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Metamath Proof Explorer

Theorem cjmulrcl

Description: A complex number times its conjugate is real. (Contributed by NM, 26-Mar-2005) (Revised by Mario Carneiro, 14-Jul-2014)

Ref Expression
Assertion cjmulrcl 
|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR )

Proof

Step Hyp Ref Expression
1 cjcj 
 |-  ( A e. CC -> ( * ` ( * ` A ) ) = A )
2 1 oveq2d 
 |-  ( A e. CC -> ( ( * ` A ) x. ( * ` ( * ` A ) ) ) = ( ( * ` A ) x. A ) )
3 cjcl 
 |-  ( A e. CC -> ( * ` A ) e. CC )
4 cjmul 
 |-  ( ( A e. CC /\ ( * ` A ) e. CC ) -> ( * ` ( A x. ( * ` A ) ) ) = ( ( * ` A ) x. ( * ` ( * ` A ) ) ) )
5 3 4 mpdan 
 |-  ( A e. CC -> ( * ` ( A x. ( * ` A ) ) ) = ( ( * ` A ) x. ( * ` ( * ` A ) ) ) )
6 mulcom 
 |-  ( ( A e. CC /\ ( * ` A ) e. CC ) -> ( A x. ( * ` A ) ) = ( ( * ` A ) x. A ) )
7 3 6 mpdan 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) = ( ( * ` A ) x. A ) )
8 2 5 7 3eqtr4d 
 |-  ( A e. CC -> ( * ` ( A x. ( * ` A ) ) ) = ( A x. ( * ` A ) ) )
9 mulcl 
 |-  ( ( A e. CC /\ ( * ` A ) e. CC ) -> ( A x. ( * ` A ) ) e. CC )
10 3 9 mpdan 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) e. CC )
11 cjreb 
 |-  ( ( A x. ( * ` A ) ) e. CC -> ( ( A x. ( * ` A ) ) e. RR <-> ( * ` ( A x. ( * ` A ) ) ) = ( A x. ( * ` A ) ) ) )
12 10 11 syl 
 |-  ( A e. CC -> ( ( A x. ( * ` A ) ) e. RR <-> ( * ` ( A x. ( * ` A ) ) ) = ( A x. ( * ` A ) ) ) )
13 8 12 mpbird 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) e. RR )