Metamath Proof Explorer


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