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Theorem assalmod 18290
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 18286 . . 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 1011 1  |-  ( W  e. AssAlg  ->  W  e.  LMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   ` cfv 5571  (class class class)co 6280   Basecbs 14843   .rcmulr 14912  Scalarcsca 14914   .scvsca 14915   Ringcrg 17520   CRingccrg 17521   LModclmod 17834  AssAlgcasa 18280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-iota 5535  df-fv 5579  df-ov 6283  df-assa 18283
This theorem is referenced by:  assa2ass  18293  issubassa  18295  assapropd  18298  aspval  18299  asplss  18300  asclrhm  18313  rnascl  18314  issubassa2  18316  aspval2  18318  assamulgscmlem1  18319  assamulgscmlem2  18320  mplmon2mul  18488  mplind  18489  matinv  19473  assaascl0  38503  assaascl1  38504
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