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Metamath Proof Explorer

Theorem mulcan2

Description: Cancellation law for multiplication. (Contributed by NM, 21-Jan-2005) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion mulcan2 ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ ∧ C ≠ 0 → A ⁢ C = B ⁢ C ↔ 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 mulcan2d ⊢ A ∈ ℂ ∧ B ∈ ℂ ∧ C ∈ ℂ ∧ C ≠ 0 → A ⁢ C = B ⁢ C ↔ A = B