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Theorem assarng 17740
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 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2467 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2467 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2467 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2467 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17735 . . 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 1009 1  |-  ( W  e. AssAlg  ->  W  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552  Scalarcsca 14554   .scvsca 14555   Ringcrg 16986   CRingccrg 16987   LModclmod 17295  AssAlgcasa 17729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5549  df-fv 5594  df-ov 6285  df-assa 17732
This theorem is referenced by:  issubassa  17744  assapropd  17747  aspval  17748  asclmul1  17759  asclmul2  17760  asclrhm  17762  rnascl  17763  aspval2  17767  assamulgscmlem1  17768  assamulgscmlem2  17769  mplind  17938  evlseu  17956  pf1subrg  18155  zlmassa  18328  matinv  18946  assaascl0  32052  assaascl1  32053
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