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Metamath Proof Explorer

Theorem abscj

Description: The absolute value of a number and its conjugate are the same. Proposition 10-3.7(b) of Gleason p. 133. (Contributed by NM, 28-Apr-2005)

Ref Expression
Assertion abscj 
|- ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) )

Proof

Step Hyp Ref Expression
1 cjcl 
 |-  ( A e. CC -> ( * ` A ) e. CC )
2 absval 
 |-  ( ( * ` A ) e. CC -> ( abs ` ( * ` A ) ) = ( sqrt ` ( ( * ` A ) x. ( * ` ( * ` A ) ) ) ) )
3 1 2 syl 
 |-  ( A e. CC -> ( abs ` ( * ` A ) ) = ( sqrt ` ( ( * ` A ) x. ( * ` ( * ` A ) ) ) ) )
4 mulcom 
 |-  ( ( A e. CC /\ ( * ` A ) e. CC ) -> ( A x. ( * ` A ) ) = ( ( * ` A ) x. A ) )
5 1 4 mpdan 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) = ( ( * ` A ) x. A ) )
6 cjcj 
 |-  ( A e. CC -> ( * ` ( * ` A ) ) = A )
7 6 oveq2d 
 |-  ( A e. CC -> ( ( * ` A ) x. ( * ` ( * ` A ) ) ) = ( ( * ` A ) x. A ) )
8 5 7 eqtr4d 
 |-  ( A e. CC -> ( A x. ( * ` A ) ) = ( ( * ` A ) x. ( * ` ( * ` A ) ) ) )
9 8 fveq2d 
 |-  ( A e. CC -> ( sqrt ` ( A x. ( * ` A ) ) ) = ( sqrt ` ( ( * ` A ) x. ( * ` ( * ` A ) ) ) ) )
10 3 9 eqtr4d 
 |-  ( A e. CC -> ( abs ` ( * ` A ) ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
11 absval 
 |-  ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
12 10 11 eqtr4d 
 |-  ( A e. CC -> ( abs ` ( * ` A ) ) = ( abs ` A ) )