Theorem rec11ii

Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999)

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
	divneq0.3	⊢ 𝐴 ≠ 0
	divneq0.4	⊢ 𝐵 ≠ 0
Assertion	rec11ii	⊢ ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		divneq0.3	⊢ 𝐴 ≠ 0
4		divneq0.4	⊢ 𝐵 ≠ 0
5	1 2	rec11i	⊢ ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )
6	3 4 5	mp2an	⊢ ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 )