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Theorem symgtrinv 27345
Description: To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Hypotheses
Ref Expression
symgtrinv.t  |-  T  =  ran  (pmTrsp `  D
)
symgtrinv.g  |-  G  =  ( SymGrp `  D )
symgtrinv.i  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
symgtrinv  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( G  gsumg  (reverse `  W )
) )

Proof of Theorem symgtrinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 symgtrinv.g . . . . . 6  |-  G  =  ( SymGrp `  D )
21symggrp 15093 . . . . 5  |-  ( D  e.  V  ->  G  e.  Grp )
3 eqid 2435 . . . . . 6  |-  (oppg `  G
)  =  (oppg `  G
)
4 symgtrinv.i . . . . . 6  |-  I  =  ( inv g `  G )
53, 4invoppggim 15146 . . . . 5  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
62, 5syl 16 . . . 4  |-  ( D  e.  V  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
7 gimghm 15041 . . . 4  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
8 ghmmhm 15006 . . . 4  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
96, 7, 83syl 19 . . 3  |-  ( D  e.  V  ->  I  e.  ( G MndHom  (oppg `  G
) ) )
10 symgtrinv.t . . . . . 6  |-  T  =  ran  (pmTrsp `  D
)
11 eqid 2435 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1210, 1, 11symgtrf 27342 . . . . 5  |-  T  C_  ( Base `  G )
13 sswrd 11727 . . . . 5  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
1412, 13ax-mp 8 . . . 4  |- Word  T  C_ Word  (
Base `  G )
1514sseli 3336 . . 3  |-  ( W  e. Word  T  ->  W  e. Word  ( Base `  G
) )
1611gsumwmhm 14780 . . 3  |-  ( ( I  e.  ( G MndHom 
(oppg `  G ) )  /\  W  e. Word  ( Base `  G
) )  ->  (
I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
179, 15, 16syl2an 464 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
1811, 4grpinvf 14839 . . . . . . . 8  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
192, 18syl 16 . . . . . . 7  |-  ( D  e.  V  ->  I : ( Base `  G
) --> ( Base `  G
) )
2019adantr 452 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  I : ( Base `  G ) --> ( Base `  G ) )
21 wrdf 11723 . . . . . . . 8  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
2221adantl 453 . . . . . . 7  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> T )
23 fss 5591 . . . . . . 7  |-  ( ( W : ( 0..^ ( # `  W
) ) --> T  /\  T  C_  ( Base `  G
) )  ->  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2422, 12, 23sylancl 644 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
25 fco 5592 . . . . . 6  |-  ( ( I : ( Base `  G ) --> ( Base `  G )  /\  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )  ->  (
I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2620, 24, 25syl2anc 643 . . . . 5  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
) : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
27 ffn 5583 . . . . 5  |-  ( ( I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
)  ->  ( I  o.  W )  Fn  (
0..^ ( # `  W
) ) )
2826, 27syl 16 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  Fn  ( 0..^ ( # `  W
) ) )
29 ffn 5583 . . . . 5  |-  ( W : ( 0..^ (
# `  W )
) --> T  ->  W  Fn  ( 0..^ ( # `  W ) ) )
3022, 29syl 16 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
31 fvco2 5790 . . . . . 6  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( I  o.  W ) `  x )  =  ( I `  ( W `
 x ) ) )
3230, 31sylan 458 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( I `
 ( W `  x ) ) )
3322ffvelrnda 5862 . . . . . . 7  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3412, 33sseldi 3338 . . . . . 6  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  ( Base `  G ) )
351, 11, 4symginv 15095 . . . . . 6  |-  ( ( W `  x )  e.  ( Base `  G
)  ->  ( I `  ( W `  x
) )  =  `' ( W `  x ) )
3634, 35syl 16 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( I `  ( W `  x )
)  =  `' ( W `  x ) )
37 eqid 2435 . . . . . . 7  |-  (pmTrsp `  D )  =  (pmTrsp `  D )
3837, 10pmtrfcnv 27337 . . . . . 6  |-  ( ( W `  x )  e.  T  ->  `' ( W `  x )  =  ( W `  x ) )
3933, 38syl 16 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  `' ( W `  x )  =  ( W `  x ) )
4032, 36, 393eqtrd 2471 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( W `
 x ) )
4128, 30, 40eqfnfvd 5822 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  =  W )
4241oveq2d 6089 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  ( I  o.  W ) )  =  ( (oppg `  G
)  gsumg  W ) )
43 grpmnd 14807 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
442, 43syl 16 . . 3  |-  ( D  e.  V  ->  G  e.  Mnd )
4511, 3gsumwrev 15152 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  (
(oppg `  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W )
) )
4644, 15, 45syl2an 464 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W
) ) )
4717, 42, 463eqtrd 2471 1  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( G  gsumg  (reverse `  W )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   `'ccnv 4869   ran crn 4871    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   0cc0 8980  ..^cfzo 11125   #chash 11608  Word cword 11707  reversecreverse 11712   Basecbs 13459    gsumg cgsu 13714   Mndcmnd 14674   Grpcgrp 14675   inv gcminusg 14676   MndHom cmhm 14726    GrpHom cghm 14993   GrpIso cgim 15034   SymGrpcsymg 15082  oppgcoppg 15131  pmTrspcpmtr 27316
This theorem is referenced by:  psgnuni  27354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-fzo 11126  df-seq 11314  df-hash 11609  df-word 11713  df-concat 11714  df-s1 11715  df-substr 11716  df-reverse 11718  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-tset 13538  df-0g 13717  df-gsum 13718  df-mnd 14680  df-mhm 14728  df-submnd 14729  df-grp 14802  df-minusg 14803  df-ghm 14994  df-gim 15036  df-symg 15083  df-oppg 15132  df-pmtr 27317
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