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Theorem psgnuni 27422
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 11421 . . . . . 6  |-  ( W  e. Word  T  ->  ( # `
 W )  e. 
NN0 )
31, 2syl 15 . . . . 5  |-  ( ph  ->  ( # `  W
)  e.  NN0 )
43nn0zd 10115 . . . 4  |-  ( ph  ->  ( # `  W
)  e.  ZZ )
5 m1expcl 11126 . . . 4  |-  ( (
# `  W )  e.  ZZ  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
64, 5syl 15 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  ZZ )
76zcnd 10118 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  e.  CC )
8 psgnuni.x . . . . . 6  |-  ( ph  ->  X  e. Word  T )
9 lencl 11421 . . . . . 6  |-  ( X  e. Word  T  ->  ( # `
 X )  e. 
NN0 )
108, 9syl 15 . . . . 5  |-  ( ph  ->  ( # `  X
)  e.  NN0 )
1110nn0zd 10115 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  ZZ )
12 m1expcl 11126 . . . 4  |-  ( (
# `  X )  e.  ZZ  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1311, 12syl 15 . . 3  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  ZZ )
1413zcnd 10118 . 2  |-  ( ph  ->  ( -u 1 ^ ( # `  X
) )  e.  CC )
15 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
16 ax-1cn 8795 . . . . 5  |-  1  e.  CC
17 ax-1ne0 8806 . . . . 5  |-  1  =/=  0
1816, 17negne0i 9121 . . . 4  |-  -u 1  =/=  0
19 expne0i 11134 . . . 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 27421 . . . . 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 11149 . . . . . 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 2317 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  ( ( -u
1 ^ ( # `  W ) )  / 
( -u 1 ^ ( # `
 X ) ) ) )
28 revcl 11479 . . . . . . . 8  |-  ( X  e. Word  T  ->  (reverse `  X )  e. Word  T
)
298, 28syl 15 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  T )
30 ccatlen 11430 . . . . . . 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 11480 . . . . . . . 8  |-  ( X  e. Word  T  ->  ( # `
 (reverse `  X
) )  =  (
# `  X )
)
338, 32syl 15 . . . . . . 7  |-  ( ph  ->  ( # `  (reverse `  X ) )  =  ( # `  X
) )
3433oveq2d 5874 . . . . . 6  |-  ( ph  ->  ( ( # `  W
)  +  ( # `  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3531, 34eqtrd 2315 . . . . 5  |-  ( ph  ->  ( # `  ( W concat  (reverse `  X )
) )  =  ( ( # `  W
)  +  ( # `  X ) ) )
3635oveq2d 5874 . . . 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 11429 . . . . . 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 5529 . . . . . . . . 9  |-  ( ph  ->  ( ( inv g `  G ) `  ( G  gsumg  W ) )  =  ( ( inv g `  G ) `  ( G  gsumg  X ) ) )
44 eqid 2283 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
4538, 37, 44symgtrinv 27413 . . . . . . . . . 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 2316 . . . . . . . 8  |-  ( ph  ->  ( G  gsumg  (reverse `  X )
)  =  ( ( inv g `  G
) `  ( G  gsumg  W ) ) )
4847oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( ( G  gsumg  W ) ( +g  `  G
) ( ( inv g `  G ) `
 ( G  gsumg  W ) ) ) )
4937symggrp 14780 . . . . . . . . 9  |-  ( D  e.  V  ->  G  e.  Grp )
5039, 49syl 15 . . . . . . . 8  |-  ( ph  ->  G  e.  Grp )
51 grpmnd 14494 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5250, 51syl 15 . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
53 eqid 2283 . . . . . . . . . . . 12  |-  ( Base `  G )  =  (
Base `  G )
5438, 37, 53symgtrf 27410 . . . . . . . . . . 11  |-  T  C_  ( Base `  G )
55 sswrd 11423 . . . . . . . . . . 11  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
5654, 55ax-mp 8 . . . . . . . . . 10  |- Word  T  C_ Word  (
Base `  G )
5756, 1sseldi 3178 . . . . . . . . 9  |-  ( ph  ->  W  e. Word  ( Base `  G ) )
5853gsumwcl 14463 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( +g  `  G )  =  ( +g  `  G )
61 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
6253, 60, 61, 44grprinv 14529 . . . . . . . 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 2315 . . . . . 6  |-  ( ph  ->  ( ( G  gsumg  W ) ( +g  `  G
) ( G  gsumg  (reverse `  X
) ) )  =  ( 0g `  G
) )
6556, 29sseldi 3178 . . . . . . 7  |-  ( ph  ->  (reverse `  X )  e. Word  ( Base `  G
) )
6653, 60gsumccat 14464 . . . . . . 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 14781 . . . . . . 7  |-  ( D  e.  V  ->  (  _I  |`  D )  =  ( 0g `  G
) )
6939, 68syl 15 . . . . . 6  |-  ( ph  ->  (  _I  |`  D )  =  ( 0g `  G ) )
7064, 67, 693eqtr4d 2325 . . . . 5  |-  ( ph  ->  ( G  gsumg  ( W concat  (reverse `  X
) ) )  =  (  _I  |`  D ) )
7137, 38, 39, 41, 70psgnunilem4 27420 . . . 4  |-  ( ph  ->  ( -u 1 ^ ( # `  ( W concat  (reverse `  X )
) ) )  =  1 )
7236, 71eqtr3d 2317 . . 3  |-  ( ph  ->  ( -u 1 ^ ( ( # `  W
)  +  ( # `  X ) ) )  =  1 )
7327, 72eqtr3d 2317 . 2  |-  ( ph  ->  ( ( -u 1 ^ ( # `  W
) )  /  ( -u 1 ^ ( # `  X ) ) )  =  1 )
747, 14, 21, 73diveq1d 9544 1  |-  ( ph  ->  ( -u 1 ^ ( # `  W
) )  =  (
-u 1 ^ ( # `
 X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152    _I cid 4304   ran crn 4690    |` cres 4691   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   -ucneg 9038    / cdiv 9423   NN0cn0 9965   ZZcz 10024   ^cexp 11104   #chash 11337  Word cword 11403   concat cconcat 11404  reversecreverse 11408   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362   inv gcminusg 14363   SymGrpcsymg 14769  pmTrspcpmtr 27384
This theorem is referenced by:  psgneu  27429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-reverse 11414  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-tset 13227  df-0g 13404  df-gsum 13405  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-ghm 14681  df-gim 14723  df-symg 14770  df-oppg 14819  df-pmtr 27385
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