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Theorem psgnuni 27525
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 11437 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 15 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 10131 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 11142 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 15 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 10134 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 11437 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 15 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 10131 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 11142 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 15 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 10134 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 9829 . . . 4  |-  -u 1  e.  CC
16 ax-1cn 8811 . . . . 5  |-  1  e.  CC
17 ax-1ne0 8822 . . . . 5  |-  1  =/=  0
1816, 17negne0i 9137 . . . 4  |-  -u 1  =/=  0
19 expne0i 11150 . . . 4  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  ( # `  X
)  e.  ZZ )  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
2015, 18, 19mp3an12 1267 . . 3  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
2111, 20syl 15 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  =/=  0
)
22 m1expaddsub 27524 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
234, 11, 22syl2anc 642 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( -u 1 ^ ( ( # `  W )  +  (
# `  X )
) ) )
24 expsub 11165 . . . . . 6  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  (
( # `  W )  e.  ZZ  /\  ( # `
 X )  e.  ZZ ) )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
2515, 18, 24mpanl12 663 . . . . 5  |-  ( ( ( # `  W
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( -u 1 ^ (
( # `  W )  -  ( # `  X
) ) )  =  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) ) )
264, 11, 25syl2anc 642 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  -  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
2723, 26eqtr3d 2330 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
28 revcl 11495 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
298, 28syl 15 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
30 ccatlen 11446 . . . . . . 7  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( # `
 ( W concat  (reverse `  X ) ) )  =  ( ( # `  W )  +  (
# `  (reverse `  X
) ) ) )
311, 29, 30syl2anc 642 . . . . . 6  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  (reverse `  X )
) ) )
32 revlen 11496 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
338, 32syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3433oveq2d 5890 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3531, 34eqtrd 2328 . . . . 5  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3635oveq2d 5890 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) ) )
37 psgnuni.g . . . . 5  |-  G  =  ( SymGrp `  D )
38 psgnuni.t . . . . 5  |-  T  =  ran  (pmTrsp `  D
)
39 psgnuni.d . . . . 5  |-  ( ph  ->  D  e.  V )
40 ccatcl 11445 . . . . . 6  |-  ( ( W  e. Word  T  /\  (reverse `  X )  e. Word  T )  ->  ( W concat  (reverse `  X )
)  e. Word  T )
411, 29, 40syl2anc 642 . . . . 5  |-  ( ph  ->  ( W concat  (reverse `  X
) )  e. Word  T
)
42 psgnuni.e . . . . . . . . . 10  |-  ( ph  ->  ( G  gsumg  W )  =  ( G  gsumg  X ) )
4342fveq2d 5545 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  ( G  gsumg  W ) )  =  ( ( inv g `  G ) `  ( G  gsumg  X ) ) )
44 eqid 2296 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
4538, 37, 44symgtrinv 27516 . . . . . . . . . 10  |-  ( ( D  e.  V  /\  X  e. Word  T )  ->  ( ( inv g `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4639, 8, 45syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  ( G  gsumg  X ) )  =  ( G  gsumg  (reverse `  X )
) )
4743, 46eqtr2d 2329 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( inv g `  G
) `  ( G  gsumg  W ) ) )
4847oveq2d 5890 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  W ) ) ) )
4937symggrp 14796 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
5039, 49syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
51 grpmnd 14510 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5250, 51syl 15 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
53 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5438, 37, 53symgtrf 27513 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
55 sswrd 11439 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5654, 55ax-mp 8 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5756, 1sseldi 3191 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5853gsumwcl 14479 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
5952, 57, 58syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  W )  e.  (
Base `  G )
)
60 eqid 2296 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
61 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6253, 60, 61, 44grprinv 14545 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( G  gsumg  W )  e.  (
Base `  G )
)  ->  ( ( G  gsumg  W ) ( +g  `  G ) ( ( inv g `  G
) `  ( G  gsumg  W ) ) )  =  ( 0g `  G
) )
6350, 59, 62syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  W ) ) )  =  ( 0g `  G ) )
6448, 63eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6556, 29sseldi 3191 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6653, 60gsumccat 14480 . . . . . . 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
) ) ) )
6752, 57, 65, 66syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) ) )
6837symgid 14797 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6939, 68syl 15 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
7064, 67, 693eqtr4d 2338 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  (  _I  |`  D ) )
7137, 38, 39, 41, 70psgnunilem4 27523 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  1 )
7236, 71eqtr3d 2330 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7327, 72eqtr3d 2330 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
747, 14, 21, 73diveq1d 9560 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165    _I cid 4320   ran crn 4706    |` cres 4707   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    - cmin 9053   -ucneg 9054    / cdiv 9439   NN0cn0 9981   ZZcz 10040   ^cexp 11120   #chash 11353  Word cword 11419   concat cconcat 11420  reversecreverse 11424   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378   inv gcminusg 14379   SymGrpcsymg 14785  pmTrspcpmtr 27487
This theorem is referenced by:  psgneu  27532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-xor 1296  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-reverse 11430  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-tset 13243  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-ghm 14697  df-gim 14739  df-symg 14786  df-oppg 14835  df-pmtr 27488
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