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Metamath Proof Explorer

Theorem slmdcmn

Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Assertion slmdcmn 
|- ( W e. SLMod -> W e. CMnd )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` W ) = ( Base ` W )
2 eqid 
 |-  ( +g ` W ) = ( +g ` W )
3 eqid 
 |-  ( .s ` W ) = ( .s ` W )
4 eqid 
 |-  ( 0g ` W ) = ( 0g ` W )
5 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid 
 |-  ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) )
7 eqid 
 |-  ( +g ` ( Scalar ` W ) ) = ( +g ` ( Scalar ` W ) )
8 eqid 
 |-  ( .r ` ( Scalar ` W ) ) = ( .r ` ( Scalar ` W ) )
9 eqid 
 |-  ( 1r ` ( Scalar ` W ) ) = ( 1r ` ( Scalar ` W ) )
10 eqid 
 |-  ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
11 1 2 3 4 5 6 7 8 9 10 isslmd 
 |-  ( W e. SLMod <-> ( W e. CMnd /\ ( Scalar ` W ) e. SRing /\ A. w e. ( Base ` ( Scalar ` W ) ) A. z e. ( Base ` ( Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( ( ( z ( .s ` W ) y ) e. ( Base ` W ) /\ ( z ( .s ` W ) ( y ( +g ` W ) x ) ) = ( ( z ( .s ` W ) y ) ( +g ` W ) ( z ( .s ` W ) x ) ) /\ ( ( w ( +g ` ( Scalar ` W ) ) z ) ( .s ` W ) y ) = ( ( w ( .s ` W ) y ) ( +g ` W ) ( z ( .s ` W ) y ) ) ) /\ ( ( ( w ( .r ` ( Scalar ` W ) ) z ) ( .s ` W ) y ) = ( w ( .s ` W ) ( z ( .s ` W ) y ) ) /\ ( ( 1r ` ( Scalar ` W ) ) ( .s ` W ) y ) = y /\ ( ( 0g ` ( Scalar ` W ) ) ( .s ` W ) y ) = ( 0g ` W ) ) ) ) )
12 11 simp1bi 
 |-  ( W e. SLMod -> W e. CMnd )