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Theorem psgnuni 16107
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 12351 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 16 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 10846 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 11989 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 10849 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 12351 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 16 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 10846 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 11989 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 16 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 10849 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 10526 . . . 4  |-  -u 1  e.  CC
16 neg1ne0 10528 . . . 4  |-  -u 1  =/=  0
17 expne0i 11997 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  X
)  e.  ZZ )  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1815, 16, 17mp3an12 1305 . . 3  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
1911, 18syl 16 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
20 m1expaddsub 16106 . . . . 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 12012 . . . . . 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 2494 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
26 revcl 12503 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
278, 26syl 16 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
28 ccatlen 12377 . . . . . . 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 12504 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
318, 30syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3231oveq2d 6206 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3329, 32eqtrd 2492 . . . . 5  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3433oveq2d 6206 . . . 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 12376 . . . . . 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 5793 . . . . . . . . 9  |-  ( ph  ->  ( ( invg `  G ) `  ( G  gsumg  W ) )  =  ( ( invg `  G ) `  ( G  gsumg  X ) ) )
42 eqid 2451 . . . . . . . . . . 11  |-  ( invg `  G )  =  ( invg `  G )
4336, 35, 42symgtrinv 16080 . . . . . . . . . 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 2493 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( invg `  G
) `  ( G  gsumg  W ) ) )
4645oveq2d 6206 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( invg `  G ) `
 ( G  gsumg  W ) ) ) )
4735symggrp 16007 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
4837, 47syl 16 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
49 grpmnd 15652 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5048, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
51 eqid 2451 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5236, 35, 51symgtrf 16077 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
53 sswrd 12344 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5452, 53ax-mp 5 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5554, 1sseldi 3452 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5651gsumwcl 15620 . . . . . . . . 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 2451 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
59 eqid 2451 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6051, 58, 59, 42grprinv 15687 . . . . . . . 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 2492 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6354, 27sseldi 3452 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6451, 58gsumccat 15621 . . . . . . 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 1219 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6635symgid 16008 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6737, 66syl 16 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
6862, 65, 673eqtr4d 2502 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  (  _I  |`  D ) )
6935, 36, 37, 39, 68psgnunilem4 16105 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  1 )
7034, 69eqtr3d 2494 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7125, 70eqtr3d 2494 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
727, 14, 19, 71diveq1d 10216 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    C_ wss 3426    _I cid 4729   ran crn 4939    |` cres 4940   ` cfv 5516  (class class class)co 6190   CCcc 9381   0cc0 9383   1c1 9384    + caddc 9386    - cmin 9696   -ucneg 9697    / cdiv 10094   NN0cn0 10680   ZZcz 10747   ^cexp 11966   #chash 12204  Word cword 12323   concat cconcat 12325  reversecreverse 12329   Basecbs 14276   +g cplusg 14340   0gc0g 14480    gsumg cgsu 14481   Mndcmnd 15511   Grpcgrp 15512   invgcminusg 15513   SymGrpcsymg 15984  pmTrspcpmtr 16049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1352  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-ot 3984  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-word 12331  df-concat 12333  df-s1 12334  df-substr 12335  df-splice 12336  df-reverse 12337  df-s2 12577  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-tset 14359  df-0g 14482  df-gsum 14483  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-subg 15780  df-ghm 15847  df-gim 15889  df-oppg 15963  df-symg 15985  df-pmtr 16050
This theorem is referenced by:  psgneu  16114  psgndiflemA  18140
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