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Theorem psgneu 17147
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 2451 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
41, 2, 3psgneldm 17144 . . . . . . . 8  |-  ( P  e.  dom  N  <->  ( P  e.  ( Base `  G
)  /\  dom  ( P 
\  _I  )  e. 
Fin ) )
54simplbi 462 . . . . . . 7  |-  ( P  e.  dom  N  ->  P  e.  ( Base `  G ) )
61, 3elbasfv 15170 . . . . . . 7  |-  ( P  e.  ( Base `  G
)  ->  D  e.  _V )
75, 6syl 17 . . . . . 6  |-  ( P  e.  dom  N  ->  D  e.  _V )
8 psgnval.t . . . . . . 7  |-  T  =  ran  (pmTrsp `  D
)
91, 8, 2psgneldm2 17145 . . . . . 6  |-  ( D  e.  _V  ->  ( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
107, 9syl 17 . . . . 5  |-  ( P  e.  dom  N  -> 
( P  e.  dom  N  <->  E. w  e. Word  T P  =  ( G  gsumg  w ) ) )
1110ibi 245 . . . 4  |-  ( P  e.  dom  N  ->  E. w  e. Word  T P  =  ( G  gsumg  w ) )
12 simpr 463 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  P  =  ( G  gsumg  w ) )
13 eqid 2451 . . . . . . 7  |-  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) )
14 ovex 6318 . . . . . . . 8  |-  ( -u
1 ^ ( # `  w ) )  e. 
_V
15 eqeq1 2455 . . . . . . . . 9  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1615anbi2d 710 . . . . . . . 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 3141 . . . . . . 7  |-  ( ( P  =  ( G 
gsumg  w )  /\  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1812, 13, 17sylancl 668 . . . . . 6  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1918ex 436 . . . . 5  |-  ( ( P  e.  dom  N  /\  w  e. Word  T )  ->  ( P  =  ( G  gsumg  w )  ->  E. s
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2019reximdva 2862 . . . 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 3067 . . 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 200 . 2  |-  ( P  e.  dom  N  ->  E. s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
24 reeanv 2958 . . . 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 732 . . . . . . . 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 770 . . . . . . . 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 771 . . . . . . . 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 772 . . . . . . . . 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 774 . . . . . . . . 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 2487 . . . . . . . 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 17140 . . . . . . 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 773 . . . . . . 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 775 . . . . . . 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 2495 . . . . . 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 436 . . . . 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 2886 . . . 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 222 . . 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 1774 . 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 2455 . . . . . 6  |-  ( s  =  t  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 w ) ) ) )
4039anbi2d 710 . . . . 5  |-  ( s  =  t  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  t  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
4140rexbidv 2901 . . . 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 6298 . . . . . . 7  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
4342eqeq2d 2461 . . . . . 6  |-  ( w  =  x  ->  ( P  =  ( G  gsumg  w )  <->  P  =  ( G  gsumg  x ) ) )
44 fveq2 5865 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
4544oveq2d 6306 . . . . . . 7  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
4645eqeq2d 2461 . . . . . 6  |-  ( w  =  x  ->  (
t  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 x ) ) ) )
4743, 46anbi12d 717 . . . . 5  |-  ( w  =  x  ->  (
( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  x )  /\  t  =  (
-u 1 ^ ( # `
 x ) ) ) ) )
4847cbvrexv 3020 . . . 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 265 . . 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 2347 . 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 670 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 188    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   E!weu 2299   E.wrex 2738   _Vcvv 3045    \ cdif 3401    _I cid 4744   dom cdm 4834   ran crn 4835   ` cfv 5582  (class class class)co 6290   Fincfn 7569   1c1 9540   -ucneg 9861   ^cexp 12272   #chash 12515  Word cword 12656   Basecbs 15121    gsumg cgsu 15339   SymGrpcsymg 17018  pmTrspcpmtr 17082  pmSgncpsgn 17130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-splice 12669  df-reverse 12670  df-s2 12944  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-tset 15209  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-subg 16814  df-ghm 16881  df-gim 16923  df-oppg 16997  df-symg 17019  df-pmtr 17083  df-psgn 17132
This theorem is referenced by:  psgnvali  17149  psgnvalii  17150  psgnfieu  17159
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