Theorem absid

Description: A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	absid	|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( A e. RR /\ 0 <_ A ) -> A e. RR )
2	1	recnd	|- ( ( A e. RR /\ 0 <_ A ) -> A e. CC )
3		absval	|- ( A e. CC -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
4	2 3	syl	|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = ( sqrt ` ( A x. ( * ` A ) ) ) )
5	1	cjred	|- ( ( A e. RR /\ 0 <_ A ) -> ( * ` A ) = A )
6	5	oveq2d	|- ( ( A e. RR /\ 0 <_ A ) -> ( A x. ( * ` A ) ) = ( A x. A ) )
7	2	sqvald	|- ( ( A e. RR /\ 0 <_ A ) -> ( A ^ 2 ) = ( A x. A ) )
8	6 7	eqtr4d	|- ( ( A e. RR /\ 0 <_ A ) -> ( A x. ( * ` A ) ) = ( A ^ 2 ) )
9	8	fveq2d	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A x. ( * ` A ) ) ) = ( sqrt ` ( A ^ 2 ) ) )
10		sqrtsq	|- ( ( A e. RR /\ 0 <_ A ) -> ( sqrt ` ( A ^ 2 ) ) = A )
11	4 9 10	3eqtrd	|- ( ( A e. RR /\ 0 <_ A ) -> ( abs ` A ) = A )