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Theorem assasca 17392
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypothesis
Ref Expression
assasca.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
assasca  |-  ( W  e. AssAlg  ->  F  e.  CRing )

Proof of Theorem assasca
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2442 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 assasca.f . . . 4  |-  F  =  (Scalar `  W )
3 eqid 2442 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
4 eqid 2442 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2442 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17386 . . 3  |-  ( W  e. AssAlg 
<->  ( ( W  e. 
LMod  /\  W  e.  Ring  /\  F  e.  CRing )  /\  A. z  e.  ( Base `  F ) 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  /\  F  e.  CRing ) )
87simp3d 1002 1  |-  ( W  e. AssAlg  ->  F  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   ` cfv 5417  (class class class)co 6090   Basecbs 14173   .rcmulr 14238  Scalarcsca 14240   .scvsca 14241   Ringcrg 16644   CRingccrg 16645   LModclmod 16947  AssAlgcasa 17380
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 4420
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093  df-assa 17383
This theorem is referenced by:  issubassa  17394  asclrhm  17411  assa2ass  30817  assamulgscmlem2  30819
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