Theorem divcan2zi

Description: A cancellation law for division. (Contributed by NM, 10-Aug-1999)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
Assertion	divcan2zi	⊢ ( 𝐵 ≠ 0 → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )
4	1 2 3	mp3an12	⊢ ( 𝐵 ≠ 0 → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )