Theorem div11d

Description: One-to-one relationship for division. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	divassd.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
	div11d.5	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) )
Assertion	div11d	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divmuld.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		divassd.4	⊢ ( 𝜑 → 𝐶 ≠ 0 )
5		div11d.5	⊢ ( 𝜑 → ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) )
6		div11	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) )
7	1 2 3 4 6	syl112anc	⊢ ( 𝜑 → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) )
8	5 7	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )