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Theorem sgnsval 26159
Description: The sign value. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypotheses
Ref Expression
sgnsval.b  |-  B  =  ( Base `  R
)
sgnsval.0  |-  .0.  =  ( 0g `  R )
sgnsval.l  |-  .<  =  ( lt `  R )
sgnsval.s  |-  S  =  (sgns `  R )
Assertion
Ref Expression
sgnsval  |-  ( ( R  e.  V  /\  X  e.  B )  ->  ( S `  X
)  =  if ( X  =  .0.  , 
0 ,  if (  .0.  .<  X , 
1 ,  -u 1
) ) )

Proof of Theorem sgnsval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sgnsval.b . . . 4  |-  B  =  ( Base `  R
)
2 sgnsval.0 . . . 4  |-  .0.  =  ( 0g `  R )
3 sgnsval.l . . . 4  |-  .<  =  ( lt `  R )
4 sgnsval.s . . . 4  |-  S  =  (sgns `  R )
51, 2, 3, 4sgnsv 26158 . . 3  |-  ( R  e.  V  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0. 
.<  x ,  1 , 
-u 1 ) ) ) )
65adantr 465 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  , 
0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) ) )
7 eqeq1 2444 . . . 4  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
8 breq2 4291 . . . . 5  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
98ifbid 3806 . . . 4  |-  ( x  =  X  ->  if (  .0.  .<  x , 
1 ,  -u 1
)  =  if (  .0.  .<  X , 
1 ,  -u 1
) )
107, 9ifbieq2d 3809 . . 3  |-  ( x  =  X  ->  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) )  =  if ( X  =  .0. 
,  0 ,  if (  .0.  .<  X , 
1 ,  -u 1
) ) )
1110adantl 466 . 2  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  x  =  X )  ->  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) )  =  if ( X  =  .0. 
,  0 ,  if (  .0.  .<  X , 
1 ,  -u 1
) ) )
12 simpr 461 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  X  e.  B )
13 c0ex 9372 . . . 4  |-  0  e.  _V
1413a1i 11 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  X  =  .0.  )  ->  0  e. 
_V )
15 1ex 9373 . . . . 5  |-  1  e.  _V
1615a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  .0.  .<  X )  ->  1  e.  _V )
17 negex 9600 . . . . 5  |-  -u 1  e.  _V
1817a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  -.  .0.  .<  X )  ->  -u 1  e.  _V )
1916, 18ifclda 3816 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  -.  X  =  .0.  )  ->  if (  .0.  .<  X , 
1 ,  -u 1
)  e.  _V )
2014, 19ifclda 3816 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  if ( X  =  .0.  ,  0 ,  if (  .0.  .<  X ,  1 ,  -u
1 ) )  e. 
_V )
216, 11, 12, 20fvmptd 5774 1  |-  ( ( R  e.  V  /\  X  e.  B )  ->  ( S `  X
)  =  if ( X  =  .0.  , 
0 ,  if (  .0.  .<  X , 
1 ,  -u 1
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   ifcif 3786   class class class wbr 4287    e. cmpt 4345   ` cfv 5413   0cc0 9274   1c1 9275   -ucneg 9588   Basecbs 14166   0gc0g 14370   ltcplt 15103  sgnscsgns 26156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-mulcl 9336  ax-i2m1 9342
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-neg 9590  df-sgns 26157
This theorem is referenced by:  sgnsf  26160
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