Theorem sq11d

Description: The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	|- ( ph -> A e. RR )
	lt2sqd.2	|- ( ph -> B e. RR )
	lt2sqd.3	|- ( ph -> 0 <_ A )
	lt2sqd.4	|- ( ph -> 0 <_ B )
	sq11d.5	|- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
Assertion	sq11d	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	|- ( ph -> A e. RR )
2		lt2sqd.2	|- ( ph -> B e. RR )
3		lt2sqd.3	|- ( ph -> 0 <_ A )
4		lt2sqd.4	|- ( ph -> 0 <_ B )
5		sq11d.5	|- ( ph -> ( A ^ 2 ) = ( B ^ 2 ) )
6		sq11	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )
7	1 3 2 4 6	syl22anc	|- ( ph -> ( ( A ^ 2 ) = ( B ^ 2 ) <-> A = B ) )
8	5 7	mpbid	|- ( ph -> A = B )