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	⊢ ( ( 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 ) )