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Theorem sgnval 27945
Description: Value of Signum function. Pronounced "signum" . See df-sgn 27944. (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 2364 . . 3  |-  ( x  =  A  ->  (
x  =  0  <->  A  =  0 ) )
2 breq1 4107 . . . 4  |-  ( x  =  A  ->  (
x  <  0  <->  A  <  0 ) )
32ifbid 3659 . . 3  |-  ( x  =  A  ->  if ( x  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 , 
-u 1 ,  1 ) )
41, 3ifbieq2d 3661 . 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 27944 . 2  |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
6 c0ex 8922 . . 3  |-  0  e.  _V
7 negex 9140 . . . 4  |-  -u 1  e.  _V
8 1ex 8923 . . . 4  |-  1  e.  _V
97, 8ifex 3699 . . 3  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  _V
106, 9ifex 3699 . 2  |-  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  e.  _V
114, 5, 10fvmpt 5685 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 1642    e. wcel 1710   ifcif 3641   class class class wbr 4104   ` cfv 5337   0cc0 8827   1c1 8828   RR*cxr 8956    < clt 8957   -ucneg 9128  sgncsgn 27943
This theorem is referenced by:  sgn0  27946  sgnp  27947  sgnn  27951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-mulcl 8889  ax-i2m1 8895
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-neg 9130  df-sgn 27944
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