Metamath Proof Explorer

Theorem mulcan

Description: Cancellation law for multiplication (full theorem form). Theorem I.7 of Apostol p. 18. (Contributed by NM, 29-Jan-1995) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion mulcan ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→C⁢A=C⁢B↔A=B

Proof

Step Hyp Ref Expression
1 simp1 ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→A∈ℂ
2 simp2 ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→B∈ℂ
3 simp3l ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→C∈ℂ
4 simp3r ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→C≠0
5 1 2 3 4 mulcand ⊢A∈ℂ∧B∈ℂ∧C∈ℂ∧C≠0→C⁢A=C⁢B↔A=B