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Metamath Proof Explorer

Theorem divmuli

Description: Relationship between division and multiplication. (Contributed by NM, 2-Feb-1995) (Revised by Mario Carneiro, 17-Feb-2014)

Ref Expression
Hypotheses divclz.1 ⊢ A ∈ ℂ
divclz.2 ⊢ B ∈ ℂ
divmulz.3 ⊢ C ∈ ℂ
divmul.4 ⊢ B ≠ 0
Assertion divmuli ⊢ A B = C ↔ B ⁢ C = A

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ A ∈ ℂ
2 divclz.2 ⊢ B ∈ ℂ
3 divmulz.3 ⊢ C ∈ ℂ
4 divmul.4 ⊢ B ≠ 0
5 1 2 3 divmulzi ⊢ B ≠ 0 → A B = C ↔ B ⁢ C = A
6 4 5 ax-mp ⊢ A B = C ↔ B ⁢ C = A