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Theorem assasca 17838
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 2441 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 assasca.f . . . 4  |-  F  =  (Scalar `  W )
3 eqid 2441 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
4 eqid 2441 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2441 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 17832 . . 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 1009 1  |-  ( W  e. AssAlg  ->  F  e.  CRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   ` cfv 5574  (class class class)co 6277   Basecbs 14504   .rcmulr 14570  Scalarcsca 14572   .scvsca 14573   Ringcrg 17066   CRingccrg 17067   LModclmod 17380  AssAlgcasa 17826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280  df-assa 17829
This theorem is referenced by:  assa2ass  17839  issubassa  17841  asclrhm  17859  assamulgscmlem2  17866
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