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Metamath Proof Explorer

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 )