Theorem rec11i

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

	Ref	Expression
Hypotheses	divclz.1	⊢ 𝐴 ∈ ℂ
	divclz.2	⊢ 𝐵 ∈ ℂ
Assertion	rec11i	⊢ ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		divclz.1	⊢ 𝐴 ∈ ℂ
2		divclz.2	⊢ 𝐵 ∈ ℂ
3		rec11	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )
4	3	an4s	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	1 2 4	mpanl12	⊢ ( ( 𝐴 ≠ 0 ∧ 𝐵 ≠ 0 ) → ( ( 1 / 𝐴 ) = ( 1 / 𝐵 ) ↔ 𝐴 = 𝐵 ) )