Metamath Proof Explorer


Theorem rec11d

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

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

Proof

Step Hyp Ref Expression
1 div1d.1 ( 𝜑𝐴 ∈ ℂ )
2 divcld.2 ( 𝜑𝐵 ∈ ℂ )
3 divne0d.3 ( 𝜑𝐴 ≠ 0 )
4 divne0d.4 ( 𝜑𝐵 ≠ 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 ( 𝜑𝐴 = 𝐵 )