Theorem rereccld

Description: Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	redivcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rereccld.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	rereccld	⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		redivcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rereccld.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		rereccl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 1 / 𝐴 ) ∈ ℝ )