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 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		nvgcl.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvgcl.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3	2	nvgrp	⊢ ( 𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp )
4	1 2	bafval	⊢ 𝑋 = ran 𝐺
5	4	grpocl	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )
6	3 5	syl3an1	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) ∈ 𝑋 )