Theorem hlipdir

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

	Ref	Expression
Hypotheses	hlipdir.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlipdir.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	hlipdir.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	hlipdir	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) = ( ( 𝐴 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlipdir.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlipdir.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		hlipdir.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
4		hlph	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD )
5	1 2 3	dipdir	⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) = ( ( 𝐴 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐶 ) ) )
6	4 5	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) = ( ( 𝐴 𝑃 𝐶 ) + ( 𝐵 𝑃 𝐶 ) ) )