Theorem hlipass

Description: Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlipass.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlipass.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	hlipass.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	hlipass	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑆 𝐵 ) 𝑃 𝐶 ) = ( 𝐴 · ( 𝐵 𝑃 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlipass.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlipass.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
3		hlipass.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4		hlph	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD )
5	1 2 3	dipass	⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑆 𝐵 ) 𝑃 𝐶 ) = ( 𝐴 · ( 𝐵 𝑃 𝐶 ) ) )
6	4 5	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝑆 𝐵 ) 𝑃 𝐶 ) = ( 𝐴 · ( 𝐵 𝑃 𝐶 ) ) )