Theorem divasszi

Description: An associative law for division. (Contributed by NM, 12-Aug-1999)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divmulz.3	⊢ 𝐶 ∈ ℂ
Assertion	divasszi	⊢ ( 𝐶 ≠ 0 → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( 𝐴 · ( 𝐵 / 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divmulz.3	⊢ 𝐶 ∈ ℂ
4		divass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( 𝐴 · ( 𝐵 / 𝐶 ) ) )
5	1 2 4	mp3an12	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( 𝐴 · ( 𝐵 / 𝐶 ) ) )
6	3 5	mpan	⊢ ( 𝐶 ≠ 0 → ( ( 𝐴 · 𝐵 ) / 𝐶 ) = ( 𝐴 · ( 𝐵 / 𝐶 ) ) )