Metamath Proof Explorer


Theorem recidd

Description: Multiplication of a number and its reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses div1d.1 ( 𝜑𝐴 ∈ ℂ )
reccld.2 ( 𝜑𝐴 ≠ 0 )
Assertion recidd ( 𝜑 → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )

Proof

Step Hyp Ref Expression
1 div1d.1 ( 𝜑𝐴 ∈ ℂ )
2 reccld.2 ( 𝜑𝐴 ≠ 0 )
3 recid ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )
4 1 2 3 syl2anc ( 𝜑 → ( 𝐴 · ( 1 / 𝐴 ) ) = 1 )