Theorem nvadd32

Description: Commutative/associative law for vector addition. (Contributed by NM, 27-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	nvadd32	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=\left(AGC\right)GB$$

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		nvgcl.2	$$⊢G={+}_v\left(U\right)$$
3	2	nvablo	$$⊢U\in \mathrm{NrmCVec}\to G\in \mathrm{AbelOp}$$
4	1 2	bafval	$$⊢X=\mathrm{ran}G$$
5	4	ablo32	$$⊢\left(G\in \mathrm{AbelOp}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=\left(AGC\right)GB$$
6	3 5	sylan	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=\left(AGC\right)GB$$