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Metamath Proof Explorer

Theorem absnid

Description: For a negative number, its absolute value is its negation. (Contributed by NM, 27-Feb-2005)

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

Proof

Step Hyp Ref Expression
1 le0neg1 
 |-  ( A e. RR -> ( A <_ 0 <-> 0 <_ -u A ) )
2 recn 
 |-  ( A e. RR -> A e. CC )
3 absneg 
 |-  ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
4 2 3 syl 
 |-  ( A e. RR -> ( abs ` -u A ) = ( abs ` A ) )
5 4 adantr 
 |-  ( ( A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = ( abs ` A ) )
6 renegcl 
 |-  ( A e. RR -> -u A e. RR )
7 absid 
 |-  ( ( -u A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = -u A )
8 6 7 sylan 
 |-  ( ( A e. RR /\ 0 <_ -u A ) -> ( abs ` -u A ) = -u A )
9 5 8 eqtr3d 
 |-  ( ( A e. RR /\ 0 <_ -u A ) -> ( abs ` A ) = -u A )
10 9 ex 
 |-  ( A e. RR -> ( 0 <_ -u A -> ( abs ` A ) = -u A ) )
11 1 10 sylbid 
 |-  ( A e. RR -> ( A <_ 0 -> ( abs ` A ) = -u A ) )
12 11 imp 
 |-  ( ( A e. RR /\ A <_ 0 ) -> ( abs ` A ) = -u A )