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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