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Theorem assarng 17397
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 2443 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2443 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2443 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2443 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2443 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17392 . . 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 1001 1  |-  ( W  e. AssAlg  ->  W  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   ` cfv 5423  (class class class)co 6096   Basecbs 14179   .rcmulr 14244  Scalarcsca 14246   .scvsca 14247   Ringcrg 16650   CRingccrg 16651   LModclmod 16953  AssAlgcasa 17386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-nul 4426
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-iota 5386  df-fv 5431  df-ov 6099  df-assa 17389
This theorem is referenced by:  issubassa  17400  assapropd  17403  aspval  17404  asclmul1  17415  asclmul2  17416  asclrhm  17417  rnascl  17418  aspval2  17422  mplind  17589  evlseu  17607  pf1subrg  17787  zlmassa  17960  matinv  18488  assaascl0  30822  assaascl1  30823  assamulgscmlem1  30825  assamulgscmlem2  30826
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