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Theorem assarng 17370
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assarng  |-  ( W  e. AssAlg  ->  W  e.  Ring )

Proof of Theorem assarng
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 17365 . . 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 457 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
87simp2d 996 1  |-  ( W  e. AssAlg  ->  W  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   ` cfv 5415  (class class class)co 6090   Basecbs 14170   .rcmulr 14235  Scalarcsca 14237   .scvsca 14238   Ringcrg 16635   CRingccrg 16636   LModclmod 16928  AssAlgcasa 17359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-nul 4418
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-assa 17362
This theorem is referenced by:  issubassa  17373  assapropd  17376  aspval  17377  asclmul1  17388  asclmul2  17389  asclrhm  17390  rnascl  17391  aspval2  17395  mplind  17560  evlseu  17578  pf1subrg  17751  zlmassa  17914  matinv  18442  assaascl0  30718  assaascl1  30719  assamulgscmlem1  30721  assamulgscmlem2  30722
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