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Theorem sgnsf 26192
Description: The sign function. (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
sgnsf  |-  ( R  e.  V  ->  S : B --> { -u 1 ,  0 ,  1 } )

Proof of Theorem sgnsf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 c0ex 9380 . . . . 5  |-  0  e.  _V
21tpid2 3989 . . . 4  |-  0  e.  { -u 1 ,  0 ,  1 }
3 1ex 9381 . . . . . 6  |-  1  e.  _V
43tpid3 3991 . . . . 5  |-  1  e.  { -u 1 ,  0 ,  1 }
5 negex 9608 . . . . . 6  |-  -u 1  e.  _V
65tpid1 3988 . . . . 5  |-  -u 1  e.  { -u 1 ,  0 ,  1 }
74, 6keepel 3857 . . . 4  |-  if (  .0.  .<  x , 
1 ,  -u 1
)  e.  { -u
1 ,  0 ,  1 }
82, 7keepel 3857 . . 3  |-  if ( x  =  .0.  , 
0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) )  e.  { -u 1 ,  0 ,  1 }
98a1i 11 . 2  |-  ( ( R  e.  V  /\  x  e.  B )  ->  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x ,  1 ,  -u
1 ) )  e. 
{ -u 1 ,  0 ,  1 } )
10 sgnsval.b . . 3  |-  B  =  ( Base `  R
)
11 sgnsval.0 . . 3  |-  .0.  =  ( 0g `  R )
12 sgnsval.l . . 3  |-  .<  =  ( lt `  R )
13 sgnsval.s . . 3  |-  S  =  (sgns `  R )
1410, 11, 12, 13sgnsv 26190 . 2  |-  ( R  e.  V  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0. 
.<  x ,  1 , 
-u 1 ) ) ) )
1510, 11, 12, 13sgnsval 26191 . . 3  |-  ( ( R  e.  V  /\  x  e.  B )  ->  ( S `  x
)  =  if ( x  =  .0.  , 
0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) )
1615, 8syl6eqel 2531 . 2  |-  ( ( R  e.  V  /\  x  e.  B )  ->  ( S `  x
)  e.  { -u
1 ,  0 ,  1 } )
179, 14, 16fmpt2d 5873 1  |-  ( R  e.  V  ->  S : B --> { -u 1 ,  0 ,  1 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3791   {ctp 3881   class class class wbr 4292   -->wf 5414   ` cfv 5418   0cc0 9282   1c1 9283   -ucneg 9596   Basecbs 14174   0gc0g 14378   ltcplt 15111  sgnscsgns 26188
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 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pr 4531  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-mulcl 9344  ax-i2m1 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-neg 9598  df-sgns 26189
This theorem is referenced by: (None)
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