Metamath Proof Explorer

Theorem rpsqrtcl

Description: The square root of a positive real is a positive real. (Contributed by NM, 22-Feb-2008)

Ref Expression
Assertion rpsqrtcl 
|- ( A e. RR+ -> ( sqrt ` A ) e. RR+ )

Proof

Step Hyp Ref Expression
1 rpre 
 |-  ( A e. RR+ -> A e. RR )
2 rpge0 
 |-  ( A e. RR+ -> 0 <_ A )
3 resqrtcl 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` A ) e. RR )
4 1 2 3 syl2anc 
 |-  ( A e. RR+ -> ( sqrt ` A ) e. RR )
5 rpgt0 
 |-  ( A e. RR+ -> 0 < A )
6 sqrtgt0 
 |-  ( ( A e. RR /\ 0 < A ) -> 0 < ( sqrt ` A ) )
7 1 5 6 syl2anc 
 |-  ( A e. RR+ -> 0 < ( sqrt ` A ) )
8 4 7 elrpd 
 |-  ( A e. RR+ -> ( sqrt ` A ) e. RR+ )