Theorem recrecd

Description: A number is equal to the reciprocal of its reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

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

Proof

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