Theorem sqrt11i

Description: The square root function is one-to-one. (Contributed by NM, 27-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	⊢ 𝐴 ∈ ℝ
	sqr11.1	⊢ 𝐵 ∈ ℝ
Assertion	sqrt11i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqr11.1	⊢ 𝐵 ∈ ℝ
3		sqrt11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	1 4	mpanl1	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = ( √ ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )