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Theorem assalmod 17313
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 2433 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2433 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2433 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4 eqid 2433 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2433 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17309 . . 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 457 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  (Scalar `  W )  e.  CRing ) )
87simp1d 993 1  |-  ( W  e. AssAlg  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   ` cfv 5406  (class class class)co 6080   Basecbs 14157   .rcmulr 14222  Scalarcsca 14224   .scvsca 14225   Ringcrg 16577   CRingccrg 16578   LModclmod 16872  AssAlgcasa 17303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-nul 4409
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-ov 6083  df-assa 17306
This theorem is referenced by:  issubassa  17317  assapropd  17320  aspval  17321  asplss  17322  asclrhm  17334  rnascl  17335  issubassa2  17337  aspval2  17339  mplmon2mul  17515  mplind  17516  matinv  18325  assaascl0  30630  assaascl1  30631  assa2ass  30632
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