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Theorem psgnfval 27423
Description: Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnfval.g  |-  G  =  ( SymGrp `  D )
psgnfval.b  |-  B  =  ( Base `  G
)
psgnfval.f  |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }
psgnfval.t  |-  T  =  ran  (pmTrsp `  D
)
psgnfval.n  |-  N  =  (pmSgn `  D )
Assertion
Ref Expression
psgnfval  |-  N  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
Distinct variable groups:    s, p, w, x    D, s, w, x    x, F    w, T    B, p
Allowed substitution hints:    B( x, w, s)    D( p)    T( x, s, p)    F( w, s, p)    G( x, w, s, p)    N( x, w, s, p)

Proof of Theorem psgnfval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 psgnfval.n . 2  |-  N  =  (pmSgn `  D )
2 fveq2 5525 . . . . . . . . . 10  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  ( SymGrp `  D )
)
3 psgnfval.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
42, 3syl6eqr 2333 . . . . . . . . 9  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  G )
54fveq2d 5529 . . . . . . . 8  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  (
Base `  G )
)
6 psgnfval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
75, 6syl6eqr 2333 . . . . . . 7  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  B )
8 rabeq 2782 . . . . . . 7  |-  ( (
Base `  ( SymGrp `  d ) )  =  B  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  {
p  e.  B  |  dom  ( p  \  _I  )  e.  Fin } )
97, 8syl 15 . . . . . 6  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  {
p  e.  B  |  dom  ( p  \  _I  )  e.  Fin } )
10 psgnfval.f . . . . . 6  |-  F  =  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }
119, 10syl6eqr 2333 . . . . 5  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  F
)
12 fveq2 5525 . . . . . . . . . 10  |-  ( d  =  D  ->  (pmTrsp `  d )  =  (pmTrsp `  D ) )
1312rneqd 4906 . . . . . . . . 9  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  ran  (pmTrsp `  D
) )
14 psgnfval.t . . . . . . . . 9  |-  T  =  ran  (pmTrsp `  D
)
1513, 14syl6eqr 2333 . . . . . . . 8  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  T )
16 wrdeq 11424 . . . . . . . 8  |-  ( ran  (pmTrsp `  d )  =  T  -> Word  ran  (pmTrsp `  d )  = Word  T
)
1715, 16syl 15 . . . . . . 7  |-  ( d  =  D  -> Word  ran  (pmTrsp `  d )  = Word  T
)
184oveq1d 5873 . . . . . . . . 9  |-  ( d  =  D  ->  (
( SymGrp `  d )  gsumg  w )  =  ( G 
gsumg  w ) )
1918eqeq2d 2294 . . . . . . . 8  |-  ( d  =  D  ->  (
x  =  ( (
SymGrp `  d )  gsumg  w )  <-> 
x  =  ( G 
gsumg  w ) ) )
2019anbi1d 685 . . . . . . 7  |-  ( d  =  D  ->  (
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( x  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
2117, 20rexeqbidv 2749 . . . . . 6  |-  ( d  =  D  ->  ( E. w  e. Word  ran  (pmTrsp `  d ) ( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2221iotabidv 5240 . . . . 5  |-  ( d  =  D  ->  ( iota s E. w  e. Word  ran  (pmTrsp `  d )
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  =  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2311, 22mpteq12dv 4098 . . . 4  |-  ( d  =  D  ->  (
x  e.  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  |->  ( iota s E. w  e. Word  ran  (pmTrsp `  d )
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
24 df-psgn 27415 . . . 4  |- pmSgn  =  ( d  e.  _V  |->  ( x  e.  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  |->  ( iota s E. w  e. Word  ran  (pmTrsp `  d )
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
25 fvex 5539 . . . . . . . 8  |-  ( Base `  G )  e.  _V
266, 25eqeltri 2353 . . . . . . 7  |-  B  e. 
_V
2726rabex 4165 . . . . . 6  |-  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  e.  _V
2810, 27eqeltri 2353 . . . . 5  |-  F  e. 
_V
2928mptex 5746 . . . 4  |-  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  e.  _V
3023, 24, 29fvmpt 5602 . . 3  |-  ( D  e.  _V  ->  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
31 fvprc 5519 . . . 4  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  (/) )
32 fvprc 5519 . . . . . . . . . . . . 13  |-  ( -.  D  e.  _V  ->  (
SymGrp `  D )  =  (/) )
333, 32syl5eq 2327 . . . . . . . . . . . 12  |-  ( -.  D  e.  _V  ->  G  =  (/) )
3433fveq2d 5529 . . . . . . . . . . 11  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  ( Base `  (/) ) )
35 base0 13185 . . . . . . . . . . 11  |-  (/)  =  (
Base `  (/) )
3634, 35syl6eqr 2333 . . . . . . . . . 10  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  (/) )
376, 36syl5eq 2327 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  B  =  (/) )
38 rabeq 2782 . . . . . . . . 9  |-  ( B  =  (/)  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  {
p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
3937, 38syl 15 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  { p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
40 rab0 3475 . . . . . . . 8  |-  { p  e.  (/)  |  dom  (
p  \  _I  )  e.  Fin }  =  (/)
4139, 40syl6eq 2331 . . . . . . 7  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  (/) )
4210, 41syl5eq 2327 . . . . . 6  |-  ( -.  D  e.  _V  ->  F  =  (/) )
43 mpteq1 4100 . . . . . 6  |-  ( F  =  (/)  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
4442, 43syl 15 . . . . 5  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
45 mpt0 5371 . . . . 5  |-  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/)
4644, 45syl6eq 2331 . . . 4  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/) )
4731, 46eqtr4d 2318 . . 3  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
4830, 47pm2.61i 156 . 2  |-  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
491, 48eqtri 2303 1  |-  N  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455    e. cmpt 4077    _I cid 4304   dom cdm 4689   ran crn 4690   iotacio 5217   ` 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:  psgnfn  27424  psgnval  27430
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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