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Metamath Proof Explorer

Theorem sn-rereccld

Description: Closure law for reciprocal. (Contributed by SN, 25-Nov-2025)

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

Proof

Step Hyp Ref Expression
1 sn-rereccld.a ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 sn-rereccld.z ⊢ ( 𝜑 → 𝐴 ≠ 0 )
3 1red ⊢ ( 𝜑 → 1 ∈ ℝ )
4 3 1 2 sn-redivcld ⊢ ( 𝜑 → ( 1 /ℝ 𝐴 ) ∈ ℝ )