Theorem sn-recgt0d

Description: The reciprocal of a positive real is positive. (Contributed by SN, 26-Nov-2025)

	Ref	Expression
Hypotheses	sn-recgt0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sn-recgt0d.z	⊢ ( 𝜑 → 0 < 𝐴 )
Assertion	sn-recgt0d	⊢ ( 𝜑 → 0 < ( 1 /ℝ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		sn-recgt0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sn-recgt0d.z	⊢ ( 𝜑 → 0 < 𝐴 )
3		sn-0lt1	⊢ 0 < 1
4	2	gt0ne0d	⊢ ( 𝜑 → 𝐴 ≠ 0 )
5	1 4	rerecid	⊢ ( 𝜑 → ( 𝐴 · ( 1 /ℝ 𝐴 ) ) = 1 )
6	3 5	breqtrrid	⊢ ( 𝜑 → 0 < ( 𝐴 · ( 1 /ℝ 𝐴 ) ) )
7	1 4	sn-rereccld	⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ∈ ℝ )
8	1 7 2	mulgt0b1d	⊢ ( 𝜑 → ( 0 < ( 1 /ℝ 𝐴 ) ↔ 0 < ( 𝐴 · ( 1 /ℝ 𝐴 ) ) ) )
9	6 8	mpbird	⊢ ( 𝜑 → 0 < ( 1 /ℝ 𝐴 ) )