Theorem mndass

Description: A monoid operation is associative. (Contributed by NM, 14-Aug-2011) (Proof shortened by AV, 8-Feb-2020)

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

Proof

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