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Theorem psgneu 27420
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 2438 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
41, 2, 3psgneldm 27417 . . . . . . . 8  |-  ( P  e.  dom  N  <->  ( P  e.  ( Base `  G
)  /\  dom  ( P 
\  _I  )  e. 
Fin ) )
54simplbi 448 . . . . . . 7  |-  ( P  e.  dom  N  ->  P  e.  ( Base `  G ) )
61, 3elbasfv 13517 . . . . . . 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 27418 . . . . . 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 234 . . . 4  |-  ( P  e.  dom  N  ->  E. w  e. Word  T P  =  ( G  gsumg  w ) )
12 simpr 449 . . . . . . 7  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  P  =  ( G  gsumg  w ) )
13 eqid 2438 . . . . . . 7  |-  ( -u
1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) )
14 ovex 6109 . . . . . . . 8  |-  ( -u
1 ^ ( # `  w ) )  e. 
_V
15 eqeq1 2444 . . . . . . . . 9  |-  ( s  =  ( -u 1 ^ ( # `  w
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  w
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1615anbi2d 686 . . . . . . . 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 3045 . . . . . . 7  |-  ( ( P  =  ( G 
gsumg  w )  /\  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  w
) ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1812, 13, 17sylancl 645 . . . . . 6  |-  ( ( ( P  e.  dom  N  /\  w  e. Word  T
)  /\  P  =  ( G  gsumg  w ) )  ->  E. s ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
1918ex 425 . . . . 5  |-  ( ( P  e.  dom  N  /\  w  e. Word  T )  ->  ( P  =  ( G  gsumg  w )  ->  E. s
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2019reximdva 2820 . . . 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 2977 . . 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 190 . 2  |-  ( P  e.  dom  N  ->  E. s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
24 reeanv 2877 . . . 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 708 . . . . . . . 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 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
) ) ) ) )  ->  w  e. Word  T )
27 simplrr 739 . . . . . . . 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 740 . . . . . . . . 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 742 . . . . . . . . 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 2472 . . . . . . . 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 27413 . . . . . . 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 741 . . . . . . 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 743 . . . . . . 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 2480 . . . . . 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 425 . . . . 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 2839 . . . 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 211 . . 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 1643 . 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 2444 . . . . . 6  |-  ( s  =  t  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 w ) ) ) )
4039anbi2d 686 . . . . 5  |-  ( s  =  t  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  t  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
4140rexbidv 2728 . . . 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 6092 . . . . . . 7  |-  ( w  =  x  ->  ( G  gsumg  w )  =  ( G  gsumg  x ) )
4342eqeq2d 2449 . . . . . 6  |-  ( w  =  x  ->  ( P  =  ( G  gsumg  w )  <->  P  =  ( G  gsumg  x ) ) )
44 fveq2 5731 . . . . . . . 8  |-  ( w  =  x  ->  ( # `
 w )  =  ( # `  x
) )
4544oveq2d 6100 . . . . . . 7  |-  ( w  =  x  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  x
) ) )
4645eqeq2d 2449 . . . . . 6  |-  ( w  =  x  ->  (
t  =  ( -u
1 ^ ( # `  w ) )  <->  t  =  ( -u 1 ^ ( # `
 x ) ) ) )
4743, 46anbi12d 693 . . . . 5  |-  ( w  =  x  ->  (
( P  =  ( G  gsumg  w )  /\  t  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  x )  /\  t  =  (
-u 1 ^ ( # `
 x ) ) ) ) )
4847cbvrexv 2935 . . . 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 254 . . 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 2322 . 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 647 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 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2283   E.wrex 2708   _Vcvv 2958    \ cdif 3319    _I cid 4496   dom cdm 4881   ran crn 4882   ` cfv 5457  (class class class)co 6084   Fincfn 7112   1c1 8996   -ucneg 9297   ^cexp 11387   #chash 11623  Word cword 11722   Basecbs 13474    gsumg cgsu 13729   SymGrpcsymg 15097  pmTrspcpmtr 27375  pmSgncpsgn 27405
This theorem is referenced by:  psgnvali  27422  psgnvalii  27423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-word 11728  df-concat 11729  df-s1 11730  df-substr 11731  df-splice 11732  df-reverse 11733  df-s2 11817  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-tset 13553  df-0g 13732  df-gsum 13733  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-mhm 14743  df-submnd 14744  df-grp 14817  df-minusg 14818  df-subg 14946  df-ghm 15009  df-gim 15051  df-symg 15098  df-oppg 15147  df-pmtr 27376  df-psgn 27406
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