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Theorem psgnval 27430
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 2289 . . . . 5  |-  ( t  =  P  ->  (
t  =  ( G 
gsumg  w )  <->  P  =  ( G  gsumg  w ) ) )
21anbi1d 685 . . . 4  |-  ( t  =  P  ->  (
( t  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( P  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
32rexbidv 2564 . . 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 5240 . 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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2283 . . . . 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 27424 . . . 4  |-  N  Fn  { x  e.  ( Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
10 fndm 5343 . . . 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 27423 . 2  |-  N  =  ( t  e.  dom  N 
|->  ( iota s E. w  e. Word  T ( t  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
14 iotaex 5236 . 2  |-  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  e.  _V
154, 13, 14fvmpt 5602 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 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    \ cdif 3149    _I cid 4304   dom cdm 4689   ran crn 4690   iotacio 5217    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Fincfn 6863   1c1 8738   -ucneg 9038   ^cexp 11104   #chash 11337  Word cword 11403   Basecbs 13148    gsumg cgsu 13401   SymGrpcsymg 14769  pmTrspcpmtr 27384  pmSgncpsgn 27414
This theorem is referenced by:  psgnvali  27431  psgnvalii  27432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-word 11409  df-slot 13152  df-base 13153  df-psgn 27415
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