Theorem sqge0

Description: The square of a real is nonnegative. (Contributed by NM, 18-Oct-1999)

	Ref	Expression
Assertion	sqge0	⊢ ( 𝐴 ∈ ℝ → 0 ≤ ( 𝐴 ↑ 2 ) )

Proof

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 ) )