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Theorem psgnvali 27363
Description: A finitary permutation has at least one representation for its 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
psgnvali  |-  ( P  e.  dom  N  ->  E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `  w
) ) ) )
Distinct variable groups:    w, G    w, N    w, P    w, T    w, D

Proof of Theorem psgnvali
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 psgnval.g . . . 4  |-  G  =  ( SymGrp `  D )
2 psgnval.t . . . 4  |-  T  =  ran  (pmTrsp `  D
)
3 psgnval.n . . . 4  |-  N  =  (pmSgn `  D )
41, 2, 3psgnval 27362 . . 3  |-  ( P  e.  dom  N  -> 
( N `  P
)  =  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
51, 2, 3psgneu 27361 . . . 4  |-  ( P  e.  dom  N  ->  E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
6 iotacl 5433 . . . 4  |-  ( E! s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  -> 
( iota 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
) ) ) } )
75, 6syl 16 . . 3  |-  ( P  e.  dom  N  -> 
( iota 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
) ) ) } )
84, 7eqeltrd 2509 . 2  |-  ( P  e.  dom  N  -> 
( N `  P
)  e.  { s  |  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) } )
9 fvex 5734 . . 3  |-  ( N `
 P )  e. 
_V
10 eqeq1 2441 . . . . 5  |-  ( s  =  ( N `  P )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( N `  P )  =  (
-u 1 ^ ( # `
 w ) ) ) )
1110anbi2d 685 . . . 4  |-  ( s  =  ( N `  P )  ->  (
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  ( N `  P )  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
1211rexbidv 2718 . . 3  |-  ( s  =  ( N `  P )  ->  ( E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
139, 12elab 3074 . 2  |-  ( ( N `  P )  e.  { s  |  E. w  e. Word  T
( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) }  <->  E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `  w
) ) ) )
148, 13sylib 189 1  |-  ( P  e.  dom  N  ->  E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  ( N `  P )  =  ( -u 1 ^ ( # `  w
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2280   {cab 2421   E.wrex 2698   dom cdm 4870   ran crn 4871   iotacio 5408   ` cfv 5446  (class class class)co 6073   1c1 8981   -ucneg 9282   ^cexp 11372   #chash 11608  Word cword 11707    gsumg cgsu 13714   SymGrpcsymg 15082  pmTrspcpmtr 27316  pmSgncpsgn 27346
This theorem is referenced by:  psgnghm  27369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-xor 1314  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-word 11713  df-concat 11714  df-s1 11715  df-substr 11716  df-splice 11717  df-reverse 11718  df-s2 11802  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-tset 13538  df-0g 13717  df-gsum 13718  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-mhm 14728  df-submnd 14729  df-grp 14802  df-minusg 14803  df-subg 14931  df-ghm 14994  df-gim 15036  df-symg 15083  df-oppg 15132  df-pmtr 27317  df-psgn 27347
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