Description: A square of a real is nonnegative. (Contributed by NM, 18-Oct-1999)
Ref | Expression | ||
---|---|---|---|
Assertion | sqge0 | ⊢ ( 𝐴 ∈ ℝ → 0 ≤ ( 𝐴 ↑ 2 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msqge0 | ⊢ ( 𝐴 ∈ ℝ → 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 | 1 4 | breqtrrd | ⊢ ( 𝐴 ∈ ℝ → 0 ≤ ( 𝐴 ↑ 2 ) ) |