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Theorem psgnvalii 27410
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 27406 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
51, 2, 3psgnval 27408 . . 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 16 . 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 449 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  e. Word  T )
8 eqidd 2438 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  =  ( G  gsumg  W ) )
9 eqidd 2438 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) )
10 oveq2 6090 . . . . . . 7  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
1110eqeq2d 2448 . . . . . 6  |-  ( w  =  W  ->  (
( G  gsumg  W )  =  ( G  gsumg  w )  <->  ( G  gsumg  W )  =  ( G 
gsumg  W ) ) )
12 fveq2 5729 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
1312oveq2d 6098 . . . . . . 7  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
1413eqeq2d 2448 . . . . . 6  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 W ) )  =  ( -u 1 ^ ( # `  w
) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) ) )
1511, 14anbi12d 693 . . . . 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 3053 . . . 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 1183 . . 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 6107 . . . . 5  |-  ( -u
1 ^ ( # `  W ) )  e. 
_V
1918a1i 11 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  e.  _V )
201, 2, 3psgneu 27407 . . . . 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 16 . . . 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 2443 . . . . . . 7  |-  ( s  =  ( -u 1 ^ ( # `  W
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
2322anbi2d 686 . . . . . 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 2727 . . . . 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 454 . . . 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 5444 . . 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 203 . 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 2469 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2282   E.wrex 2707   _Vcvv 2957   dom cdm 4879   ran crn 4880   iotacio 5417   ` cfv 5455  (class class class)co 6082   1c1 8992   -ucneg 9293   ^cexp 11383   #chash 11619  Word cword 11718    gsumg cgsu 13725   SymGrpcsymg 15093  pmTrspcpmtr 27362  pmSgncpsgn 27392
This theorem is referenced by:  psgnpmtr  27411  psgnghm  27415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-xor 1315  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-ot 3825  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-card 7827  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fzo 11137  df-seq 11325  df-exp 11384  df-hash 11620  df-word 11724  df-concat 11725  df-s1 11726  df-substr 11727  df-splice 11728  df-reverse 11729  df-s2 11813  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-tset 13549  df-0g 13728  df-gsum 13729  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-mhm 14739  df-submnd 14740  df-grp 14813  df-minusg 14814  df-subg 14942  df-ghm 15005  df-gim 15047  df-symg 15094  df-oppg 15143  df-pmtr 27363  df-psgn 27393
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