Metamath Proof Explorer


Theorem sqrtltd

Description: Square root is strictly monotonic. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses resqrcld.1
|- ( ph -> A e. RR )
resqrcld.2
|- ( ph -> 0 <_ A )
sqr11d.3
|- ( ph -> B e. RR )
sqr11d.4
|- ( ph -> 0 <_ B )
Assertion sqrtltd
|- ( ph -> ( A < B <-> ( sqrt ` A ) < ( sqrt ` B ) ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1
 |-  ( ph -> A e. RR )
2 resqrcld.2
 |-  ( ph -> 0 <_ A )
3 sqr11d.3
 |-  ( ph -> B e. RR )
4 sqr11d.4
 |-  ( ph -> 0 <_ B )
5 sqrtlt
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( A < B <-> ( sqrt ` A ) < ( sqrt ` B ) ) )
6 1 2 3 4 5 syl22anc
 |-  ( ph -> ( A < B <-> ( sqrt ` A ) < ( sqrt ` B ) ) )