Theorem divcan1zi

Description: A cancellation law for division. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
Assertion	divcan1zi	⊢ ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divcan1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 )
4	1 2 3	mp3an12	⊢ ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 )