Metamath Proof Explorer

Theorem sgnp

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

Ref Expression
Assertion sgnp 
|- ( ( A e. RR* /\ 0 < A ) -> ( sgn ` A ) = 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* /\ 0 < A ) -> ( sgn ` A ) = if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) )
3 0xr 
 |-  0 e. RR*
4 xrltne 
 |-  ( ( 0 e. RR* /\ A e. RR* /\ 0 < A ) -> A =/= 0 )
5 3 4 mp3an1 
 |-  ( ( A e. RR* /\ 0 < A ) -> A =/= 0 )
6 5 neneqd 
 |-  ( ( A e. RR* /\ 0 < A ) -> -. A = 0 )
7 6 iffalsed 
 |-  ( ( A e. RR* /\ 0 < A ) -> if ( A = 0 , 0 , if ( A < 0 , -u 1 , 1 ) ) = if ( A < 0 , -u 1 , 1 ) )
8 xrltnsym 
 |-  ( ( 0 e. RR* /\ A e. RR* ) -> ( 0 < A -> -. A < 0 ) )
9 3 8 mpan 
 |-  ( A e. RR* -> ( 0 < A -> -. A < 0 ) )
10 9 imp 
 |-  ( ( A e. RR* /\ 0 < A ) -> -. A < 0 )
11 10 iffalsed 
 |-  ( ( A e. RR* /\ 0 < A ) -> if ( A < 0 , -u 1 , 1 ) = 1 )
12 2 7 11 3eqtrd 
 |-  ( ( A e. RR* /\ 0 < A ) -> ( sgn ` A ) = 1 )