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Theorem psgnvalii 16661
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 16657 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
51, 2, 3psgnval 16659 . . 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 461 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  e. Word  T )
8 eqidd 2458 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  =  ( G  gsumg  W ) )
9 eqidd 2458 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) )
10 oveq2 6304 . . . . . . 7  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
1110eqeq2d 2471 . . . . . 6  |-  ( w  =  W  ->  (
( G  gsumg  W )  =  ( G  gsumg  w )  <->  ( G  gsumg  W )  =  ( G 
gsumg  W ) ) )
12 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
1312oveq2d 6312 . . . . . . 7  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
1413eqeq2d 2471 . . . . . 6  |-  ( w  =  W  ->  (
( -u 1 ^ ( # `
 W ) )  =  ( -u 1 ^ ( # `  w
) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) ) )
1511, 14anbi12d 710 . . . . 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 3210 . . . 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 1226 . . 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 6324 . . . . 5  |-  ( -u
1 ^ ( # `  W ) )  e. 
_V
1918a1i 11 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  e.  _V )
201, 2, 3psgneu 16658 . . . . 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 2461 . . . . . . 7  |-  ( s  =  ( -u 1 ^ ( # `  W
) )  ->  (
s  =  ( -u
1 ^ ( # `  w ) )  <->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 w ) ) ) )
2322anbi2d 703 . . . . . 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 2968 . . . . 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 466 . . . 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 5582 . . 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 210 . 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 2498 1  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( N `  ( G  gsumg  W ) )  =  ( -u 1 ^ ( # `  W
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E!weu 2283   E.wrex 2808   _Vcvv 3109   dom cdm 5008   ran crn 5009   iotacio 5555   ` cfv 5594  (class class class)co 6296   1c1 9510   -ucneg 9825   ^cexp 12169   #chash 12408  Word cword 12538    gsumg cgsu 14858   SymGrpcsymg 16529  pmTrspcpmtr 16593  pmSgncpsgn 16641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1364  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-seq 12111  df-exp 12170  df-hash 12409  df-word 12546  df-lsw 12547  df-concat 12548  df-s1 12549  df-substr 12550  df-splice 12551  df-reverse 12552  df-s2 12825  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-tset 14731  df-0g 14859  df-gsum 14860  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-subg 16325  df-ghm 16392  df-gim 16434  df-oppg 16508  df-symg 16530  df-pmtr 16594  df-psgn 16643
This theorem is referenced by:  psgnpmtr  16662  psgn0fv0  16663  psgnsn  16672  psgnprfval1  16674  psgnghm  18743
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