Theorem msqsqrtd

Description: Square root theorem. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	$$⊢\phi \to A\in ℂ$$
Assertion	msqsqrtd	$$⊢\phi \to \sqrt{A}\sqrt{A}=A$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2	1	sqrtcld	$$⊢\phi \to \sqrt{A}\in ℂ$$
3	2	sqvald	$$⊢\phi \to {\sqrt{A}}^2=\sqrt{A}\sqrt{A}$$
4	1	sqsqrtd	$$⊢\phi \to {\sqrt{A}}^2=A$$
5	3 4	eqtr3d	$$⊢\phi \to \sqrt{A}\sqrt{A}=A$$