Theorem lmodvnegid

Description: Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodvnegid.v	$$⊢V={\mathrm{Base}}_W$$
	lmodvnegid.p	$$⊢\underset{\dot{}}{+}={+}_W$$
	lmodvnegid.z	$$⊢\underset{\dot{}}{0}=0_W$$
	lmodvnegid.n	$$⊢N={inv}_g\left(W\right)$$
Assertion	lmodvnegid	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\right)\to X\underset{\dot{}}{+}N\left(X\right)=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		lmodvnegid.v	$$⊢V={\mathrm{Base}}_W$$
2		lmodvnegid.p	$$⊢\underset{\dot{}}{+}={+}_W$$
3		lmodvnegid.z	$$⊢\underset{\dot{}}{0}=0_W$$
4		lmodvnegid.n	$$⊢N={inv}_g\left(W\right)$$
5		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
6	1 2 3 4	grprinv	$$⊢\left(W\in \mathrm{Grp}\wedge X\in V\right)\to X\underset{\dot{}}{+}N\left(X\right)=\underset{\dot{}}{0}$$
7	5 6	sylan	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\right)\to X\underset{\dot{}}{+}N\left(X\right)=\underset{\dot{}}{0}$$