Theorem slmdass

Description: Semiring left module vector sum is associative. (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmdvacl.v	$$⊢V={\mathrm{Base}}_W$$
	slmdvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
Assertion	slmdass	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(X\in V\wedge Y\in V\wedge Z\in V\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		slmdvacl.v	$$⊢V={\mathrm{Base}}_W$$
2		slmdvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
3		slmdmnd	$$⊢W\in \mathrm{SLMod}\to W\in \mathrm{Mnd}$$
4	1 2	mndass	$$⊢\left(W\in \mathrm{Mnd}\wedge \left(X\in V\wedge Y\in V\wedge Z\in V\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$
5	3 4	sylan	$$⊢\left(W\in \mathrm{SLMod}\wedge \left(X\in V\wedge Y\in V\wedge Z\in V\right)\right)\to \left(X\underset{\dot{}}{+}Y\right)\underset{\dot{}}{+}Z=X\underset{\dot{}}{+}\left(Y\underset{\dot{}}{+}Z\right)$$