Metamath Proof Explorer

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 e. CC -> ( ( sqrt ` A ) ^ 2 ) = A )

Proof

Step Hyp Ref Expression
1 sqrtthlem 
 |-  ( A e. CC -> ( ( ( sqrt ` A ) ^ 2 ) = A /\ 0 <_ ( Re ` ( sqrt ` A ) ) /\ ( _i x. ( sqrt ` A ) ) e/ RR+ ) )
2 1 simp1d 
 |-  ( A e. CC -> ( ( sqrt ` A ) ^ 2 ) = A )