Theorem sqrt11i

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

	Ref	Expression
Hypotheses	sqrtthi.1	$$⊢A\in ℝ$$
	sqr11.1	$$⊢B\in ℝ$$
Assertion	sqrt11i	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(\sqrt{A}=\sqrt{B}\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	$$⊢A\in ℝ$$
2		sqr11.1	$$⊢B\in ℝ$$
3		sqrt11	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(\sqrt{A}=\sqrt{B}\leftrightarrow A=B\right)$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge 0\le B\right)\to \left(\sqrt{A}=\sqrt{B}\leftrightarrow A=B\right)$$
5	1 4	mpanl1	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(\sqrt{A}=\sqrt{B}\leftrightarrow A=B\right)$$