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Theorem assasca 16111
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 2316 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 assasca.f . . . 4  |-  F  =  (Scalar `  W )
3 eqid 2316 . . . 4  |-  ( Base `  F )  =  (
Base `  F )
4 eqid 2316 . . . 4  |-  ( .s
`  W )  =  ( .s `  W
)
5 eqid 2316 . . . 4  |-  ( .r
`  W )  =  ( .r `  W
)
61, 2, 3, 4, 5isassa 16105 . . 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 446 . 2  |-  ( W  e. AssAlg  ->  ( W  e. 
LMod  /\  W  e.  Ring  /\  F  e.  CRing ) )
87simp3d 969 1  |-  ( W  e. AssAlg  ->  F  e.  CRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   A.wral 2577   ` cfv 5292  (class class class)co 5900   Basecbs 13195   .rcmulr 13256  Scalarcsca 13258   .scvsca 13259   Ringcrg 15386   CRingccrg 15387   LModclmod 15676  AssAlgcasa 16099
This theorem is referenced by:  issubassa  16113  asclrhm  16130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903  df-assa 16102
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