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Metamath Proof Explorer

Theorem msq11

Description: The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999) (Revised by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion msq11 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 le2msq ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 · 𝐴 ) ≤ ( 𝐵 · 𝐵 ) ) )
2 le2msq ⊢ ( ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 · 𝐵 ) ≤ ( 𝐴 · 𝐴 ) ) )
3 2 ancoms ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 · 𝐵 ) ≤ ( 𝐴 · 𝐴 ) ) )
4 1 3 anbi12d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ↔ ( ( 𝐴 · 𝐴 ) ≤ ( 𝐵 · 𝐵 ) ∧ ( 𝐵 · 𝐵 ) ≤ ( 𝐴 · 𝐴 ) ) ) )
5 simpll ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 𝐴 ∈ ℝ )
6 simprl ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → 𝐵 ∈ ℝ )
7 5 6 letri3d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) )
8 5 5 remulcld ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐴 · 𝐴 ) ∈ ℝ )
9 6 6 remulcld ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( 𝐵 · 𝐵 ) ∈ ℝ )
10 8 9 letri3d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ ( ( 𝐴 · 𝐴 ) ≤ ( 𝐵 · 𝐵 ) ∧ ( 𝐵 · 𝐵 ) ≤ ( 𝐴 · 𝐴 ) ) ) )
11 4 7 10 3bitr4rd ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )