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 | hldir | ⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐵 ) 𝑆 𝐶 ) = ( ( 𝐴 𝑆 𝐶 ) 𝐺 ( 𝐵 𝑆 𝐶 ) ) ) |
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 | nvdir | ⊢ ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐵 ) 𝑆 𝐶 ) = ( ( 𝐴 𝑆 𝐶 ) 𝐺 ( 𝐵 𝑆 𝐶 ) ) ) |
6 | 4 5 | sylan | ⊢ ( ( 𝑈 ∈ CHilOLD ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐴 + 𝐵 ) 𝑆 𝐶 ) = ( ( 𝐴 𝑆 𝐶 ) 𝐺 ( 𝐵 𝑆 𝐶 ) ) ) |