Metamath Proof Explorer

Theorem cjmulge0d

Description: A complex number times its conjugate is nonnegative. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis recld.1 
|- ( ph -> A e. CC )
Assertion cjmulge0d 
|- ( ph -> 0 <_ ( A x. ( * ` A ) ) )

Proof

Step Hyp Ref Expression
1 recld.1 
 |-  ( ph -> A e. CC )
2 cjmulge0 
 |-  ( A e. CC -> 0 <_ ( A x. ( * ` A ) ) )
3 1 2 syl 
 |-  ( ph -> 0 <_ ( A x. ( * ` A ) ) )