Theorem subdid

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulm1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	subdid	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		mulnegd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subdid.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		subdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 · ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) − ( 𝐴 · 𝐶 ) ) )