Theorem cmn12

Description: Commutative/associative law for commutative monoids. (Contributed by Stefan O'Rear, 5-Sep-2015) (Revised by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	ablcom.b	|- B = ( Base ` G )
	ablcom.p	|- .+ = ( +g ` G )
Assertion	cmn12	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		ablcom.b	|- B = ( Base ` G )
2		ablcom.p	|- .+ = ( +g ` G )
3		cmnmnd	|- ( G e. CMnd -> G e. Mnd )
4	3	adantr	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> G e. Mnd )
5		simpr1	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )
6		simpr2	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )
7		simpr3	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )
8	1 2	cmncom	|- ( ( G e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
9	8	3adant3r3	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ Y ) = ( Y .+ X ) )
10	1 2 4 5 6 7 9	mnd12g	|- ( ( G e. CMnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .+ ( Y .+ Z ) ) = ( Y .+ ( X .+ Z ) ) )