MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psgnvalii Structured version   Unicode version

Theorem psgnvalii 16015
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 16011 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  e.  dom  N )
51, 2, 3psgnval 16013 . . 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 2444 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( G  gsumg  W )  =  ( G  gsumg  W ) )
9 eqidd 2444 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 W ) ) )
10 oveq2 6099 . . . . . . 7  |-  ( w  =  W  ->  ( G  gsumg  w )  =  ( G  gsumg  W ) )
1110eqeq2d 2454 . . . . . 6  |-  ( w  =  W  ->  (
( G  gsumg  W )  =  ( G  gsumg  w )  <->  ( G  gsumg  W )  =  ( G 
gsumg  W ) ) )
12 fveq2 5691 . . . . . . . 8  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
1312oveq2d 6107 . . . . . . 7  |-  ( w  =  W  ->  ( -u 1 ^ ( # `  w ) )  =  ( -u 1 ^ ( # `  W
) ) )
1413eqeq2d 2454 . . . . . 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 3073 . . . 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 1216 . . 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 6116 . . . . 5  |-  ( -u
1 ^ ( # `  W ) )  e. 
_V
1918a1i 11 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( -u 1 ^ ( # `  W
) )  e.  _V )
201, 2, 3psgneu 16012 . . . . 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 2449 . . . . . . 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 2736 . . . . 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 5406 . . 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 2475 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 1369    e. wcel 1756   E!weu 2253   E.wrex 2716   _Vcvv 2972   dom cdm 4840   ran crn 4841   iotacio 5379   ` cfv 5418  (class class class)co 6091   1c1 9283   -ucneg 9596   ^cexp 11865   #chash 12103  Word cword 12221    gsumg cgsu 14379   SymGrpcsymg 15882  pmTrspcpmtr 15947  pmSgncpsgn 15995
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-reverse 12235  df-s2 12475  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-tset 14257  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-subg 15678  df-ghm 15745  df-gim 15787  df-oppg 15861  df-symg 15883  df-pmtr 15948  df-psgn 15997
This theorem is referenced by:  psgnpmtr  16016  psgn0fv0  16017  psgnprfval1  16026  psgnghm  18010  psgnsn  30771
  Copyright terms: Public domain W3C validator