Theorem divmul2

Description: Relationship between division and multiplication. (Contributed by NM, 7-Feb-2006)

	Ref	Expression
Assertion	divmul2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		divmul	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ ( 𝐶 · 𝐵 ) = 𝐴 ) )
2		eqcom	⊢ ( ( 𝐶 · 𝐵 ) = 𝐴 ↔ 𝐴 = ( 𝐶 · 𝐵 ) )
3	1 2	bitrdi	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) )