Theorem divcan3i

Description: A cancellation law for division. (Contributed by NM, 16-Feb-1995)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divcl.3	⊢ 𝐵 ≠ 0
Assertion	divcan3i	⊢ ( ( 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	divcan3zi	⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 )
5	3 4	ax-mp	⊢ ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴