Theorem nvrcan

Description: Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nvgcl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	nvgcl.2	$$⊢G={+}_v\left(U\right)$$
Assertion	nvrcan	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvgcl.2	$$⊢G={+}_v\left(U\right)$$
3	2	nvgrp	$$⊢U\in \mathrm{NrmCVec}\to G\in \mathrm{GrpOp}$$
4	1 2	bafval	$$⊢X=\mathrm{ran}G$$
5	4	grporcan	$$⊢\left(G\in \mathrm{GrpOp}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$
6	3 5	sylan	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$