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Theorem sgnval 13130
Description: Value of Signum function. Pronounced "signum" . See df-sgn 13129. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
sgnval  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )

Proof of Theorem sgnval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2433 . . 3  |-  ( x  =  A  ->  (
x  =  0  <->  A  =  0 ) )
2 breq1 4429 . . . 4  |-  ( x  =  A  ->  (
x  <  0  <->  A  <  0 ) )
32ifbid 3937 . . 3  |-  ( x  =  A  ->  if ( x  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 , 
-u 1 ,  1 ) )
41, 3ifbieq2d 3940 . 2  |-  ( x  =  A  ->  if ( x  =  0 ,  0 ,  if ( x  <  0 ,  -u 1 ,  1 ) )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
5 df-sgn 13129 . 2  |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
6 c0ex 9636 . . 3  |-  0  e.  _V
7 negex 9872 . . . 4  |-  -u 1  e.  _V
8 1ex 9637 . . . 4  |-  1  e.  _V
97, 8ifex 3983 . . 3  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  _V
106, 9ifex 3983 . 2  |-  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  e.  _V
114, 5, 10fvmpt 5964 1  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   ifcif 3915   class class class wbr 4426   ` cfv 5601   0cc0 9538   1c1 9539   RR*cxr 9673    < clt 9674   -ucneg 9860  sgncsgn 13128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-mulcl 9600  ax-i2m1 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-neg 9862  df-sgn 13129
This theorem is referenced by:  sgn0  13131  sgnp  13132  sgnn  13136  sgnneg  29199  sgn3da  29200
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