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Theorem psgnfval 27391
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 5720 . . . . . . . . . 10  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  ( SymGrp `  D )
)
3 psgnfval.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
42, 3syl6eqr 2485 . . . . . . . . 9  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  G )
54fveq2d 5724 . . . . . . . 8  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  (
Base `  G )
)
6 psgnfval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
75, 6syl6eqr 2485 . . . . . . 7  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  B )
8 rabeq 2942 . . . . . . 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 2485 . . . . 5  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  F
)
12 fveq2 5720 . . . . . . . . . 10  |-  ( d  =  D  ->  (pmTrsp `  d )  =  (pmTrsp `  D ) )
1312rneqd 5089 . . . . . . . . 9  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  ran  (pmTrsp `  D
) )
14 psgnfval.t . . . . . . . . 9  |-  T  =  ran  (pmTrsp `  D
)
1513, 14syl6eqr 2485 . . . . . . . 8  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  T )
16 wrdeq 11730 . . . . . . . 8  |-  ( ran  (pmTrsp `  d )  =  T  -> Word  ran  (pmTrsp `  d )  = Word  T
)
1715, 16syl 16 . . . . . . 7  |-  ( d  =  D  -> Word  ran  (pmTrsp `  d )  = Word  T
)
184oveq1d 6088 . . . . . . . . 9  |-  ( d  =  D  ->  (
( SymGrp `  d )  gsumg  w )  =  ( G 
gsumg  w ) )
1918eqeq2d 2446 . . . . . . . 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 2909 . . . . . 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 5431 . . . . 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 4279 . . . 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 27383 . . . 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 5734 . . . . . . . 8  |-  ( Base `  G )  e.  _V
266, 25eqeltri 2505 . . . . . . 7  |-  B  e. 
_V
2726rabex 4346 . . . . . 6  |-  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  e.  _V
2810, 27eqeltri 2505 . . . . 5  |-  F  e. 
_V
2928mptex 5958 . . . 4  |-  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  e.  _V
3023, 24, 29fvmpt 5798 . . 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 5714 . . . 4  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  (/) )
32 fvprc 5714 . . . . . . . . . . . . 13  |-  ( -.  D  e.  _V  ->  (
SymGrp `  D )  =  (/) )
333, 32syl5eq 2479 . . . . . . . . . . . 12  |-  ( -.  D  e.  _V  ->  G  =  (/) )
3433fveq2d 5724 . . . . . . . . . . 11  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  ( Base `  (/) ) )
35 base0 13498 . . . . . . . . . . 11  |-  (/)  =  (
Base `  (/) )
3634, 35syl6eqr 2485 . . . . . . . . . 10  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  (/) )
376, 36syl5eq 2479 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  B  =  (/) )
38 rabeq 2942 . . . . . . . . 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 3640 . . . . . . . 8  |-  { p  e.  (/)  |  dom  (
p  \  _I  )  e.  Fin }  =  (/)
4139, 40syl6eq 2483 . . . . . . 7  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  (/) )
4210, 41syl5eq 2479 . . . . . 6  |-  ( -.  D  e.  _V  ->  F  =  (/) )
4342mpteq1d 4282 . . . . 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 5564 . . . . 5  |-  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/)
4543, 44syl6eq 2483 . . . 4  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/) )
4631, 45eqtr4d 2470 . . 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 2455 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 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309   (/)c0 3620    e. cmpt 4258    _I cid 4485   dom cdm 4870   ran crn 4871   iotacio 5408   ` cfv 5446  (class class class)co 6073   Fincfn 7101   1c1 8983   -ucneg 9284   ^cexp 11374   #chash 11610  Word cword 11709   Basecbs 13461    gsumg cgsu 13716   SymGrpcsymg 15084  pmTrspcpmtr 27352  pmSgncpsgn 27382
This theorem is referenced by:  psgnfn  27392  psgnval  27398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-word 11715  df-slot 13465  df-base 13466  df-psgn 27383
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