Theorem nvdir

Description: Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		nvdi.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		nvdi.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		nvdi.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		eqid	⊢ ( 1st ‘ 𝑈 ) = ( 1st ‘ 𝑈 )
5	4	nvvc	⊢ ( 𝑈 ∈ NrmCVec → ( 1st ‘ 𝑈 ) ∈ CVecOLD )
6	2	vafval	⊢ 𝐺 = ( 1st ‘ ( 1st ‘ 𝑈 ) )
7	3	smfval	⊢ 𝑆 = ( 2nd ‘ ( 1st ‘ 𝑈 ) )
8	1 2	bafval	⊢ 𝑋 = ran 𝐺
9	6 7 8	vcdir	⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐵 ) 𝑆 𝐶 ) = ( ( 𝐴 𝑆 𝐶 ) 𝐺 ( 𝐵 𝑆 𝐶 ) ) )
10	5 9	sylan	⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐵 ) 𝑆 𝐶 ) = ( ( 𝐴 𝑆 𝐶 ) 𝐺 ( 𝐵 𝑆 𝐶 ) ) )