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Theorem sgnsv 27870
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 3043 . . 3  |-  ( R  e.  V  ->  R  e.  _V )
3 fveq2 5774 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 sgnsval.b . . . . . 6  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2441 . . . . 5  |-  ( r  =  R  ->  ( Base `  r )  =  B )
6 fveq2 5774 . . . . . . . . 9  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
7 sgnsval.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  R )
86, 7syl6eqr 2441 . . . . . . . 8  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
98adantr 463 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( 0g `  r
)  =  .0.  )
109eqeq2d 2396 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( x  =  ( 0g `  r )  <-> 
x  =  .0.  )
)
11 fveq2 5774 . . . . . . . . . 10  |-  ( r  =  R  ->  ( lt `  r )  =  ( lt `  R
) )
12 sgnsval.l . . . . . . . . . 10  |-  .<  =  ( lt `  R )
1311, 12syl6eqr 2441 . . . . . . . . 9  |-  ( r  =  R  ->  ( lt `  r )  = 
.<  )
1413adantr 463 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( lt `  r
)  =  .<  )
15 eqidd 2383 . . . . . . . 8  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  x  =  x )
169, 14, 15breq123d 4381 . . . . . . 7  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  -> 
( ( 0g `  r ) ( lt
`  r ) x  <-> 
.0.  .<  x ) )
1716ifbid 3879 . . . . . 6  |-  ( ( r  =  R  /\  x  e.  ( Base `  r ) )  ->  if ( ( 0g `  r ) ( lt
`  r ) x ,  1 ,  -u
1 )  =  if (  .0.  .<  x ,  1 ,  -u
1 ) )
1810, 17ifbieq2d 3882 . . . . 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 4444 . . . 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 27869 . . . 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 5784 . . . . 5  |-  ( Base `  r )  e.  _V
2221mptex 6044 . . . 4  |-  ( x  e.  ( Base `  r
)  |->  if ( x  =  ( 0g `  r ) ,  0 ,  if ( ( 0g `  r ) ( lt `  r
) x ,  1 ,  -u 1 ) ) )  e.  _V
2319, 20, 22fvmpt3i 5861 . . 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 2435 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 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   ifcif 3857   class class class wbr 4367    |-> cmpt 4425   ` cfv 5496   0cc0 9403   1c1 9404   -ucneg 9719   Basecbs 14634   0gc0g 14847   ltcplt 15687  sgnscsgns 27868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-sgns 27869
This theorem is referenced by:  sgnsval  27871  sgnsf  27872
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