Metamath Proof Explorer


Theorem sqr11d

Description: The square root function is one-to-one. (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 )
sqrt11d.5
|- ( ph -> ( sqrt ` A ) = ( sqrt ` B ) )
Assertion sqr11d
|- ( ph -> A = 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 sqrt11d.5
 |-  ( ph -> ( sqrt ` A ) = ( sqrt ` B ) )
6 sqrt11
 |-  ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )
7 1 2 3 4 6 syl22anc
 |-  ( ph -> ( ( sqrt ` A ) = ( sqrt ` B ) <-> A = B ) )
8 5 7 mpbid
 |-  ( ph -> A = B )