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Theorem psgnval 27100
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 2394 . . . . 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 2671 . . 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 5380 . 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 2388 . . 3  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2388 . . . . 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 27094 . . . 4  |-  N  Fn  { x  e.  ( Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
10 fndm 5485 . . . 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 27093 . 2  |-  N  =  ( t  e.  dom  N 
|->  ( iota s E. w  e. Word  T ( t  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
14 iotaex 5376 . 2  |-  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  e.  _V
154, 13, 14fvmpt 5746 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 1649    e. wcel 1717   E.wrex 2651   {crab 2654    \ cdif 3261    _I cid 4435   dom cdm 4819   ran crn 4820   iotacio 5357    Fn wfn 5390   ` cfv 5395  (class class class)co 6021   Fincfn 7046   1c1 8925   -ucneg 9225   ^cexp 11310   #chash 11546  Word cword 11645   Basecbs 13397    gsumg cgsu 13652   SymGrpcsymg 15020  pmTrspcpmtr 27054  pmSgncpsgn 27084
This theorem is referenced by:  psgnvali  27101  psgnvalii  27102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-word 11651  df-slot 13401  df-base 13402  df-psgn 27085
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