Theorem subdii

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 26-Nov-1994)

	Ref	Expression
Hypotheses	mulm1.1	⊢ 𝐴 ∈ ℂ
	mulneg.2	⊢ 𝐵 ∈ ℂ
	subdi.3	⊢ 𝐶 ∈ ℂ
Assertion	subdii	⊢ ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	⊢ 𝐴 ∈ ℂ
2		mulneg.2	⊢ 𝐵 ∈ ℂ
3		subdi.3	⊢ 𝐶 ∈ ℂ
4		subdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )
5	1 2 3 4	mp3an	⊢ ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) )