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Theorem sgnval 28232
Description: Value of Signum function. Pronounced "signum" . See df-sgn 28231. (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 2410 . . 3  |-  ( x  =  A  ->  (
x  =  0  <->  A  =  0 ) )
2 breq1 4175 . . . 4  |-  ( x  =  A  ->  (
x  <  0  <->  A  <  0 ) )
32ifbid 3717 . . 3  |-  ( x  =  A  ->  if ( x  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 , 
-u 1 ,  1 ) )
41, 3ifbieq2d 3719 . 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 28231 . 2  |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
6 c0ex 9041 . . 3  |-  0  e.  _V
7 negex 9260 . . . 4  |-  -u 1  e.  _V
8 1ex 9042 . . . 4  |-  1  e.  _V
97, 8ifex 3757 . . 3  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  _V
106, 9ifex 3757 . 2  |-  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  e.  _V
114, 5, 10fvmpt 5765 1  |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   ifcif 3699   class class class wbr 4172   ` cfv 5413   0cc0 8946   1c1 8947   RR*cxr 9075    < clt 9076   -ucneg 9248  sgncsgn 28230
This theorem is referenced by:  sgn0  28233  sgnp  28234  sgnn  28238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-mulcl 9008  ax-i2m1 9014
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-neg 9250  df-sgn 28231
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