This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cjmulval

Description: A complex number times its conjugate. (Contributed by NM, 1-Feb-2007) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion cjmulval 
|- ( A e. CC -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )

Proof

Step Hyp Ref Expression
1 recl 
 |-  ( A e. CC -> ( Re ` A ) e. RR )
2 1 recnd 
 |-  ( A e. CC -> ( Re ` A ) e. CC )
3 2 sqvald 
 |-  ( A e. CC -> ( ( Re ` A ) ^ 2 ) = ( ( Re ` A ) x. ( Re ` A ) ) )
4 imcl 
 |-  ( A e. CC -> ( Im ` A ) e. RR )
5 4 recnd 
 |-  ( A e. CC -> ( Im ` A ) e. CC )
6 5 sqvald 
 |-  ( A e. CC -> ( ( Im ` A ) ^ 2 ) = ( ( Im ` A ) x. ( Im ` A ) ) )
7 3 6 oveq12d 
 |-  ( A e. CC -> ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) = ( ( ( Re ` A ) x. ( Re ` A ) ) + ( ( Im ` A ) x. ( Im ` A ) ) ) )
8 ipcnval 
 |-  ( ( A e. CC /\ A e. CC ) -> ( Re ` ( A x. ( * ` A ) ) ) = ( ( ( Re ` A ) x. ( Re ` A ) ) + ( ( Im ` A ) x. ( Im ` A ) ) ) )
9 8 anidms 
 |-  ( A e. CC -> ( Re ` ( A x. ( * ` A ) ) ) = ( ( ( Re ` A ) x. ( Re ` A ) ) + ( ( Im ` A ) x. ( Im ` A ) ) ) )
10 cjmulrcl 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) e. RR )
11 rere 
 |-  ( ( A x. ( * ` A ) ) e. RR -> ( Re ` ( A x. ( * ` A ) ) ) = ( A x. ( * ` A ) ) )
12 10 11 syl 
 |-  ( A e. CC -> ( Re ` ( A x. ( * ` A ) ) ) = ( A x. ( * ` A ) ) )
13 7 9 12 3eqtr2rd 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) = ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) )