Theorem absre

Description: Absolute value of a real number. (Contributed by NM, 17-Mar-2005)

	Ref	Expression
Assertion	absre	|- ( A e. RR -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		absval	|- ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
3	1 2	syl	|- ( A e. RR -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
4	1	sqvald	|- ( A e. RR -> ( A ^ 2 ) = ( A x. A ) )
5		cjre	|- ( A e. RR -> ( * ` A ) = A )
6	5	oveq2d	|- ( A e. RR -> ( A x. ( * ` A ) ) = ( A x. A ) )
7	4 6	eqtr4d	|- ( A e. RR -> ( A ^ 2 ) = ( A x. ( * ` A ) ) )
8	7	fveq2d	|- ( A e. RR -> ( sqrt ` ( A ^ 2 ) ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
9	3 8	eqtr4d	|- ( A e. RR -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) )