Theorem sq11d

Description: The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	lt2sqd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	lt2sqd.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	lt2sqd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
	sq11d.5	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) )
Assertion	sq11d	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		lt2sqd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		lt2sqd.3	⊢ ( 𝜑 → 0 ≤ 𝐴 )
4		lt2sqd.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		sq11d.5	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) )
6		sq11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ 𝐴 = 𝐵 ) )
7	1 3 2 4 6	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 ↑ 2 ) = ( 𝐵 ↑ 2 ) ↔ 𝐴 = 𝐵 ) )
8	5 7	mpbid	⊢ ( 𝜑 → 𝐴 = 𝐵 )