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Theorem signswrid 29440
Description: The zero-skipping operation propagages non-zeros. (Contributed by Thierry Arnoux, 11-Oct-2018.)
Hypotheses
Ref Expression
signsw.p  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
signsw.w  |-  W  =  { <. ( Base `  ndx ) ,  { -u 1 ,  0 ,  1 } >. ,  <. ( +g  `  ndx ) , 
.+^  >. }
Assertion
Ref Expression
signswrid  |-  ( X  e.  { -u 1 ,  0 ,  1 }  ->  ( X  .+^  0 )  =  X )
Distinct variable group:    a, b, X
Allowed substitution hints:    .+^ ( a, b)    W( a, b)

Proof of Theorem signswrid
StepHypRef Expression
1 c0ex 9634 . . . 4  |-  0  e.  _V
21tpid2 4085 . . 3  |-  0  e.  { -u 1 ,  0 ,  1 }
3 signsw.p . . . 4  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
43signspval 29434 . . 3  |-  ( ( X  e.  { -u
1 ,  0 ,  1 }  /\  0  e.  { -u 1 ,  0 ,  1 } )  ->  ( X  .+^  0 )  =  if ( 0  =  0 ,  X ,  0 ) )
52, 4mpan2 676 . 2  |-  ( X  e.  { -u 1 ,  0 ,  1 }  ->  ( X  .+^  0 )  =  if ( 0  =  0 ,  X ,  0 ) )
6 eqid 2450 . . 3  |-  0  =  0
7 iftrue 3886 . . 3  |-  ( 0  =  0  ->  if ( 0  =  0 ,  X ,  0 )  =  X )
86, 7mp1i 13 . 2  |-  ( X  e.  { -u 1 ,  0 ,  1 }  ->  if (
0  =  0 ,  X ,  0 )  =  X )
95, 8eqtrd 2484 1  |-  ( X  e.  { -u 1 ,  0 ,  1 }  ->  ( X  .+^  0 )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1443    e. wcel 1886   ifcif 3880   {cpr 3969   {ctp 3971   <.cop 3973   ` cfv 5581  (class class class)co 6288    |-> cmpt2 6290   0cc0 9536   1c1 9537   -ucneg 9858   ndxcnx 15111   Basecbs 15114   +g cplusg 15183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-mulcl 9598  ax-i2m1 9604
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293
This theorem is referenced by:  signstfveq0  29459
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