Theorem sqn0rp

Description: The square of a nonzero real is a positive real. (Contributed by AV, 5-Mar-2023)

	Ref	Expression
Assertion	sqn0rp	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑ 2 ) ∈ ℝ+ )

Proof

Step	Hyp	Ref	Expression
1		resqcl	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ↑ 2 ) ∈ ℝ )
2	1	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑ 2 ) ∈ ℝ )
3		sqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 ↑ 2 ) )
4	2 3	elrpd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑ 2 ) ∈ ℝ+ )