Metamath Proof Explorer

Theorem msqgt0d

Description: A nonzero square is positive. Theorem I.20 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 
|- ( ph -> A e. RR )
msqgt0d.2 
|- ( ph -> A =/= 0 )
Assertion msqgt0d 
|- ( ph -> 0 < ( A x. A ) )

Proof

Step Hyp Ref Expression
1 leidd.1 
 |-  ( ph -> A e. RR )
2 msqgt0d.2 
 |-  ( ph -> A =/= 0 )
3 msqgt0 
 |-  ( ( A e. RR /\ A =/= 0 ) -> 0 < ( A x. A ) )
4 1 2 3 syl2anc 
 |-  ( ph -> 0 < ( A x. A ) )