Theorem hlcom

Description: Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hladdf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hladdf.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
Assertion	hlcom	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		hladdf.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hladdf.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
4	1 2	nvcom	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐴 ) )
5	3 4	syl3an1	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 𝐵 ) = ( 𝐵 𝐺 𝐴 ) )