Theorem msqgt0i

Description: A nonzero square is positive. Theorem I.20 of Apostol p. 20. (Contributed by NM, 17-Jan-1997) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	lt2.1	⊢ 𝐴 ∈ ℝ
Assertion	msqgt0i	⊢ ( 𝐴 ≠ 0 → 0 < ( 𝐴 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		lt2.1	⊢ 𝐴 ∈ ℝ
2		msqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 · 𝐴 ) )
3	1 2	mpan	⊢ ( 𝐴 ≠ 0 → 0 < ( 𝐴 · 𝐴 ) )