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Theorem assalmod 16379
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 2436 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2436 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2436 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2436 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2436 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 16375 . . 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 1652    e. wcel 1725   A.wral 2705   ` cfv 5454  (class class class)co 6081   Basecbs 13469   .rcmulr 13530  Scalarcsca 13532   .scvsca 13533   Ringcrg 15660   CRingccrg 15661   LModclmod 15950  AssAlgcasa 16369
This theorem is referenced by:  issubassa  16383  assapropd  16386  aspval  16387  asplss  16388  asclrhm  16400  rnascl  16401  issubassa2  16403  aspval2  16405  mplmon2mul  16561  mplind  16562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-assa 16372
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