Theorem srgmgp

Description: A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018)

	Ref	Expression
Hypothesis	srgmgp.g	|- G = ( mulGrp ` R )
Assertion	srgmgp	|- ( R e. SRing -> G e. Mnd )

Proof

Step	Hyp	Ref	Expression
1		srgmgp.g	|- G = ( mulGrp ` R )
2		eqid	|- ( Base ` R ) = ( Base ` R )
3		eqid	|- ( +g ` R ) = ( +g ` R )
4		eqid	|- ( .r ` R ) = ( .r ` R )
5		eqid	|- ( 0g ` R ) = ( 0g ` R )
6	2 1 3 4 5	issrg	|- ( R e. SRing <-> ( R e. CMnd /\ G e. Mnd /\ A. x e. ( Base ` R ) ( A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) /\ ( ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) /\ ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) ) ) )
7	6	simp2bi	|- ( R e. SRing -> G e. Mnd )