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Theorem sgnsval 27955
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 27954 . . 3  |-  ( R  e.  V  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0. 
.<  x ,  1 , 
-u 1 ) ) ) )
65adantr 463 . 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 4443 . . . . 5  |-  ( x  =  X  ->  (  .0.  .<  x  <->  .0.  .<  X ) )
98ifbid 3951 . . . 4  |-  ( x  =  X  ->  if (  .0.  .<  x , 
1 ,  -u 1
)  =  if (  .0.  .<  X , 
1 ,  -u 1
) )
107, 9ifbieq2d 3954 . . 3  |-  ( x  =  X  ->  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) )  =  if ( X  =  .0. 
,  0 ,  if (  .0.  .<  X , 
1 ,  -u 1
) ) )
1110adantl 464 . 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 459 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  X  e.  B )
13 c0ex 9579 . . . 4  |-  0  e.  _V
1413a1i 11 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  X  =  .0.  )  ->  0  e. 
_V )
15 1ex 9580 . . . . 5  |-  1  e.  _V
1615a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  .0.  .<  X )  ->  1  e.  _V )
17 negex 9809 . . . . 5  |-  -u 1  e.  _V
1817a1i 11 . . . 4  |-  ( ( ( ( R  e.  V  /\  X  e.  B )  /\  -.  X  =  .0.  )  /\  -.  .0.  .<  X )  ->  -u 1  e.  _V )
1916, 18ifclda 3961 . . 3  |-  ( ( ( R  e.  V  /\  X  e.  B
)  /\  -.  X  =  .0.  )  ->  if (  .0.  .<  X , 
1 ,  -u 1
)  e.  _V )
2014, 19ifclda 3961 . 2  |-  ( ( R  e.  V  /\  X  e.  B )  ->  if ( X  =  .0.  ,  0 ,  if (  .0.  .<  X ,  1 ,  -u
1 ) )  e. 
_V )
216, 11, 12, 20fvmptd 5936 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570   0cc0 9481   1c1 9482   -ucneg 9797   Basecbs 14719   0gc0g 14932   ltcplt 15772  sgnscsgns 27952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-mulcl 9543  ax-i2m1 9549
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-neg 9799  df-sgns 27953
This theorem is referenced by: (None)
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