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Theorem psgnfval 27094
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 5670 . . . . . . . . . 10  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  ( SymGrp `  D )
)
3 psgnfval.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
42, 3syl6eqr 2439 . . . . . . . . 9  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  G )
54fveq2d 5674 . . . . . . . 8  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  (
Base `  G )
)
6 psgnfval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
75, 6syl6eqr 2439 . . . . . . 7  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  B )
8 rabeq 2895 . . . . . . 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 16 . . . . . 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 2439 . . . . 5  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  F
)
12 fveq2 5670 . . . . . . . . . 10  |-  ( d  =  D  ->  (pmTrsp `  d )  =  (pmTrsp `  D ) )
1312rneqd 5039 . . . . . . . . 9  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  ran  (pmTrsp `  D
) )
14 psgnfval.t . . . . . . . . 9  |-  T  =  ran  (pmTrsp `  D
)
1513, 14syl6eqr 2439 . . . . . . . 8  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  T )
16 wrdeq 11667 . . . . . . . 8  |-  ( ran  (pmTrsp `  d )  =  T  -> Word  ran  (pmTrsp `  d )  = Word  T
)
1715, 16syl 16 . . . . . . 7  |-  ( d  =  D  -> Word  ran  (pmTrsp `  d )  = Word  T
)
184oveq1d 6037 . . . . . . . . 9  |-  ( d  =  D  ->  (
( SymGrp `  d )  gsumg  w )  =  ( G 
gsumg  w ) )
1918eqeq2d 2400 . . . . . . . 8  |-  ( d  =  D  ->  (
x  =  ( (
SymGrp `  d )  gsumg  w )  <-> 
x  =  ( G 
gsumg  w ) ) )
2019anbi1d 686 . . . . . . 7  |-  ( d  =  D  ->  (
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( x  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
2117, 20rexeqbidv 2862 . . . . . 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 5381 . . . . 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 4230 . . . 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 27086 . . . 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 5684 . . . . . . . 8  |-  ( Base `  G )  e.  _V
266, 25eqeltri 2459 . . . . . . 7  |-  B  e. 
_V
2726rabex 4297 . . . . . 6  |-  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  e.  _V
2810, 27eqeltri 2459 . . . . 5  |-  F  e. 
_V
2928mptex 5907 . . . 4  |-  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  e.  _V
3023, 24, 29fvmpt 5747 . . 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 5664 . . . 4  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  (/) )
32 fvprc 5664 . . . . . . . . . . . . 13  |-  ( -.  D  e.  _V  ->  (
SymGrp `  D )  =  (/) )
333, 32syl5eq 2433 . . . . . . . . . . . 12  |-  ( -.  D  e.  _V  ->  G  =  (/) )
3433fveq2d 5674 . . . . . . . . . . 11  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  ( Base `  (/) ) )
35 base0 13435 . . . . . . . . . . 11  |-  (/)  =  (
Base `  (/) )
3634, 35syl6eqr 2439 . . . . . . . . . 10  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  (/) )
376, 36syl5eq 2433 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  B  =  (/) )
38 rabeq 2895 . . . . . . . . 9  |-  ( B  =  (/)  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  {
p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
3937, 38syl 16 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  { p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
40 rab0 3593 . . . . . . . 8  |-  { p  e.  (/)  |  dom  (
p  \  _I  )  e.  Fin }  =  (/)
4139, 40syl6eq 2437 . . . . . . 7  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  (/) )
4210, 41syl5eq 2433 . . . . . 6  |-  ( -.  D  e.  _V  ->  F  =  (/) )
4342mpteq1d 4233 . . . . 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
) ) ) ) ) )
44 mpt0 5514 . . . . 5  |-  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/)
4543, 44syl6eq 2437 . . . 4  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/) )
4631, 45eqtr4d 2424 . . 3  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
4730, 46pm2.61i 158 . 2  |-  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
481, 47eqtri 2409 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 359    = wceq 1649    e. wcel 1717   E.wrex 2652   {crab 2655   _Vcvv 2901    \ cdif 3262   (/)c0 3573    e. cmpt 4209    _I cid 4436   dom cdm 4820   ran crn 4821   iotacio 5358   ` cfv 5396  (class class class)co 6022   Fincfn 7047   1c1 8926   -ucneg 9226   ^cexp 11311   #chash 11547  Word cword 11646   Basecbs 13398    gsumg cgsu 13653   SymGrpcsymg 15021  pmTrspcpmtr 27055  pmSgncpsgn 27085
This theorem is referenced by:  psgnfn  27095  psgnval  27101
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-word 11652  df-slot 13402  df-base 13403  df-psgn 27086
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