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Theorem symgtrinv 27413
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 14780 . . . . 5  |-  ( D  e.  V  ->  G  e.  Grp )
3 eqid 2283 . . . . . 6  |-  (oppg `  G
)  =  (oppg `  G
)
4 symgtrinv.i . . . . . 6  |-  I  =  ( inv g `  G )
53, 4invoppggim 14833 . . . . 5  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
62, 5syl 15 . . . 4  |-  ( D  e.  V  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
7 gimghm 14728 . . . 4  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
8 ghmmhm 14693 . . . 4  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
96, 7, 83syl 18 . . 3  |-  ( D  e.  V  ->  I  e.  ( G MndHom  (oppg `  G
) ) )
10 symgtrinv.t . . . . . 6  |-  T  =  ran  (pmTrsp `  D
)
11 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1210, 1, 11symgtrf 27410 . . . . 5  |-  T  C_  ( Base `  G )
13 sswrd 11423 . . . . 5  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
1412, 13ax-mp 8 . . . 4  |- Word  T  C_ Word  (
Base `  G )
1514sseli 3176 . . 3  |-  ( W  e. Word  T  ->  W  e. Word  ( Base `  G
) )
1611gsumwmhm 14467 . . 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 463 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
1811, 4grpinvf 14526 . . . . . . . 8  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
192, 18syl 15 . . . . . . 7  |-  ( D  e.  V  ->  I : ( Base `  G
) --> ( Base `  G
) )
2019adantr 451 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  I : ( Base `  G ) --> ( Base `  G ) )
21 wrdf 11419 . . . . . . . 8  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
2221adantl 452 . . . . . . 7  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> T )
23 fss 5397 . . . . . . 7  |-  ( ( W : ( 0..^ ( # `  W
) ) --> T  /\  T  C_  ( Base `  G
) )  ->  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2422, 12, 23sylancl 643 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
25 fco 5398 . . . . . 6  |-  ( ( I : ( Base `  G ) --> ( Base `  G )  /\  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )  ->  (
I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2620, 24, 25syl2anc 642 . . . . 5  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
) : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
27 ffn 5389 . . . . 5  |-  ( ( I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
)  ->  ( I  o.  W )  Fn  (
0..^ ( # `  W
) ) )
2826, 27syl 15 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  Fn  ( 0..^ ( # `  W
) ) )
29 ffn 5389 . . . . 5  |-  ( W : ( 0..^ (
# `  W )
) --> T  ->  W  Fn  ( 0..^ ( # `  W ) ) )
3022, 29syl 15 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
31 fvco2 5594 . . . . . 6  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( I  o.  W ) `  x )  =  ( I `  ( W `
 x ) ) )
3230, 31sylan 457 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( I `
 ( W `  x ) ) )
33 ffvelrn 5663 . . . . . . . 8  |-  ( ( W : ( 0..^ ( # `  W
) ) --> T  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3422, 33sylan 457 . . . . . . 7  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3512, 34sseldi 3178 . . . . . 6  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  ( Base `  G ) )
361, 11, 4symginv 14782 . . . . . 6  |-  ( ( W `  x )  e.  ( Base `  G
)  ->  ( I `  ( W `  x
) )  =  `' ( W `  x ) )
3735, 36syl 15 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( I `  ( W `  x )
)  =  `' ( W `  x ) )
38 eqid 2283 . . . . . . 7  |-  (pmTrsp `  D )  =  (pmTrsp `  D )
3938, 10pmtrfcnv 27405 . . . . . 6  |-  ( ( W `  x )  e.  T  ->  `' ( W `  x )  =  ( W `  x ) )
4034, 39syl 15 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  `' ( W `  x )  =  ( W `  x ) )
4132, 37, 403eqtrd 2319 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( W `
 x ) )
4228, 30, 41eqfnfvd 5625 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  =  W )
4342oveq2d 5874 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  ( I  o.  W ) )  =  ( (oppg `  G
)  gsumg  W ) )
44 grpmnd 14494 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
452, 44syl 15 . . 3  |-  ( D  e.  V  ->  G  e.  Mnd )
4611, 3gsumwrev 14839 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  (
(oppg `  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W )
) )
4745, 15, 46syl2an 463 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W
) ) )
4817, 43, 473eqtrd 2319 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 358    = wceq 1623    e. wcel 1684    C_ wss 3152   `'ccnv 4688   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   0cc0 8737  ..^cfzo 10870   #chash 11337  Word cword 11403  reversecreverse 11408   Basecbs 13148    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362   inv gcminusg 14363   MndHom cmhm 14413    GrpHom cghm 14680   GrpIso cgim 14721   SymGrpcsymg 14769  oppgcoppg 14818  pmTrspcpmtr 27384
This theorem is referenced by:  psgnuni  27422
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-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-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-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-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-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-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-reverse 11414  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-ghm 14681  df-gim 14723  df-symg 14770  df-oppg 14819  df-pmtr 27385
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