Theorem rerecrecd

Description: A number is equal to the reciprocal of its reciprocal. (Contributed by SN, 2-Apr-2026)

	Ref	Expression
Hypotheses	sn-rereccld.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sn-rereccld.z	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	rerecrecd	⊢ ( 𝜑 → ( 1 /ℝ ( 1 /ℝ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sn-rereccld.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sn-rereccld.z	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3	1 2	rerecid2d	⊢ ( 𝜑 → ( ( 1 /ℝ 𝐴 ) · 𝐴 ) = 1 )
4		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
5	1 2	sn-rereccld	⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ∈ ℝ )
6	1 2	rerecne0d	⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ≠ 0 )
7	4 1 5 6	redivmuld	⊢ ( 𝜑 → ( ( 1 /ℝ ( 1 /ℝ 𝐴 ) ) = 𝐴 ↔ ( ( 1 /ℝ 𝐴 ) · 𝐴 ) = 1 ) )
8	3 7	mpbird	⊢ ( 𝜑 → ( 1 /ℝ ( 1 /ℝ 𝐴 ) ) = 𝐴 )