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 ) ) ) |
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 ) ) ) |