Metamath Proof Explorer


Theorem sqrtthi

Description: Square root theorem. Theorem I.35 of Apostol p. 29. (Contributed by NM, 26-May-1999) (Revised by Mario Carneiro, 6-Sep-2013)

Ref Expression
Hypothesis sqrtthi.1
|- A e. RR
Assertion sqrtthi
|- ( 0 <_ A -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )

Proof

Step Hyp Ref Expression
1 sqrtthi.1
 |-  A e. RR
2 remsqsqrt
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )
3 1 2 mpan
 |-  ( 0 <_ A -> ( ( sqrt ` A ) x. ( sqrt ` A ) ) = A )