Metamath Proof Explorer


Theorem sqr11d

Description: The square root function is one-to-one. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
sqr11d.3 ( 𝜑𝐵 ∈ ℝ )
sqr11d.4 ( 𝜑 → 0 ≤ 𝐵 )
sqrt11d.5 ( 𝜑 → ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) )
Assertion sqr11d ( 𝜑𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ( 𝜑𝐴 ∈ ℝ )
2 resqrcld.2 ( 𝜑 → 0 ≤ 𝐴 )
3 sqr11d.3 ( 𝜑𝐵 ∈ ℝ )
4 sqr11d.4 ( 𝜑 → 0 ≤ 𝐵 )
5 sqrt11d.5 ( 𝜑 → ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) )
6 sqrt11 ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
7 1 2 3 4 6 syl22anc ( 𝜑 → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
8 5 7 mpbid ( 𝜑𝐴 = 𝐵 )