Theorem lt2sq

Description: The square function on nonnegative reals is strictly monotonic. (Contributed by NM, 24-Feb-2006)

	Ref	Expression
Assertion	lt2sq	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lt2msq	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 · 𝐴 ) < ( 𝐵 · 𝐵 ) ) )
2		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3		recn	⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4		sqval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
5		sqval	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 ) )
6	4 5	breqan12d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ↔ ( 𝐴 · 𝐴 ) < ( 𝐵 · 𝐵 ) ) )
7	2 3 6	syl2an	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ↔ ( 𝐴 · 𝐴 ) < ( 𝐵 · 𝐵 ) ) )
8	7	ad2ant2r	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ↔ ( 𝐴 · 𝐴 ) < ( 𝐵 · 𝐵 ) ) )
9	1 8	bitr4d	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )