Theorem sq11

Description: The square function is one-to-one for nonnegative reals. (Contributed by NM, 8-Apr-2001) (Proof shortened by Mario Carneiro, 28-May-2016)

	Ref	Expression
Assertion	sq11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 𝐴 ∈ ℝ )
2	1	recnd	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 𝐴 ∈ ℂ )
3		sqval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
4	2 3	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
5		simpl	⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → 𝐵 ∈ ℝ )
6	5	recnd	⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → 𝐵 ∈ ℂ )
7		sqval	⊢ ( 𝐵 ∈ ℂ → ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 ) )
8	6 7	syl	⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 ) )
9	4 8	eqeqan12d	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ) )
10		msq11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
11	9 10	bitrd	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ 𝐴 = 𝐵 ) )