Theorem sqr11d

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

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

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		resqrcld.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		sqr11d.3	⊢ ( 𝜑 → 𝐵 ∈ 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	⊢ ( 𝜑 → 𝐴 = 𝐵 )