Metamath Proof Explorer


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 ) )