Theorem sqrtlt

Description: Square root is strictly monotonic. Closed form of sqrtlti . (Contributed by Scott Fenton, 17-Apr-2014) (Proof shortened by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	sqrtlt	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtle	⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) → ( 𝐵 ≤ 𝐴 ↔ ( √ ‘ 𝐵 ) ≤ ( √ ‘ 𝐴 ) ) )
2	1	ancoms	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐵 ≤ 𝐴 ↔ ( √ ‘ 𝐵 ) ≤ ( √ ‘ 𝐴 ) ) )
3	2	notbid	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ¬ 𝐵 ≤ 𝐴 ↔ ¬ ( √ ‘ 𝐵 ) ≤ ( √ ‘ 𝐴 ) ) )
4		simpll	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 𝐴 ∈ ℝ )
5		simprl	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 𝐵 ∈ ℝ )
6	4 5	ltnled	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴 ) )
7		resqrtcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
8	7	adantr	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( √ ‘ 𝐴 ) ∈ ℝ )
9		resqrtcl	⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( √ ‘ 𝐵 ) ∈ ℝ )
10	9	adantl	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( √ ‘ 𝐵 ) ∈ ℝ )
11	8 10	ltnled	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ↔ ¬ ( √ ‘ 𝐵 ) ≤ ( √ ‘ 𝐴 ) ) )
12	3 6 11	3bitr4d	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 < 𝐵 ↔ ( √ ‘ 𝐴 ) < ( √ ‘ 𝐵 ) ) )