Metamath Proof Explorer


Theorem subdiri

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 ( ( 𝐴𝐵 ) · 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 ) → ( ( 𝐴𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) ) )
5 1 2 3 4 mp3an ( ( 𝐴𝐵 ) · 𝐶 ) = ( ( 𝐴 · 𝐶 ) − ( 𝐵 · 𝐶 ) )