Metamath Proof Explorer

Theorem le2sq

Description: The square function on nonnegative reals is monotonic. (Contributed by NM, 18-Oct-1999)

Ref Expression
Assertion le2sq ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑ 2 ) ≤ ( 𝐵 ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 le2msq ⊢ ( ( ( 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 ) ) )