Theorem absval

Description: The absolute value (modulus) of a complex number. Proposition 10-3.7(a) of Gleason p. 133. (Contributed by NM, 27-Jul-1999) (Revised by Mario Carneiro, 7-Nov-2013)

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

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( x = A -> ( * ` x ) = ( * ` A ) )
2		oveq12	|- ( ( x = A /\ ( * ` x ) = ( * ` A ) ) -> ( x x. ( * ` x ) ) = ( A x. ( * ` A ) ) )
3	1 2	mpdan	|- ( x = A -> ( x x. ( * ` x ) ) = ( A x. ( * ` A ) ) )
4	3	fveq2d	|- ( x = A -> ( sqrt ` ( x x. ( * ` x ) ) ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
5		df-abs	|- abs = ( x e. CC |-> ( sqrt ` ( x x. ( * ` x ) ) ) )
6		fvex	|- ( sqrt ` ( A x. ( * ` A ) ) ) e. _V
7	4 5 6	fvmpt	|- ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )