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Theorem psgneu 16005
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 2441 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
41, 2, 3psgneldm 16002 . . . . . . . 8  |-  ( P  e.  dom  N  <->  ( P  e.  ( Base `  G
)  /\  dom  ( P 
\  _I  )  e. 
Fin ) )
54simplbi 457 . . . . . . 7  |-  ( P  e.  dom  N  ->  P  e.  ( Base `  G ) )
61, 3elbasfv 14217 . . . . . . 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 16003 . . . . . 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 458 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  P  =  ( G  gsumg  w ) )
13 eqid 2441 . . . . . . 7  |-  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) )
14 ovex 6115 . . . . . . . 8  |-  ( -u
1 ^ ( # `  w ) )  e. 
_V
15 eqeq1 2447 . . . . . . . . 9  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1615anbi2d 698 . . . . . . . 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 3061 . . . . . . 7  |-  ( ( P  =  ( G 
gsumg  w )  /\  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1812, 13, 17sylancl 657 . . . . . 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 2826 . . . 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 2990 . . 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 2886 . . . 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 720 . . . . . . . 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 754 . . . . . . . 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 755 . . . . . . . 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 756 . . . . . . . . 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 758 . . . . . . . . 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 2475 . . . . . . . 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 15998 . . . . . . 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 757 . . . . . . 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 759 . . . . . . 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 2483 . . . . . 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 2846 . . . 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 1691 . 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 2447 . . . . . 6  |-  ( s  =  t  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 w ) ) ) )
4039anbi2d 698 . . . . 5  |-  ( s  =  t  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  t  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
4140rexbidv 2734 . . . 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 6098 . . . . . . 7  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
4342eqeq2d 2452 . . . . . 6  |-  ( w  =  x  ->  ( P  =  ( G  gsumg  w )  <->  P  =  ( G  gsumg  x ) ) )
44 fveq2 5688 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
4544oveq2d 6106 . . . . . . 7  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
4645eqeq2d 2452 . . . . . 6  |-  ( w  =  x  ->  (
t  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 x ) ) ) )
4743, 46anbi12d 705 . . . . 5  |-  ( w  =  x  ->  (
( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  x )  /\  t  =  (
-u 1 ^ ( # `
 x ) ) ) ) )
4847cbvrexv 2946 . . . 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 2323 . 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 659 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 1362    = wceq 1364   E.wex 1591    e. wcel 1761   E!weu 2259   E.wrex 2714   _Vcvv 2970    \ cdif 3322    _I cid 4627   dom cdm 4836   ran crn 4837   ` cfv 5415  (class class class)co 6090   Fincfn 7306   1c1 9279   -ucneg 9592   ^cexp 11861   #chash 12099  Word cword 12217   Basecbs 14170    gsumg cgsu 14375   SymGrpcsymg 15875  pmTrspcpmtr 15940  pmSgncpsgn 15988
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-xor 1346  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-reverse 12231  df-s2 12471  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-tset 14253  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-subg 15671  df-ghm 15738  df-gim 15780  df-oppg 15854  df-symg 15876  df-pmtr 15941  df-psgn 15990
This theorem is referenced by:  psgnvali  16007  psgnvalii  16008  psgnfieu  16017
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