Theorem divcan1i

Description: A cancellation law for division. (Contributed by NM, 18-May-1999)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divcl.3	⊢ 𝐵 ≠ 0
Assertion	divcan1i	⊢ ( ( 𝐴 / 𝐵 ) · 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 3	divcli	⊢ ( 𝐴 / 𝐵 ) ∈ ℂ
5	1 2 3	divcan2i	⊢ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴
6	2 4 5	mulcomli	⊢ ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴