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Theorem signspval 28137
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 3976 . 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 3938 . . . 4  |-  ( a  =  X  ->  if ( b  =  0 ,  a ,  b )  =  if ( b  =  0 ,  X ,  b ) )
3 eqeq1 2466 . . . . 5  |-  ( b  =  Y  ->  (
b  =  0  <->  Y  =  0 ) )
4 id 22 . . . . 5  |-  ( b  =  Y  ->  b  =  Y )
53, 4ifbieq2d 3959 . . . 4  |-  ( b  =  Y  ->  if ( b  =  0 ,  X ,  b )  =  if ( Y  =  0 ,  X ,  Y ) )
6 signsw.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
72, 5, 6ovmpt2g 6414 . . 3  |-  ( ( 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 ) )
873expa 1191 . 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 ) )
91, 8mpdan 668 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 1374    e. wcel 1762   ifcif 3934   {ctp 4026  (class class class)co 6277    |-> cmpt2 6279   0cc0 9483   1c1 9484   -ucneg 9797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282
This theorem is referenced by:  signsw0glem  28138  signswmnd  28142  signswrid  28143  signswlid  28144  signswn0  28145  signswch  28146
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