Theorem msq11i

Description: The square of a nonnegative number is a one-to-one function. (Contributed by NM, 29-Jul-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
Assertion	msq11i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		msq11	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	1 4	mpanl1	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( 𝐴 · 𝐴 ) = ( 𝐵 · 𝐵 ) ↔ 𝐴 = 𝐵 ) )