Theorem lmodlcan

Description: Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

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

Proof

Step	Hyp	Ref	Expression
1		lmodvacl.v	$$⊢V={\mathrm{Base}}_W$$
2		lmodvacl.a	$$⊢\underset{\dot{}}{+}={+}_W$$
3		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
4	1 2	grplcan	$$⊢\left(W\in \mathrm{Grp}\wedge \left(X\in V\wedge Y\in V\wedge Z\in V\right)\right)\to \left(Z\underset{\dot{}}{+}X=Z\underset{\dot{}}{+}Y\leftrightarrow X=Y\right)$$
5	3 4	sylan	$$⊢\left(W\in \mathrm{LMod}\wedge \left(X\in V\wedge Y\in V\wedge Z\in V\right)\right)\to \left(Z\underset{\dot{}}{+}X=Z\underset{\dot{}}{+}Y\leftrightarrow X=Y\right)$$