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Theorem sgnval 28592
Description: Value of Signum function. Pronounced "signum" . See df-sgn 28591. (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 2444 . . 3  |-  ( x  =  A  ->  (
x  =  0  <->  A  =  0 ) )
2 breq1 4218 . . . 4  |-  ( x  =  A  ->  (
x  <  0  <->  A  <  0 ) )
32ifbid 3759 . . 3  |-  ( x  =  A  ->  if ( x  <  0 ,  -u 1 ,  1 )  =  if ( A  <  0 , 
-u 1 ,  1 ) )
41, 3ifbieq2d 3761 . 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 28591 . 2  |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
6 c0ex 9090 . . 3  |-  0  e.  _V
7 negex 9309 . . . 4  |-  -u 1  e.  _V
8 1ex 9091 . . . 4  |-  1  e.  _V
97, 8ifex 3799 . . 3  |-  if ( A  <  0 , 
-u 1 ,  1 )  e.  _V
106, 9ifex 3799 . 2  |-  if ( A  =  0 ,  0 ,  if ( A  <  0 , 
-u 1 ,  1 ) )  e.  _V
114, 5, 10fvmpt 5809 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 1653    e. wcel 1726   ifcif 3741   class class class wbr 4215   ` cfv 5457   0cc0 8995   1c1 8996   RR*cxr 9124    < clt 9125   -ucneg 9297  sgncsgn 28590
This theorem is referenced by:  sgn0  28593  sgnp  28594  sgnn  28598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-mulcl 9057  ax-i2m1 9063
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-neg 9299  df-sgn 28591
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