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Theorem psgnvalii 26580
Description: Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
Hypotheses
Ref Expression
psgnval.g  |-  G  =  ( SymGrp `  D )
psgnval.t  |-  T  =  ran  (pmTrsp `  D
)
psgnval.n  |-  N  =  (pmSgn `  D )
Assertion
Ref Expression
psgnvalii  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg  W ) )  =  ( -u 1 ^ ( # `  W
) ) )

Proof of Theorem psgnvalii
Dummy variables  s  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnval.g . . . 4  |-  G  =  ( SymGrp `  D )
2 psgnval.t . . . 4  |-  T  =  ran  (pmTrsp `  D
)
3 psgnval.n . . . 4  |-  N  =  (pmSgn `  D )
41, 2, 3psgneldm2i 26576 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
51, 2, 3psgnval 26578 . . 3  |-  ( ( G  gsumg  W )  e.  dom  N  ->  ( N `  ( G  gsumg  W ) )  =  ( iota s E. w  e. Word  T ( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
64, 5syl 15 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg  W ) )  =  ( iota s E. w  e. Word  T ( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) ) )
7 simpr 447 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  e. Word  T )
8 eqidd 2317 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  =  ( G  gsumg  W ) )
9 eqidd 2317 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) )
10 oveq2 5908 . . . . . . 7  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
1110eqeq2d 2327 . . . . . 6  |-  ( w  =  W  ->  (
( G  gsumg  W )  =  ( G  gsumg  w )  <->  ( G  gsumg  W )  =  ( G 
gsumg  W ) ) )
12 fveq2 5563 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
1312oveq2d 5916 . . . . . . 7  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
1413eqeq2d 2327 . . . . . 6  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 W ) )  =  ( -u 1 ^ ( # `  w
) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) ) )
1511, 14anbi12d 691 . . . . 5  |-  ( w  =  W  ->  (
( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 w ) ) )  <->  ( ( G 
gsumg  W )  =  ( G  gsumg  W )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  W
) ) ) ) )
1615rspcev 2918 . . . 4  |-  ( ( W  e. Word  T  /\  ( ( G  gsumg  W )  =  ( G  gsumg  W )  /\  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) ) )  ->  E. w  e. Word  T ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  w
) ) ) )
177, 8, 9, 16syl12anc 1180 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  E. w  e. Word  T
( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
18 ovex 5925 . . . . 5  |-  ( -u
1 ^ ( # `  W ) )  e. 
_V
1918a1i 10 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  e.  _V )
201, 2, 3psgneu 26577 . . . . 5  |-  ( ( G  gsumg  W )  e.  dom  N  ->  E! s E. w  e. Word  T ( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
214, 20syl 15 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  E! s E. w  e. Word  T ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )
22 eqeq1 2322 . . . . . . 7  |-  ( s  =  ( -u 1 ^ ( # `  W
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
2322anbi2d 684 . . . . . 6  |-  ( s  =  ( -u 1 ^ ( # `  W
) )  ->  (
( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) )  <->  ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2423rexbidv 2598 . . . . 5  |-  ( s  =  ( -u 1 ^ ( # `  W
) )  ->  ( E. w  e. Word  T ( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2524adantl 452 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  s  =  (
-u 1 ^ ( # `
 W ) ) )  ->  ( E. w  e. Word  T (
( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) )  <->  E. w  e. Word  T ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  w
) ) ) ) )
2619, 21, 25iota2d 5281 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( E. w  e. Word  T ( ( G 
gsumg  W )  =  ( G  gsumg  w )  /\  ( -u 1 ^ ( # `  W ) )  =  ( -u 1 ^ ( # `  w
) ) )  <->  ( iota s E. w  e. Word  T
( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  (
-u 1 ^ ( # `
 w ) ) ) )  =  (
-u 1 ^ ( # `
 W ) ) ) )
2717, 26mpbid 201 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( iota s E. w  e. Word  T ( ( G  gsumg  W )  =  ( G  gsumg  w )  /\  s  =  ( -u 1 ^ ( # `  w
) ) ) )  =  ( -u 1 ^ ( # `  W
) ) )
286, 27eqtrd 2348 1  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg  W ) )  =  ( -u 1 ^ ( # `  W
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E!weu 2176   E.wrex 2578   _Vcvv 2822   dom cdm 4726   ran crn 4727   iotacio 5254   ` cfv 5292  (class class class)co 5900   1c1 8783   -ucneg 9083   ^cexp 11151   #chash 11384  Word cword 11450    gsumg cgsu 13450   SymGrpcsymg 14818  pmTrspcpmtr 26532  pmSgncpsgn 26562
This theorem is referenced by:  psgnpmtr  26581  psgnghm  26585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-ot 3684  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-n0 10013  df-z 10072  df-uz 10278  df-rp 10402  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-hash 11385  df-word 11456  df-concat 11457  df-s1 11458  df-substr 11459  df-splice 11460  df-reverse 11461  df-s2 11545  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-tset 13274  df-0g 13453  df-gsum 13454  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-mhm 14464  df-submnd 14465  df-grp 14538  df-minusg 14539  df-subg 14667  df-ghm 14730  df-gim 14772  df-symg 14819  df-oppg 14868  df-pmtr 26533  df-psgn 26563
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