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	|- ( ph -> A e. CC )
Assertion	msqsqrtd	|- ( ph -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2	1	sqrtcld	|- ( ph -> ( sqrt ` A ) e. CC )
3	2	sqvald	|- ( ph -> ( ( sqrt ` A ) ^ 2 ) = ( ( sqrt ` A ) x. ( sqrt ` A ) ) )
4	1	sqsqrtd	|- ( ph -> ( ( sqrt ` A ) ^ 2 ) = A )
5	3 4	eqtr3d	|- ( ph -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )