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Theorem sgnsv 27549
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 3127 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5872 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 sgnsval.b . . . . . 6  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2526 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fveq2 5872 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
7 sgnsval.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
86, 7syl6eqr 2526 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
98adantr 465 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( 0g `  r
)  =  .0.  )
109eqeq2d 2481 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( x  =  ( 0g `  r )  <-> 
x  =  .0.  )
)
11 fveq2 5872 . . . . . . . . . 10  |-  ( r  =  R  ->  ( lt `  r )  =  ( lt `  R
) )
12 sgnsval.l . . . . . . . . . 10  |-  .<  =  ( lt `  R )
1311, 12syl6eqr 2526 . . . . . . . . 9  |-  ( r  =  R  ->  ( lt `  r )  = 
.<  )
1413adantr 465 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( lt `  r
)  =  .<  )
15 eqidd 2468 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  x  =  x )
169, 14, 15breq123d 4467 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( ( 0g `  r ) ( lt
`  r ) x  <-> 
.0.  .<  x ) )
1716ifbid 3967 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  if ( ( 0g `  r ) ( lt
`  r ) x ,  1 ,  -u
1 )  =  if (  .0.  .<  x ,  1 ,  -u
1 ) )
1810, 17ifbieq2d 3970 . . . . 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 4530 . . . 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 27548 . . . 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 5882 . . . . 5  |-  ( Base `  r )  e.  _V
2221mptex 6142 . . . 4  |-  ( x  e.  ( Base `  r
)  |->  if ( x  =  ( 0g `  r ) ,  0 ,  if ( ( 0g `  r ) ( lt `  r
) x ,  1 ,  -u 1 ) ) )  e.  _V
2319, 20, 22fvmpt3i 5961 . . 3  |-  ( R  e.  _V  ->  (sgns `  R )  =  ( x  e.  B  |->  if ( x  =  .0. 
,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) ) )
242, 23syl 16 . 2  |-  ( R  e.  V  ->  (sgns `  R )  =  ( x  e.  B  |->  if ( x  =  .0. 
,  0 ,  if (  .0.  .<  x , 
1 ,  -u 1
) ) ) )
251, 24syl5eq 2520 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 1379    e. wcel 1767   _Vcvv 3118   ifcif 3945   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594   0cc0 9504   1c1 9505   -ucneg 9818   Basecbs 14506   0gc0g 14711   ltcplt 15444  sgnscsgns 27547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-sgns 27548
This theorem is referenced by:  sgnsval  27550  sgnsf  27551
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