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Theorem psgnfval 16011
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 5696 . . . . . . . . . 10  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  ( SymGrp `  D )
)
3 psgnfval.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
42, 3syl6eqr 2493 . . . . . . . . 9  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  G )
54fveq2d 5700 . . . . . . . 8  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  (
Base `  G )
)
6 psgnfval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
75, 6syl6eqr 2493 . . . . . . 7  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  B )
8 rabeq 2971 . . . . . . 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 2493 . . . . 5  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  F
)
12 fveq2 5696 . . . . . . . . . 10  |-  ( d  =  D  ->  (pmTrsp `  d )  =  (pmTrsp `  D ) )
1312rneqd 5072 . . . . . . . . 9  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  ran  (pmTrsp `  D
) )
14 psgnfval.t . . . . . . . . 9  |-  T  =  ran  (pmTrsp `  D
)
1513, 14syl6eqr 2493 . . . . . . . 8  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  T )
16 wrdeq 12256 . . . . . . . 8  |-  ( ran  (pmTrsp `  d )  =  T  -> Word  ran  (pmTrsp `  d )  = Word  T
)
1715, 16syl 16 . . . . . . 7  |-  ( d  =  D  -> Word  ran  (pmTrsp `  d )  = Word  T
)
184oveq1d 6111 . . . . . . . . 9  |-  ( d  =  D  ->  (
( SymGrp `  d )  gsumg  w )  =  ( G 
gsumg  w ) )
1918eqeq2d 2454 . . . . . . . 8  |-  ( d  =  D  ->  (
x  =  ( (
SymGrp `  d )  gsumg  w )  <-> 
x  =  ( G 
gsumg  w ) ) )
2019anbi1d 704 . . . . . . 7  |-  ( d  =  D  ->  (
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( x  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
2117, 20rexeqbidv 2937 . . . . . 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 5407 . . . . 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 4375 . . . 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 16002 . . . 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 5706 . . . . . . . 8  |-  ( Base `  G )  e.  _V
266, 25eqeltri 2513 . . . . . . 7  |-  B  e. 
_V
2726rabex 4448 . . . . . 6  |-  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  e.  _V
2810, 27eqeltri 2513 . . . . 5  |-  F  e. 
_V
2928mptex 5953 . . . 4  |-  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  e.  _V
3023, 24, 29fvmpt 5779 . . 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 5690 . . . 4  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  (/) )
32 fvprc 5690 . . . . . . . . . . . . 13  |-  ( -.  D  e.  _V  ->  (
SymGrp `  D )  =  (/) )
333, 32syl5eq 2487 . . . . . . . . . . . 12  |-  ( -.  D  e.  _V  ->  G  =  (/) )
3433fveq2d 5700 . . . . . . . . . . 11  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  ( Base `  (/) ) )
35 base0 14218 . . . . . . . . . . 11  |-  (/)  =  (
Base `  (/) )
3634, 35syl6eqr 2493 . . . . . . . . . 10  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  (/) )
376, 36syl5eq 2487 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  B  =  (/) )
38 rabeq 2971 . . . . . . . . 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 3663 . . . . . . . 8  |-  { p  e.  (/)  |  dom  (
p  \  _I  )  e.  Fin }  =  (/)
4139, 40syl6eq 2491 . . . . . . 7  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  (/) )
4210, 41syl5eq 2487 . . . . . 6  |-  ( -.  D  e.  _V  ->  F  =  (/) )
4342mpteq1d 4378 . . . . 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 5543 . . . . 5  |-  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/)
4543, 44syl6eq 2491 . . . 4  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/) )
4631, 45eqtr4d 2478 . . 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 164 . 2  |-  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
481, 47eqtri 2463 1  |-  N  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2721   {crab 2724   _Vcvv 2977    \ cdif 3330   (/)c0 3642    e. cmpt 4355    _I cid 4636   dom cdm 4845   ran crn 4846   iotacio 5384   ` cfv 5423  (class class class)co 6096   Fincfn 7315   1c1 9288   -ucneg 9601   ^cexp 11870   #chash 12108  Word cword 12226   Basecbs 14179    gsumg cgsu 14384   SymGrpcsymg 15887  pmTrspcpmtr 15952  pmSgncpsgn 16000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-word 12234  df-slot 14183  df-base 14184  df-psgn 16002
This theorem is referenced by:  psgnfn  16012  psgnval  16018  psgnfvalfi  16024
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