Theorem sqgt0

Description: The square of a nonzero real is positive. (Contributed by NM, 8-Sep-2007)

	Ref	Expression
Assertion	sqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		msqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 · 𝐴 ) )
2		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3		sqval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
4	2 3	syl	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
6	1 5	breqtrrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 ↑ 2 ) )