Metamath Proof Explorer

Theorem assaring

Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014)

Ref Expression
Assertion assaring 
|- ( W e. AssAlg -> W e. Ring )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` W ) = ( Base ` W )
2 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
3 eqid 
 |-  ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) )
4 eqid 
 |-  ( .s ` W ) = ( .s ` W )
5 eqid 
 |-  ( .r ` W ) = ( .r ` W )
6 1 2 3 4 5 isassa 
 |-  ( 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 ) ) ) ) )
7 6 simplbi 
 |-  ( W e. AssAlg -> ( W e. LMod /\ W e. Ring /\ ( Scalar ` W ) e. CRing ) )
8 7 simp2d 
 |-  ( W e. AssAlg -> W e. Ring )