MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psgneu Structured version   Unicode version

Theorem psgneu 16404
Description: A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnval.g  |-  G  =  ( SymGrp `  D )
psgnval.t  |-  T  =  ran  (pmTrsp `  D
)
psgnval.n  |-  N  =  (pmSgn `  D )
Assertion
Ref Expression
psgneu  |-  ( P  e.  dom  N  ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
Distinct variable groups:    w, s, G    N, s, w    P, s, w    T, s, w    D, s, w

Proof of Theorem psgneu
Dummy variables  t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnval.g . . . . . . . . 9  |-  G  =  ( SymGrp `  D )
2 psgnval.n . . . . . . . . 9  |-  N  =  (pmSgn `  D )
3 eqid 2467 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
41, 2, 3psgneldm 16401 . . . . . . . 8  |-  ( P  e.  dom  N  <->  ( P  e.  ( Base `  G
)  /\  dom  ( P 
\  _I  )  e. 
Fin ) )
54simplbi 460 . . . . . . 7  |-  ( P  e.  dom  N  ->  P  e.  ( Base `  G ) )
61, 3elbasfv 14554 . . . . . . 7  |-  ( P  e.  ( Base `  G
)  ->  D  e.  _V )
75, 6syl 16 . . . . . 6  |-  ( P  e.  dom  N  ->  D  e.  _V )
8 psgnval.t . . . . . . 7  |-  T  =  ran  (pmTrsp `  D
)
91, 8, 2psgneldm2 16402 . . . . . 6  |-  ( D  e.  _V  ->  ( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
107, 9syl 16 . . . . 5  |-  ( P  e.  dom  N  -> 
( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
1110ibi 241 . . . 4  |-  ( P  e.  dom  N  ->  E. w  e. Word  T P  =  ( G  gsumg  w ) )
12 simpr 461 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  P  =  ( G  gsumg  w ) )
13 eqid 2467 . . . . . . 7  |-  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) )
14 ovex 6320 . . . . . . . 8  |-  ( -u
1 ^ ( # `  w ) )  e. 
_V
15 eqeq1 2471 . . . . . . . . 9  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1615anbi2d 703 . . . . . . . 8  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
1714, 16spcev 3210 . . . . . . 7  |-  ( ( P  =  ( G 
gsumg  w )  /\  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1812, 13, 17sylancl 662 . . . . . 6  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1918ex 434 . . . . 5  |-  ( ( P  e.  dom  N  /\  w  e. Word  T )  ->  ( P  =  ( G  gsumg  w )  ->  E. s
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2019reximdva 2942 . . . 4  |-  ( P  e.  dom  N  -> 
( E. w  e. Word  T P  =  ( G  gsumg  w )  ->  E. w  e. Word  T E. s ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2111, 20mpd 15 . . 3  |-  ( P  e.  dom  N  ->  E. w  e. Word  T E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
22 rexcom4 3138 . . 3  |-  ( E. w  e. Word  T E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. s E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
2321, 22sylib 196 . 2  |-  ( P  e.  dom  N  ->  E. s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
24 reeanv 3034 . . . 4  |-  ( E. w  e. Word  T E. x  e. Word  T (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  <-> 
( E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  E. x  e. Word  T ( P  =  ( G 
gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )
257ad2antrr 725 . . . . . . . 8  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  D  e.  _V )
26 simplrl 759 . . . . . . . 8  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  w  e. Word  T )
27 simplrr 760 . . . . . . . 8  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  x  e. Word  T )
28 simprll 761 . . . . . . . . 9  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  P  =  ( G  gsumg  w ) )
29 simprrl 763 . . . . . . . . 9  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  P  =  ( G  gsumg  x ) )
3028, 29eqtr3d 2510 . . . . . . . 8  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  ( G  gsumg  w )  =  ( G 
gsumg  x ) )
311, 8, 25, 26, 27, 30psgnuni 16397 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 x ) ) )
32 simprlr 762 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  s  =  ( -u 1 ^ ( # `
 w ) ) )
33 simprrr 764 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  t  =  ( -u 1 ^ ( # `
 x ) ) )
3431, 32, 333eqtr4d 2518 . . . . . 6  |-  ( ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T
) )  /\  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )  ->  s  =  t )
3534ex 434 . . . . 5  |-  ( ( P  e.  dom  N  /\  ( w  e. Word  T  /\  x  e. Word  T ) )  ->  ( (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  ->  s  =  t ) )
3635rexlimdvva 2966 . . . 4  |-  ( P  e.  dom  N  -> 
( E. w  e. Word  T E. x  e. Word  T
( ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  ->  s  =  t ) )
3724, 36syl5bir 218 . . 3  |-  ( P  e.  dom  N  -> 
( ( E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  E. x  e. Word  T ( P  =  ( G 
gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  ->  s  =  t ) )
3837alrimivv 1696 . 2  |-  ( P  e.  dom  N  ->  A. s A. t ( ( E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  E. x  e. Word  T ( P  =  ( G 
gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  ->  s  =  t ) )
39 eqeq1 2471 . . . . . 6  |-  ( s  =  t  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 w ) ) ) )
4039anbi2d 703 . . . . 5  |-  ( s  =  t  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  t  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
4140rexbidv 2978 . . . 4  |-  ( s  =  t  ->  ( E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) ) ) )
42 oveq2 6303 . . . . . . 7  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
4342eqeq2d 2481 . . . . . 6  |-  ( w  =  x  ->  ( P  =  ( G  gsumg  w )  <->  P  =  ( G  gsumg  x ) ) )
44 fveq2 5872 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
4544oveq2d 6311 . . . . . . 7  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
4645eqeq2d 2481 . . . . . 6  |-  ( w  =  x  ->  (
t  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 x ) ) ) )
4743, 46anbi12d 710 . . . . 5  |-  ( w  =  x  ->  (
( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  x )  /\  t  =  (
-u 1 ^ ( # `
 x ) ) ) ) )
4847cbvrexv 3094 . . . 4  |-  ( E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. x  e. Word  T ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )
4941, 48syl6bb 261 . . 3  |-  ( s  =  t  ->  ( E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. x  e. Word  T ( P  =  ( G  gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) ) )
5049eu4 2340 . 2  |-  ( E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( E. s E. w  e. Word  T
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  A. s A. t ( ( E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  /\  E. x  e. Word  T ( P  =  ( G 
gsumg  x )  /\  t  =  ( -u 1 ^ ( # `  x
) ) ) )  ->  s  =  t ) ) )
5123, 38, 50sylanbrc 664 1  |-  ( P  e.  dom  N  ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767   E!weu 2275   E.wrex 2818   _Vcvv 3118    \ cdif 3478    _I cid 4796   dom cdm 5005   ran crn 5006   ` cfv 5594  (class class class)co 6295   Fincfn 7528   1c1 9505   -ucneg 9818   ^cexp 12146   #chash 12385  Word cword 12515   Basecbs 14507    gsumg cgsu 14713   SymGrpcsymg 16274  pmTrspcpmtr 16339  pmSgncpsgn 16387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12793  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-tset 14591  df-0g 14714  df-gsum 14715  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-subg 16070  df-ghm 16137  df-gim 16179  df-oppg 16253  df-symg 16275  df-pmtr 16340  df-psgn 16389
This theorem is referenced by:  psgnvali  16406  psgnvalii  16407  psgnfieu  16416
  Copyright terms: Public domain W3C validator