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 ) ) |
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 ) ) |