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 )