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Metamath Proof Explorer

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