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 ) )