Metamath Proof Explorer


Theorem divmulzi

Description: Relationship between division and multiplication. (Contributed by NM, 8-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)

Ref Expression
Hypotheses divclz.1 𝐴 ∈ ℂ
divclz.2 𝐵 ∈ ℂ
divmulz.3 𝐶 ∈ ℂ
Assertion divmulzi ( 𝐵 ≠ 0 → ( ( 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 ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) )
5 1 3 4 mp3an12 ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) )
6 2 5 mpan ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) )