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Metamath Proof Explorer

Theorem recreci

Description: A number is equal to the reciprocal of its reciprocal. Theorem I.10 of Apostol p. 18. (Contributed by NM, 9-Feb-1995)

Ref Expression
Hypotheses divclz.1 ⊢ 𝐴 ∈ ℂ
reccl.2 ⊢ 𝐴 ≠ 0
Assertion recreci ( 1 / ( 1 / 𝐴 ) ) = 𝐴

Proof

Step Hyp Ref Expression
1 divclz.1 ⊢ 𝐴 ∈ ℂ
2 reccl.2 ⊢ 𝐴 ≠ 0
3 recrec ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( 1 / 𝐴 ) ) = 𝐴 )
4 1 2 3 mp2an ⊢ ( 1 / ( 1 / 𝐴 ) ) = 𝐴