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Theorem sgnsval 26335
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 26334 . . 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 2458 . . . 4  |-  ( x  =  X  ->  (
x  =  .0.  <->  X  =  .0.  ) )
8 breq2 4403 . . . . 5  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
98ifbid 3918 . . . 4  |-  ( x  =  X  ->  if (  .0.  .<  x , 
1 ,  -u 1
)  =  if (  .0.  .<  X , 
1 ,  -u 1
) )
107, 9ifbieq2d 3921 . . 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 9490 . . . 4  |-  0  e.  _V
1413a1i 11 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  X  =  .0.  )  ->  0  e. 
_V )
15 1ex 9491 . . . . 5  |-  1  e.  _V
1615a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  .0.  .<  X )  ->  1  e.  _V )
17 negex 9718 . . . . 5  |-  -u 1  e.  _V
1817a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  -.  .0.  .<  X )  ->  -u 1  e.  _V )
1916, 18ifclda 3928 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  -.  X  =  .0.  )  ->  if (  .0.  .<  X , 
1 ,  -u 1
)  e.  _V )
2014, 19ifclda 3928 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  if ( X  =  .0.  ,  0 ,  if (  .0.  .<  X ,  1 ,  -u
1 ) )  e. 
_V )
216, 11, 12, 20fvmptd 5887 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 1370    e. wcel 1758   _Vcvv 3076   ifcif 3898   class class class wbr 4399    |-> cmpt 4457   ` cfv 5525   0cc0 9392   1c1 9393   -ucneg 9706   Basecbs 14291   0gc0g 14496   ltcplt 15229  sgnscsgns 26332
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-rep 4510  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-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  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-iun 4280  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-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-neg 9708  df-sgns 26333
This theorem is referenced by:  sgnsf  26336
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