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Theorem psgnuni 16320
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 12524 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 16 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 10960 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 12153 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 10963 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 12524 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 16 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 10960 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 12153 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 10963 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 10635 . . . 4  |-  -u 1  e.  CC
16 neg1ne0 10637 . . . 4  |-  -u 1  =/=  0
17 expne0i 12162 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  X
)  e.  ZZ )  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1815, 16, 17mp3an12 1314 . . 3  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1911, 18syl 16 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
20 m1expaddsub 16319 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
214, 11, 20syl2anc 661 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( -u 1 ^ ( ( # `  W )  +  (
# `  X )
) ) )
22 expsub 12177 . . . . . 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 682 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
244, 11, 23syl2anc 661 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
2521, 24eqtr3d 2510 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
26 revcl 12694 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
278, 26syl 16 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
28 ccatlen 12555 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( # `
 ( W concat  (reverse `  X ) ) )  =  ( ( # `  W )  +  (
# `  (reverse `  X
) ) ) )
291, 27, 28syl2anc 661 . . . . . 6  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  (reverse `  X )
) ) )
30 revlen 12695 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
318, 30syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3231oveq2d 6298 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3329, 32eqtrd 2508 . . . . 5  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3433oveq2d 6298 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (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 12554 . . . . . 6  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( W concat  (reverse `  X )
)  e. Word  T )
391, 27, 38syl2anc 661 . . . . 5  |-  ( ph  ->  ( W concat  (reverse `  X
) )  e. Word  T
)
40 psgnuni.e . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
4140fveq2d 5868 . . . . . . . . 9  |-  ( ph  ->  ( ( invg `  G ) `  ( G  gsumg  W ) )  =  ( ( invg `  G ) `  ( G  gsumg  X ) ) )
42 eqid 2467 . . . . . . . . . . 11  |-  ( invg `  G )  =  ( invg `  G )
4336, 35, 42symgtrinv 16293 . . . . . . . . . 10  |-  ( ( D  e.  V  /\  X  e. Word  T )  ->  ( ( invg `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4437, 8, 43syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ( invg `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4541, 44eqtr2d 2509 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( invg `  G
) `  ( G  gsumg  W ) ) )
4645oveq2d 6298 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  W ) ) ) )
4735symggrp 16220 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
4837, 47syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
49 grpmnd 15863 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5048, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
51 eqid 2467 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5236, 35, 51symgtrf 16290 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
53 sswrd 12517 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5452, 53ax-mp 5 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5554, 1sseldi 3502 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5651gsumwcl 15831 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
5750, 55, 56syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
58 eqid 2467 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
59 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6051, 58, 59, 42grprinv 15898 . . . . . . . 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 661 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  W ) ) )  =  ( 0g `  G ) )
6246, 61eqtrd 2508 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6354, 27sseldi 3502 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6451, 58gsumccat 15832 . . . . . . 7  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
)  /\  (reverse `  X
)  e. Word  ( Base `  G ) )  -> 
( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6550, 55, 63, 64syl3anc 1228 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6635symgid 16221 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6737, 66syl 16 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
6862, 65, 673eqtr4d 2518 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  (  _I  |`  D ) )
6935, 36, 37, 39, 68psgnunilem4 16318 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  1 )
7034, 69eqtr3d 2510 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7125, 70eqtr3d 2510 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
727, 14, 19, 71diveq1d 10324 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476    _I cid 4790   ran crn 5000    |` cres 5001   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    - cmin 9801   -ucneg 9802    / cdiv 10202   NN0cn0 10791   ZZcz 10860   ^cexp 12130   #chash 12369  Word cword 12496   concat cconcat 12498  reversecreverse 12502   Basecbs 14486   +g cplusg 14551   0gc0g 14691    gsumg cgsu 14692   Mndcmnd 15722   Grpcgrp 15723   invgcminusg 15724   SymGrpcsymg 16197  pmTrspcpmtr 16262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12072  df-exp 12131  df-hash 12370  df-word 12504  df-concat 12506  df-s1 12507  df-substr 12508  df-splice 12509  df-reverse 12510  df-s2 12772  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-tset 14570  df-0g 14693  df-gsum 14694  df-mnd 15728  df-mhm 15777  df-submnd 15778  df-grp 15858  df-minusg 15859  df-subg 15993  df-ghm 16060  df-gim 16102  df-oppg 16176  df-symg 16198  df-pmtr 16263
This theorem is referenced by:  psgneu  16327  psgndiflemA  18404
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