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 )