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Theorem psgnval 27533
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 2302 . . . . 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 2577 . . 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 5256 . 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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
7 eqid 2296 . . . . 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 27527 . . . 4  |-  N  Fn  { x  e.  ( Base `  G )  |  dom  ( x  \  _I  )  e.  Fin }
10 fndm 5359 . . . 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 27526 . 2  |-  N  =  ( t  e.  dom  N 
|->  ( iota s E. w  e. Word  T ( t  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
14 iotaex 5252 . 2  |-  ( iota s E. w  e. Word  T ( P  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  e.  _V
154, 13, 14fvmpt 5618 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 1632    e. wcel 1696   E.wrex 2557   {crab 2560    \ cdif 3162    _I cid 4320   dom cdm 4705   ran crn 4706   iotacio 5233    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Fincfn 6879   1c1 8754   -ucneg 9054   ^cexp 11120   #chash 11353  Word cword 11419   Basecbs 13164    gsumg cgsu 13417   SymGrpcsymg 14785  pmTrspcpmtr 27487  pmSgncpsgn 27517
This theorem is referenced by:  psgnvali  27534  psgnvalii  27535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-word 11425  df-slot 13168  df-base 13169  df-psgn 27518
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