Metamath Proof Explorer

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 
|- ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) ^ 2 ) = A )

Proof

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