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Metamath Proof Explorer

Theorem sqrt11

Description: The square root function is one-to-one. (Contributed by Scott Fenton, 11-Jun-2013)

Ref Expression
Assertion sqrt11 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 resqrtcl ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
2 sqrtge0 ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 0 ≤ ( √ ‘ 𝐴 ) )
3 1 2 jca ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐴 ) ) )
4 resqrtcl ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( √ ‘ 𝐵 ) ∈ ℝ )
5 sqrtge0 ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → 0 ≤ ( √ ‘ 𝐵 ) )
6 4 5 jca ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐵 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐵 ) ) )
7 sq11 ⊢ ( ( ( ( √ ‘ 𝐴 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐴 ) ) ∧ ( ( √ ‘ 𝐵 ) ∈ ℝ ∧ 0 ≤ ( √ ‘ 𝐵 ) ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ) )
8 3 6 7 syl2an ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ) )
9 resqrtth ⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
10 resqrtth ⊢ ( ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐵 ) ↑ 2 ) = 𝐵 )
11 9 10 eqeqan12d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐵 ) ↑ 2 ) ↔ 𝐴 = 𝐵 ) )
12 8 11 bitr3d ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )