Theorem divcan2i

Description: A cancellation law for division. (Contributed by NM, 9-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divcl.3	⊢ 𝐵 ≠ 0
Assertion	divcan2i	⊢ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divcl.3	⊢ 𝐵 ≠ 0
4	1 2	divcan2zi	⊢ ( 𝐵 ≠ 0 → ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 )
5	3 4	ax-mp	⊢ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴