Metamath Proof Explorer


Theorem recdivd

Description: The reciprocal of a ratio. (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 )
Assertion recdivd ( 𝜑 → ( 1 / ( 𝐴 / 𝐵 ) ) = ( 𝐵 / 𝐴 ) )

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 recdiv ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( 1 / ( 𝐴 / 𝐵 ) ) = ( 𝐵 / 𝐴 ) )
6 1 3 2 4 5 syl22anc ( 𝜑 → ( 1 / ( 𝐴 / 𝐵 ) ) = ( 𝐵 / 𝐴 ) )