Theorem hldi

Description: Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hldi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hldi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	hldi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
Assertion	hldi	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		hldi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hldi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		hldi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		hlnv	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec )
5	1 2 3	nvdi	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) )
6	4 5	sylan	⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) )