Theorem msqgt0d

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

	Ref	Expression
Hypotheses	leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	msqgt0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	msqgt0d	⊢ ( 𝜑 → 0 < ( 𝐴 · 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		msqgt0d.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		msqgt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → 0 < ( 𝐴 · 𝐴 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 0 < ( 𝐴 · 𝐴 ) )