689  assaascl1  32690
  Copyright terms: Public domain W3C validator