Theorem mndassd

Description: A monoid operation is associative. (Contributed by Thierry Arnoux, 3-Aug-2025)

	Ref	Expression
Hypotheses	mndassd.1	|- B = ( Base ` G )
	mndassd.2	|- .+ = ( +g ` G )
	mndassd.3	|- ( ph -> G e. Mnd )
	mndassd.4	|- ( ph -> X e. B )
	mndassd.5	|- ( ph -> Y e. B )
	mndassd.6	|- ( ph -> Z e. B )
Assertion	mndassd	|- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		mndassd.1	|- B = ( Base ` G )
2		mndassd.2	|- .+ = ( +g ` G )
3		mndassd.3	|- ( ph -> G e. Mnd )
4		mndassd.4	|- ( ph -> X e. B )
5		mndassd.5	|- ( ph -> Y e. B )
6		mndassd.6	|- ( ph -> Z e. B )
7	1 2	mndass	|- ( ( G e. Mnd /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )
8	3 4 5 6 7	syl13anc	|- ( ph -> ( ( X .+ Y ) .+ Z ) = ( X .+ ( Y .+ Z ) ) )