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 ) )