Theorem resqrtth

Description: Square root theorem over the reals. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 9-Jul-2013)

	Ref	Expression
Assertion	resqrtth	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to {\sqrt{A}}^2=A$$

Proof

Step	Hyp	Ref	Expression
1		resqrtthlem	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left({\sqrt{A}}^2=A\wedge 0\le \Re \left(\sqrt{A}\right)\wedge \mathrm{i}\sqrt{A}\notin {ℝ}^{+}\right)$$
2	1	simp1d	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to {\sqrt{A}}^2=A$$