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Metamath Proof Explorer

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 ) )