Theorem sqrtth

Description: Square root theorem over the complex numbers. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 10-Jul-2013)

	Ref	Expression
Assertion	sqrtth	$$⊢A\in ℂ\to {\sqrt{A}}^2=A$$

Proof

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