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Metamath Proof Explorer

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 )