Theorem grpass

Description: A group operation is associative. (Contributed by NM, 14-Aug-2011)

	Ref	Expression
Hypotheses	grpcl.b	$$⊢B={\mathrm{Base}}_G$$
	grpcl.p	$$⊢\underset{\dot{}}{+}={+}_G$$
Assertion	grpass	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		grpcl.b	$$⊢B={\mathrm{Base}}_G$$
2		grpcl.p	$$⊢\underset{\dot{}}{+}={+}_G$$
3		grpmnd	$$⊢G\in \mathrm{Grp}\to G\in \mathrm{Mnd}$$
4	1 2	mndass	$$⊢\left(G\in \mathrm{Mnd}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$
5	3 4	sylan	$$⊢\left(G\in \mathrm{Grp}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$