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

Theorem psgnuni 16848
Description: If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Hypotheses
Ref Expression
psgnuni.g  |-  G  =  ( SymGrp `  D )
psgnuni.t  |-  T  =  ran  (pmTrsp `  D
)
psgnuni.d  |-  ( ph  ->  D  e.  V )
psgnuni.w  |-  ( ph  ->  W  e. Word  T )
psgnuni.x  |-  ( ph  ->  X  e. Word  T )
psgnuni.e  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
Assertion
Ref Expression
psgnuni  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )

Proof of Theorem psgnuni
StepHypRef Expression
1 psgnuni.w . . . . . 6  |-  ( ph  ->  W  e. Word  T )
2 lencl 12614 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 17 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 11006 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 12233 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 17 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 11009 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 12614 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 17 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 11006 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 12233 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 17 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 11009 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 10680 . . . 4  |-  -u 1  e.  CC
16 neg1ne0 10682 . . . 4  |-  -u 1  =/=  0
17 expne0i 12242 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  X
)  e.  ZZ )  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1815, 16, 17mp3an12 1316 . . 3  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1911, 18syl 17 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
20 m1expaddsub 16847 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
214, 11, 20syl2anc 659 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( -u 1 ^ ( ( # `  W )  +  (
# `  X )
) ) )
22 expsub 12258 . . . . . 6  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  (
( # `  W )  e.  ZZ  /\  ( # `
 X )  e.  ZZ ) )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
2315, 16, 22mpanl12 680 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
244, 11, 23syl2anc 659 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
2521, 24eqtr3d 2445 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
26 revcl 12791 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
278, 26syl 17 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
28 ccatlen 12648 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( # `
 ( W ++  (reverse `  X ) ) )  =  ( ( # `  W )  +  (
# `  (reverse `  X
) ) ) )
291, 27, 28syl2anc 659 . . . . . 6  |-  ( ph  ->  ( # `  ( W ++  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  (reverse `  X )
) ) )
30 revlen 12792 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
318, 30syl 17 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3231oveq2d 6294 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3329, 32eqtrd 2443 . . . . 5  |-  ( ph  ->  ( # `  ( W ++  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3433oveq2d 6294 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W ++  (reverse `  X )
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
35 psgnuni.g . . . . 5  |-  G  =  ( SymGrp `  D )
36 psgnuni.t . . . . 5  |-  T  =  ran  (pmTrsp `  D
)
37 psgnuni.d . . . . 5  |-  ( ph  ->  D  e.  V )
38 ccatcl 12647 . . . . . 6  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( W ++  (reverse `  X )
)  e. Word  T )
391, 27, 38syl2anc 659 . . . . 5  |-  ( ph  ->  ( W ++  (reverse `  X
) )  e. Word  T
)
40 psgnuni.e . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
4140fveq2d 5853 . . . . . . . . 9  |-  ( ph  ->  ( ( invg `  G ) `  ( G  gsumg  W ) )  =  ( ( invg `  G ) `  ( G  gsumg  X ) ) )
42 eqid 2402 . . . . . . . . . . 11  |-  ( invg `  G )  =  ( invg `  G )
4336, 35, 42symgtrinv 16821 . . . . . . . . . 10  |-  ( ( D  e.  V  /\  X  e. Word  T )  ->  ( ( invg `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4437, 8, 43syl2anc 659 . . . . . . . . 9  |-  ( ph  ->  ( ( invg `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4541, 44eqtr2d 2444 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( invg `  G
) `  ( G  gsumg  W ) ) )
4645oveq2d 6294 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  W ) ) ) )
4735symggrp 16749 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
4837, 47syl 17 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
49 grpmnd 16386 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5048, 49syl 17 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
51 eqid 2402 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5236, 35, 51symgtrf 16818 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
53 sswrd 12606 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5452, 53ax-mp 5 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5554, 1sseldi 3440 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5651gsumwcl 16332 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
5750, 55, 56syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
58 eqid 2402 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
59 eqid 2402 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6051, 58, 59, 42grprinv 16421 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( G  gsumg  W )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  W ) ( +g  `  G ) ( ( invg `  G
) `  ( G  gsumg  W ) ) )  =  ( 0g `  G
) )
6148, 57, 60syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  W ) ) )  =  ( 0g `  G ) )
6246, 61eqtrd 2443 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6354, 27sseldi 3440 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6451, 58gsumccat 16333 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
)  /\  (reverse `  X
)  e. Word  ( Base `  G ) )  -> 
( G  gsumg  ( W ++  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6550, 55, 63, 64syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W ++  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6635symgid 16750 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6737, 66syl 17 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
6862, 65, 673eqtr4d 2453 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W ++  (reverse `  X
) ) )  =  (  _I  |`  D ) )
6935, 36, 37, 39, 68psgnunilem4 16846 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W ++  (reverse `  X )
) ) )  =  1 )
7034, 69eqtr3d 2445 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7125, 70eqtr3d 2445 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
727, 14, 19, 71diveq1d 10369 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598    C_ wss 3414    _I cid 4733   ran crn 4824    |` cres 4825   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    - cmin 9841   -ucneg 9842    / cdiv 10247   NN0cn0 10836   ZZcz 10905   ^cexp 12210   #chash 12452  Word cword 12583   ++ cconcat 12585  reversecreverse 12589   Basecbs 14841   +g cplusg 14909   0gc0g 15054    gsumg cgsu 15055   Mndcmnd 16243   Grpcgrp 16377   invgcminusg 16378   SymGrpcsymg 16726  pmTrspcpmtr 16790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-xor 1367  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-ot 3981  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-word 12591  df-lsw 12592  df-concat 12593  df-s1 12594  df-substr 12595  df-splice 12596  df-reverse 12597  df-s2 12869  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-tset 14928  df-0g 15056  df-gsum 15057  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-subg 16522  df-ghm 16589  df-gim 16631  df-oppg 16705  df-symg 16727  df-pmtr 16791
This theorem is referenced by:  psgneu  16855  psgndiflemA  18935
  Copyright terms: Public domain W3C validator