MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sgnval Structured version   Unicode version

Theorem sgnval 12694
Description: Value of Signum function. Pronounced "signum" . See df-sgn 12693. (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 2458 . . 3  |-  ( x  =  A  ->  (
x  =  0  <->  A  =  0 ) )
2 breq1 4402 . . . 4  |-  ( x  =  A  ->  (
x  <  0  <->  A  <  0 ) )
32ifbid 3918 . . 3  |-  ( x  =  A  ->  if ( x  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 , 
-u 1 ,  1 ) )
41, 3ifbieq2d 3921 . 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 12693 . 2  |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
6 c0ex 9490 . . 3  |-  0  e.  _V
7 negex 9718 . . . 4  |-  -u 1  e.  _V
8 1ex 9491 . . . 4  |-  1  e.  _V
97, 8ifex 3965 . . 3  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  _V
106, 9ifex 3965 . 2  |-  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  e.  _V
114, 5, 10fvmpt 5882 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 1370    e. wcel 1758   ifcif 3898   class class class wbr 4399   ` cfv 5525   0cc0 9392   1c1 9393   RR*cxr 9527    < clt 9528   -ucneg 9706  sgncsgn 12692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-mulcl 9454  ax-i2m1 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-iota 5488  df-fun 5527  df-fv 5533  df-ov 6202  df-neg 9708  df-sgn 12693
This theorem is referenced by:  sgn0  12695  sgnp  12696  sgnn  12700  sgnneg  27066  sgn3da  27067
  Copyright terms: Public domain W3C validator