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 = ( BaseSet ` U )
	nvgcl.2	|- G = ( +v ` U )
Assertion	nvgcl	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G B ) e. X )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	|- X = ( BaseSet ` U )
2		nvgcl.2	|- G = ( +v ` U )
3	2	nvgrp	|- ( U e. NrmCVec -> G e. GrpOp )
4	1 2	bafval	|- X = ran G
5	4	grpocl	|- ( ( G e. GrpOp /\ A e. X /\ B e. X ) -> ( A G B ) e. X )
6	3 5	syl3an1	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G B ) e. X )