Metamath Proof Explorer
Description: Distribution of multiplication over subtraction. Theorem I.5 of
Apostol p. 18. (Contributed by NM, 8-May-1999)
|
|
Ref |
Expression |
|
Hypotheses |
mulm1.1 |
⊢ 𝐴 ∈ ℂ |
|
|
mulneg.2 |
⊢ 𝐵 ∈ ℂ |
|
|
subdi.3 |
⊢ 𝐶 ∈ ℂ |
|
Assertion |
subdiri |
⊢ ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mulm1.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
mulneg.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
subdi.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
subdir |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) ) |
5 |
1 2 3 4
|
mp3an |
⊢ ( ( 𝐴 − 𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) |