Theorem sqgt0d

Description: The square of a nonzero real is positive. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	sqgt0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	sqgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		sqgt0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		sqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 ↑ 2 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 0 < ( 𝐴 ↑ 2 ) )