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Theorem assalmod 16334
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 2404 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2404 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2404 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2404 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2404 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 16330 . . 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 447 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
87simp1d 969 1  |-  ( W  e. AssAlg  ->  W  e.  LMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   Ringcrg 15615   CRingccrg 15616   LModclmod 15905  AssAlgcasa 16324
This theorem is referenced by:  issubassa  16338  assapropd  16341  aspval  16342  asplss  16343  asclrhm  16355  rnascl  16356  issubassa2  16358  aspval2  16360  mplmon2mul  16516  mplind  16517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-assa 16327
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