Theorem hlass

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

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

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	nvass	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) )
5	3 4	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝐺 𝐶 ) = ( 𝐴 𝐺 ( 𝐵 𝐺 𝐶 ) ) )