Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  psgnval Structured version   Unicode version

Theorem psgnval 27398
Description: Value of the permutation sign function. (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
psgnval  |-  ( P  e.  dom  N  -> 
( N `  P
)  =  ( iota 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 psgnval
Dummy variables  t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2441 . . . . 5  |-  ( t  =  P  ->  (
t  =  ( G 
gsumg  w )  <->  P  =  ( G  gsumg  w ) ) )
21anbi1d 686 . . . 4  |-  ( t  =  P  ->  (
( t  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
32rexbidv 2718 . . 3  |-  ( t  =  P  ->  ( E. w  e. Word  T ( t  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
43iotabidv 5431 . 2  |-  ( t  =  P  ->  ( iota s E. w  e. Word  T ( t  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  =  ( iota s E. w  e. Word  T ( P  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
5 psgnval.g . . 3  |-  G  =  ( SymGrp `  D )
6 eqid 2435 . . 3  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2435 . . . . 5  |-  { x  e.  ( Base `  G
)  |  dom  (
x  \  _I  )  e.  Fin }  =  {
x  e.  ( Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
8 psgnval.n . . . . 5  |-  N  =  (pmSgn `  D )
95, 6, 7, 8psgnfn 27392 . . . 4  |-  N  Fn  { x  e.  ( Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
10 fndm 5536 . . . 4  |-  ( N  Fn  { x  e.  ( Base `  G
)  |  dom  (
x  \  _I  )  e.  Fin }  ->  dom  N  =  { x  e.  ( Base `  G
)  |  dom  (
x  \  _I  )  e.  Fin } )
119, 10ax-mp 8 . . 3  |-  dom  N  =  { x  e.  (
Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
12 psgnval.t . . 3  |-  T  =  ran  (pmTrsp `  D
)
135, 6, 11, 12, 8psgnfval 27391 . 2  |-  N  =  ( t  e.  dom  N 
|->  ( iota s E. w  e. Word  T ( t  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
14 iotaex 5427 . 2  |-  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  e.  _V
154, 13, 14fvmpt 5798 1  |-  ( P  e.  dom  N  -> 
( N `  P
)  =  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    \ cdif 3309    _I cid 4485   dom cdm 4870   ran crn 4871   iotacio 5408    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8983   -ucneg 9284   ^cexp 11374   #chash 11610  Word cword 11709   Basecbs 13461    gsumg cgsu 13716   SymGrpcsymg 15084  pmTrspcpmtr 27352  pmSgncpsgn 27382
This theorem is referenced by:  psgnvali  27399  psgnvalii  27400
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 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  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-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-uni 4008  df-int 4043  df-iun 4087  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-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-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-word 11715  df-slot 13465  df-base 13466  df-psgn 27383
  Copyright terms: Public domain W3C validator