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Metamath Proof Explorer

Theorem nvdi

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 nvdi ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 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 vcdi ⊢ ( ( ( 1st ‘ 𝑈 ) ∈ CVecOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) )
10 5 9 sylan ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( 𝐴 𝑆 ( 𝐵 𝐺 𝐶 ) ) = ( ( 𝐴 𝑆 𝐵 ) 𝐺 ( 𝐴 𝑆 𝐶 ) ) )