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Theorem psgnfval 17153
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 5870 . . . . . . . . . 10  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  ( SymGrp `  D )
)
3 psgnfval.g . . . . . . . . . 10  |-  G  =  ( SymGrp `  D )
42, 3syl6eqr 2505 . . . . . . . . 9  |-  ( d  =  D  ->  ( SymGrp `
 d )  =  G )
54fveq2d 5874 . . . . . . . 8  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  (
Base `  G )
)
6 psgnfval.b . . . . . . . 8  |-  B  =  ( Base `  G
)
75, 6syl6eqr 2505 . . . . . . 7  |-  ( d  =  D  ->  ( Base `  ( SymGrp `  d
) )  =  B )
8 rabeq 3040 . . . . . . 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 17 . . . . . 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 2505 . . . . 5  |-  ( d  =  D  ->  { p  e.  ( Base `  ( SymGrp `
 d ) )  |  dom  ( p 
\  _I  )  e. 
Fin }  =  F
)
12 fveq2 5870 . . . . . . . . . 10  |-  ( d  =  D  ->  (pmTrsp `  d )  =  (pmTrsp `  D ) )
1312rneqd 5065 . . . . . . . . 9  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  ran  (pmTrsp `  D
) )
14 psgnfval.t . . . . . . . . 9  |-  T  =  ran  (pmTrsp `  D
)
1513, 14syl6eqr 2505 . . . . . . . 8  |-  ( d  =  D  ->  ran  (pmTrsp `  d )  =  T )
16 wrdeq 12696 . . . . . . . 8  |-  ( ran  (pmTrsp `  d )  =  T  -> Word  ran  (pmTrsp `  d )  = Word  T
)
1715, 16syl 17 . . . . . . 7  |-  ( d  =  D  -> Word  ran  (pmTrsp `  d )  = Word  T
)
184oveq1d 6310 . . . . . . . . 9  |-  ( d  =  D  ->  (
( SymGrp `  d )  gsumg  w )  =  ( G 
gsumg  w ) )
1918eqeq2d 2463 . . . . . . . 8  |-  ( d  =  D  ->  (
x  =  ( (
SymGrp `  d )  gsumg  w )  <-> 
x  =  ( G 
gsumg  w ) ) )
2019anbi1d 712 . . . . . . 7  |-  ( d  =  D  ->  (
( x  =  ( ( SymGrp `  d )  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( x  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) ) )
2117, 20rexeqbidv 3004 . . . . . 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 5570 . . . . 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 4484 . . . 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 17144 . . . 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 5880 . . . . . . 7  |-  ( Base `  G )  e.  _V
266, 25eqeltri 2527 . . . . . 6  |-  B  e. 
_V
2710, 26rabex2 4559 . . . . 5  |-  F  e. 
_V
2827mptex 6141 . . . 4  |-  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  e.  _V
2923, 24, 28fvmpt 5953 . . 3  |-  ( D  e.  _V  ->  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
30 fvprc 5864 . . . 4  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  (/) )
31 fvprc 5864 . . . . . . . . . . . . 13  |-  ( -.  D  e.  _V  ->  (
SymGrp `  D )  =  (/) )
323, 31syl5eq 2499 . . . . . . . . . . . 12  |-  ( -.  D  e.  _V  ->  G  =  (/) )
3332fveq2d 5874 . . . . . . . . . . 11  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  ( Base `  (/) ) )
34 base0 15174 . . . . . . . . . . 11  |-  (/)  =  (
Base `  (/) )
3533, 34syl6eqr 2505 . . . . . . . . . 10  |-  ( -.  D  e.  _V  ->  (
Base `  G )  =  (/) )
366, 35syl5eq 2499 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  B  =  (/) )
37 rabeq 3040 . . . . . . . . 9  |-  ( B  =  (/)  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  {
p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
3836, 37syl 17 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  { p  e.  (/)  |  dom  ( p  \  _I  )  e.  Fin } )
39 rab0 3755 . . . . . . . 8  |-  { p  e.  (/)  |  dom  (
p  \  _I  )  e.  Fin }  =  (/)
4038, 39syl6eq 2503 . . . . . . 7  |-  ( -.  D  e.  _V  ->  { p  e.  B  |  dom  ( p  \  _I  )  e.  Fin }  =  (/) )
4110, 40syl5eq 2499 . . . . . 6  |-  ( -.  D  e.  _V  ->  F  =  (/) )
4241mpteq1d 4487 . . . . 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
) ) ) ) ) )
43 mpt0 5710 . . . . 5  |-  ( x  e.  (/)  |->  ( iota s E. w  e. Word  T ( x  =  ( G 
gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/)
4442, 43syl6eq 2503 . . . 4  |-  ( -.  D  e.  _V  ->  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )  =  (/) )
4530, 44eqtr4d 2490 . . 3  |-  ( -.  D  e.  _V  ->  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) ) )
4629, 45pm2.61i 168 . 2  |-  (pmSgn `  D )  =  ( x  e.  F  |->  ( iota s E. w  e. Word  T ( x  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
471, 46eqtri 2475 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 371    = wceq 1446    e. wcel 1889   E.wrex 2740   {crab 2743   _Vcvv 3047    \ cdif 3403   (/)c0 3733    |-> cmpt 4464    _I cid 4747   dom cdm 4837   ran crn 4838   iotacio 5547   ` cfv 5585  (class class class)co 6295   Fincfn 7574   1c1 9545   -ucneg 9866   ^cexp 12279   #chash 12522  Word cword 12663   Basecbs 15133    gsumg cgsu 15351   SymGrpcsymg 17030  pmTrspcpmtr 17094  pmSgncpsgn 17142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-slot 15137  df-base 15138  df-psgn 17144
This theorem is referenced by:  psgnfn  17154  psgnval  17160  psgnfvalfi  17166
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