Metamath Proof Explorer
Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypotheses |
hvdistr1.1 |
⊢ 𝐴 ∈ ℂ |
|
|
hvdistr1.2 |
⊢ 𝐵 ∈ ℋ |
|
|
hvdistr1.3 |
⊢ 𝐶 ∈ ℋ |
|
Assertion |
hvsubdistr1i |
⊢ ( 𝐴 ·ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) −ℎ ( 𝐴 ·ℎ 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
hvdistr1.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
hvdistr1.2 |
⊢ 𝐵 ∈ ℋ |
3 |
|
hvdistr1.3 |
⊢ 𝐶 ∈ ℋ |
4 |
|
hvsubdistr1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝐴 ·ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) −ℎ ( 𝐴 ·ℎ 𝐶 ) ) ) |
5 |
1 2 3 4
|
mp3an |
⊢ ( 𝐴 ·ℎ ( 𝐵 −ℎ 𝐶 ) ) = ( ( 𝐴 ·ℎ 𝐵 ) −ℎ ( 𝐴 ·ℎ 𝐶 ) ) |