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Theorem assarng 16077
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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2296 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2296 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 16072 . . 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 446 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
87simp2d 968 1  |-  ( W  e. AssAlg  ->  W  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228   Ringcrg 15353   CRingccrg 15354   LModclmod 15643  AssAlgcasa 16066
This theorem is referenced by:  issubassa  16080  assapropd  16083  aspval  16084  asclmul1  16095  asclmul2  16096  asclrhm  16097  rnascl  16098  aspval2  16102  mplind  16259  zlmassa  16494  evlseu  19416  pf1subrg  19447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-assa 16069
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