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Theorem signspval 28382
Description: The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.)
Hypothesis
Ref Expression
signsw.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
Assertion
Ref Expression
signspval  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
Distinct variable groups:    a, b, X    Y, a, b
Allowed substitution hints:    .+^ ( a, b)

Proof of Theorem signspval
StepHypRef Expression
1 ifcl 3968 . 2  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  if ( Y  =  0 ,  X ,  Y )  e.  { -u 1 ,  0 ,  1 } )
2 ifeq1 3930 . . 3  |-  ( a  =  X  ->  if ( b  =  0 ,  a ,  b )  =  if ( b  =  0 ,  X ,  b ) )
3 eqeq1 2447 . . . 4  |-  ( b  =  Y  ->  (
b  =  0  <->  Y  =  0 ) )
4 id 22 . . . 4  |-  ( b  =  Y  ->  b  =  Y )
53, 4ifbieq2d 3951 . . 3  |-  ( b  =  Y  ->  if ( b  =  0 ,  X ,  b )  =  if ( Y  =  0 ,  X ,  Y ) )
6 signsw.p . . 3  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
72, 5, 6ovmpt2g 6422 . 2  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 }  /\  if ( Y  =  0 ,  X ,  Y )  e.  { -u 1 ,  0 ,  1 } )  -> 
( X  .+^  Y )  =  if ( Y  =  0 ,  X ,  Y ) )
81, 7mpd3an3 1326 1  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  Y  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^ 
Y )  =  if ( Y  =  0 ,  X ,  Y
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   ifcif 3926   {ctp 4018  (class class class)co 6281    |-> cmpt2 6283   0cc0 9495   1c1 9496   -ucneg 9811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286
This theorem is referenced by:  signsw0glem  28383  signswmnd  28387  signswrid  28388  signswlid  28389  signswn0  28390  signswch  28391
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