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Theorem sgnsv 26190
Description: The sign mapping. (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
sgnsv  |-  ( R  e.  V  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0. 
.<  x ,  1 , 
-u 1 ) ) ) )
Distinct variable groups:    x,  .0.    x, 
.<    x, B    x, R    x, V
Allowed substitution hint:    S( x)

Proof of Theorem sgnsv
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 sgnsval.s . 2  |-  S  =  (sgns `  R )
2 elex 2981 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5691 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 sgnsval.b . . . . . 6  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2493 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fveq2 5691 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
7 sgnsval.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
86, 7syl6eqr 2493 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
98adantr 465 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( 0g `  r
)  =  .0.  )
109eqeq2d 2454 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( x  =  ( 0g `  r )  <-> 
x  =  .0.  )
)
11 fveq2 5691 . . . . . . . . . 10  |-  ( r  =  R  ->  ( lt `  r )  =  ( lt `  R
) )
12 sgnsval.l . . . . . . . . . 10  |-  .<  =  ( lt `  R )
1311, 12syl6eqr 2493 . . . . . . . . 9  |-  ( r  =  R  ->  ( lt `  r )  = 
.<  )
1413adantr 465 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( lt `  r
)  =  .<  )
15 eqidd 2444 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  x  =  x )
169, 14, 15breq123d 4306 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( ( 0g `  r ) ( lt
`  r ) x  <-> 
.0.  .<  x ) )
1716ifbid 3811 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  if ( ( 0g `  r ) ( lt
`  r ) x ,  1 ,  -u
1 )  =  if (  .0.  .<  x ,  1 ,  -u
1 ) )
1810, 17ifbieq2d 3814 . . . . 5  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  if ( x  =  ( 0g `  r ) ,  0 ,  if ( ( 0g `  r ) ( lt
`  r ) x ,  1 ,  -u
1 ) )  =  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x ,  1 ,  -u
1 ) ) )
195, 18mpteq12dva 4369 . . . 4  |-  ( r  =  R  ->  (
x  e.  ( Base `  r )  |->  if ( x  =  ( 0g
`  r ) ,  0 ,  if ( ( 0g `  r
) ( lt `  r ) x ,  1 ,  -u 1
) ) )  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0.  .<  x ,  1 ,  -u
1 ) ) ) )
20 df-sgns 26189 . . . 4  |- sgns  =  (
r  e.  _V  |->  ( x  e.  ( Base `  r )  |->  if ( x  =  ( 0g
`  r ) ,  0 ,  if ( ( 0g `  r
) ( lt `  r ) x ,  1 ,  -u 1
) ) ) )
21 fvex 5701 . . . . . 6  |-  ( Base `  r )  e.  _V
2221mptex 5948 . . . . 5  |-  ( x  e.  ( Base `  r
)  |->  if ( x  =  ( 0g `  r ) ,  0 ,  if ( ( 0g `  r ) ( lt `  r
) x ,  1 ,  -u 1 ) ) )  e.  _V
2322a1i 11 . . . 4  |-  ( r  e.  _V  ->  (
x  e.  ( Base `  r )  |->  if ( x  =  ( 0g
`  r ) ,  0 ,  if ( ( 0g `  r
) ( lt `  r ) x ,  1 ,  -u 1
) ) )  e. 
_V )
2419, 20, 23fvmpt3 5777 . . 3  |-  ( R  e.  _V  ->  (sgns `  R )  =  ( x  e.  B  |->  if ( x  =  .0. 
,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) ) )
252, 24syl 16 . 2  |-  ( R  e.  V  ->  (sgns `  R )  =  ( x  e.  B  |->  if ( x  =  .0. 
,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) ) )
261, 25syl5eq 2487 1  |-  ( R  e.  V  ->  S  =  ( x  e.  B  |->  if ( x  =  .0.  ,  0 ,  if (  .0. 
.<  x ,  1 , 
-u 1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2972   ifcif 3791   class class class wbr 4292    e. cmpt 4350   ` 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
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 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-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-sgns 26189
This theorem is referenced by:  sgnsval  26191  sgnsf  26192
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