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Theorem psgneu 27297
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 2404 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
41, 2, 3psgneldm 27294 . . . . . . . 8  |-  ( P  e.  dom  N  <->  ( P  e.  ( Base `  G
)  /\  dom  ( P 
\  _I  )  e. 
Fin ) )
54simplbi 447 . . . . . . 7  |-  ( P  e.  dom  N  ->  P  e.  ( Base `  G ) )
61, 3elbasfv 13467 . . . . . . 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 27295 . . . . . 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 233 . . . 4  |-  ( P  e.  dom  N  ->  E. w  e. Word  T P  =  ( G  gsumg  w ) )
12 simpr 448 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  P  =  ( G  gsumg  w ) )
13 eqid 2404 . . . . . . 7  |-  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) )
14 ovex 6065 . . . . . . . 8  |-  ( -u
1 ^ ( # `  w ) )  e. 
_V
15 eqeq1 2410 . . . . . . . . 9  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1615anbi2d 685 . . . . . . . 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 3003 . . . . . . 7  |-  ( ( P  =  ( G 
gsumg  w )  /\  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1812, 13, 17sylancl 644 . . . . . 6  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1918ex 424 . . . . 5  |-  ( ( P  e.  dom  N  /\  w  e. Word  T )  ->  ( P  =  ( G  gsumg  w )  ->  E. s
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2019reximdva 2778 . . . 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 2935 . . 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 189 . 2  |-  ( P  e.  dom  N  ->  E. s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
24 reeanv 2835 . . . 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 707 . . . . . . . 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 737 . . . . . . . 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 738 . . . . . . . 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 739 . . . . . . . . 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 741 . . . . . . . . 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 2438 . . . . . . . 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 27290 . . . . . . 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 740 . . . . . . 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 742 . . . . . . 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 2446 . . . . . 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 424 . . . . 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 2797 . . . 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 210 . . 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 1639 . 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 2410 . . . . . 6  |-  ( s  =  t  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 w ) ) ) )
4039anbi2d 685 . . . . 5  |-  ( s  =  t  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  t  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
4140rexbidv 2687 . . . 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 6048 . . . . . . 7  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
4342eqeq2d 2415 . . . . . 6  |-  ( w  =  x  ->  ( P  =  ( G  gsumg  w )  <->  P  =  ( G  gsumg  x ) ) )
44 fveq2 5687 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
4544oveq2d 6056 . . . . . . 7  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
4645eqeq2d 2415 . . . . . 6  |-  ( w  =  x  ->  (
t  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 x ) ) ) )
4743, 46anbi12d 692 . . . . 5  |-  ( w  =  x  ->  (
( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  x )  /\  t  =  (
-u 1 ^ ( # `
 x ) ) ) ) )
4847cbvrexv 2893 . . . 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 253 . . 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 2293 . 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 646 1  |-  ( P  e.  dom  N  ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   E!weu 2254   E.wrex 2667   _Vcvv 2916    \ cdif 3277    _I cid 4453   dom cdm 4837   ran crn 4838   ` cfv 5413  (class class class)co 6040   Fincfn 7068   1c1 8947   -ucneg 9248   ^cexp 11337   #chash 11573  Word cword 11672   Basecbs 13424    gsumg cgsu 13679   SymGrpcsymg 15047  pmTrspcpmtr 27252  pmSgncpsgn 27282
This theorem is referenced by:  psgnvali  27299  psgnvalii  27300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1311  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-word 11678  df-concat 11679  df-s1 11680  df-substr 11681  df-splice 11682  df-reverse 11683  df-s2 11767  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-tset 13503  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-subg 14896  df-ghm 14959  df-gim 15001  df-symg 15048  df-oppg 15097  df-pmtr 27253  df-psgn 27283
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