Theorem nvgcl

Description: Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007) (New usage is discouraged.)

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

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	grpocl	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\wedge B\in X\right)\to AGB\in X$$
6	3 5	syl3an1	$$⊢\left(U\in \mathrm{NrmCVec}\wedge A\in X\wedge B\in X\right)\to AGB\in X$$