Theorem lmod0vid

Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	0vlid.v	$$⊢V={\mathrm{Base}}_W$$
	0vlid.a	$$⊢\underset{\dot{}}{+}={+}_W$$
	0vlid.z	$$⊢\underset{\dot{}}{0}=0_W$$
Assertion	lmod0vid	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\right)\to \left(X\underset{\dot{}}{+}X=X\leftrightarrow \underset{\dot{}}{0}=X\right)$$

Proof

Step	Hyp	Ref	Expression
1		0vlid.v	$$⊢V={\mathrm{Base}}_W$$
2		0vlid.a	$$⊢\underset{\dot{}}{+}={+}_W$$
3		0vlid.z	$$⊢\underset{\dot{}}{0}=0_W$$
4		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
5	1 2 3	grpid	$$⊢\left(W\in \mathrm{Grp}\wedge X\in V\right)\to \left(X\underset{\dot{}}{+}X=X\leftrightarrow \underset{\dot{}}{0}=X\right)$$
6	4 5	sylan	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\right)\to \left(X\underset{\dot{}}{+}X=X\leftrightarrow \underset{\dot{}}{0}=X\right)$$