Theorem divdiri

Description: Distribution of division over addition. (Contributed by NM, 16-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divmulz.3	⊢ 𝐶 ∈ ℂ
	divass.4	⊢ 𝐶 ≠ 0
Assertion	divdiri	⊢ ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divmulz.3	⊢ 𝐶 ∈ ℂ
4		divass.4	⊢ 𝐶 ≠ 0
5	1 2 3	divdirzi	⊢ ( 𝐶 ≠ 0 → ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) ) )
6	4 5	ax-mp	⊢ ( ( 𝐴 + 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) + ( 𝐵 / 𝐶 ) )