Theorem rec11d

Description: Reciprocal is one-to-one. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	divne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	divne0d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
	rec11d.5	⊢ ( 𝜑 → ( 1 / 𝐴 ) = ( 1 / 𝐵 ) )
Assertion	rec11d	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		div1d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		divcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		divne0d.3	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		divne0d.4	⊢ ( 𝜑 → 𝐵 ≠ 0 )
5		rec11d.5	⊢ ( 𝜑 → ( 1 / 𝐴 ) = ( 1 / 𝐵 ) )
6		rec11	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7	1 3 2 4 6	syl22anc	⊢ ( 𝜑 → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )
8	5 7	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )