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Theorem assalmod 17394
Description: An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.)
Assertion
Ref Expression
assalmod  |-  ( W  e. AssAlg  ->  W  e.  LMod )

Proof of Theorem assalmod
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 17390 . . 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 ) )
87simp1d 1000 1  |-  ( W  e. AssAlg  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2718   ` cfv 5421  (class class class)co 6094   Basecbs 14177   .rcmulr 14242  Scalarcsca 14244   .scvsca 14245   Ringcrg 16648   CRingccrg 16649   LModclmod 16951  AssAlgcasa 17384
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 4424
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 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-iota 5384  df-fv 5429  df-ov 6097  df-assa 17387
This theorem is referenced by:  issubassa  17398  assapropd  17401  aspval  17402  asplss  17403  asclrhm  17415  rnascl  17416  issubassa2  17418  aspval2  17420  mplmon2mul  17586  mplind  17587  matinv  18486  assaascl0  30820  assaascl1  30821  assa2ass  30822  assamulgscmlem1  30823  assamulgscmlem2  30824
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