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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 )