Theorem sqrtmsq2i

Description: Relationship between square root and squares. (Contributed by NM, 31-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	$$⊢A\in ℝ$$
	sqr11.1	$$⊢B\in ℝ$$
Assertion	sqrtmsq2i	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(\sqrt{A}=B\leftrightarrow A=BB\right)$$

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	$$⊢A\in ℝ$$
2		sqr11.1	$$⊢B\in ℝ$$
3		sqrtsq2	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(\sqrt{A}=B\leftrightarrow A=B^2\right)$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge 0\le B\right)\to \left(\sqrt{A}=B\leftrightarrow A=B^2\right)$$
5	1 4	mpanl1	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(\sqrt{A}=B\leftrightarrow A=B^2\right)$$
6	2	recni	$$⊢B\in ℂ$$
7	6	sqvali	$$⊢B^2=BB$$
8	7	eqeq2i	$$⊢A=B^2\leftrightarrow A=BB$$
9	5 8	bitrdi	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(\sqrt{A}=B\leftrightarrow A=BB\right)$$