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Theorem signsw0glem 29435
Description: Neutral element property of  .+^. (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
signsw0glem  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
Distinct variable group:    a, b, u
Allowed substitution hints:    .+^ ( u, a,
b)

Proof of Theorem signsw0glem
StepHypRef Expression
1 c0ex 9634 . . . . . 6  |-  0  e.  _V
21tpid2 4085 . . . . 5  |-  0  e.  { -u 1 ,  0 ,  1 }
3 signsw.p . . . . . 6  |-  .+^  =  ( a  e.  { -u
1 ,  0 ,  1 } ,  b  e.  { -u 1 ,  0 ,  1 }  |->  if ( b  =  0 ,  a ,  b ) )
43signspval 29434 . . . . 5  |-  ( ( 0  e.  { -u
1 ,  0 ,  1 }  /\  u  e.  { -u 1 ,  0 ,  1 } )  ->  ( 0 
.+^  u )  =  if ( u  =  0 ,  0 ,  u ) )
52, 4mpan 675 . . . 4  |-  ( u  e.  { -u 1 ,  0 ,  1 }  ->  ( 0 
.+^  u )  =  if ( u  =  0 ,  0 ,  u ) )
6 iftrue 3886 . . . . . 6  |-  ( u  =  0  ->  if ( u  =  0 ,  0 ,  u
)  =  0 )
7 id 22 . . . . . 6  |-  ( u  =  0  ->  u  =  0 )
86, 7eqtr4d 2487 . . . . 5  |-  ( u  =  0  ->  if ( u  =  0 ,  0 ,  u
)  =  u )
9 iffalse 3889 . . . . 5  |-  ( -.  u  =  0  ->  if ( u  =  0 ,  0 ,  u
)  =  u )
108, 9pm2.61i 168 . . . 4  |-  if ( u  =  0 ,  0 ,  u )  =  u
115, 10syl6eq 2500 . . 3  |-  ( u  e.  { -u 1 ,  0 ,  1 }  ->  ( 0 
.+^  u )  =  u )
123signspval 29434 . . . . 5  |-  ( ( u  e.  { -u
1 ,  0 ,  1 }  /\  0  e.  { -u 1 ,  0 ,  1 } )  ->  ( u  .+^  0 )  =  if ( 0  =  0 ,  u ,  0 ) )
132, 12mpan2 676 . . . 4  |-  ( u  e.  { -u 1 ,  0 ,  1 }  ->  ( u  .+^  0 )  =  if ( 0  =  0 ,  u ,  0 ) )
14 eqid 2450 . . . . 5  |-  0  =  0
1514iftruei 3887 . . . 4  |-  if ( 0  =  0 ,  u ,  0 )  =  u
1613, 15syl6eq 2500 . . 3  |-  ( u  e.  { -u 1 ,  0 ,  1 }  ->  ( u  .+^  0 )  =  u )
1711, 16jca 535 . 2  |-  ( u  e.  { -u 1 ,  0 ,  1 }  ->  ( (
0  .+^  u )  =  u  /\  ( u 
.+^  0 )  =  u ) )
1817rgen 2746 1  |-  A. u  e.  { -u 1 ,  0 ,  1 }  ( ( 0  .+^  u )  =  u  /\  ( u  .+^  0 )  =  u )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1443    e. wcel 1886   A.wral 2736   ifcif 3880   {ctp 3971  (class class class)co 6288    |-> cmpt2 6290   0cc0 9536   1c1 9537   -ucneg 9858
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:  signsw0g  29438  signswmnd  29439
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