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Metamath Proof Explorer

Theorem lt2sqd

Description: The square function on nonnegative reals is strictly monotonic. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses sqgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
lt2sqd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
lt2sqd.3 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
lt2sqd.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion lt2sqd ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )

Proof

Step Hyp Ref Expression
1 sqgt0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 lt2sqd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 lt2sqd.3 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
4 lt2sqd.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
5 lt2sq ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )
6 1 3 2 4 5 syl22anc ⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 ↑ 2 ) < ( 𝐵 ↑ 2 ) ) )