Metamath Proof Explorer

Theorem sgnn

Description: The signum of a negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015)

Ref Expression
Assertion sgnn 
|- ( ( A e. RR* /\ A < 0 ) -> ( sgn ` A ) = -u 1 )

Proof

Step Hyp Ref Expression
1 sgnval 
 |-  ( A e. RR* -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
2 1 adantr 
 |-  ( ( A e. RR* /\ A < 0 ) -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
3 0xr 
 |-  0 e. RR*
4 xrltne 
 |-  ( ( A e. RR* /\ 0 e. RR* /\ A < 0 ) -> 0 =/= A )
5 3 4 mp3an2 
 |-  ( ( A e. RR* /\ A < 0 ) -> 0 =/= A )
6 nesym 
 |-  ( 0 =/= A <-> -. A = 0 )
7 5 6 sylib 
 |-  ( ( A e. RR* /\ A < 0 ) -> -. A = 0 )
8 7 iffalsed 
 |-  ( ( A e. RR* /\ A < 0 ) -> if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , -u 1 , 1 ) )
9 iftrue 
 |-  ( A < 0 -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
10 9 adantl 
 |-  ( ( A e. RR* /\ A < 0 ) -> if ( A < 0 , -u 1 , 1 ) = -u 1 )
11 2 8 10 3eqtrd 
 |-  ( ( A e. RR* /\ A < 0 ) -> ( sgn ` A ) = -u 1 )