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Theorem symgtrinv 27082
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 15030 . . . . 5  |-  ( D  e.  V  ->  G  e.  Grp )
3 eqid 2387 . . . . . 6  |-  (oppg `  G
)  =  (oppg `  G
)
4 symgtrinv.i . . . . . 6  |-  I  =  ( inv g `  G )
53, 4invoppggim 15083 . . . . 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 14978 . . . 4  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
8 ghmmhm 14943 . . . 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 2387 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
1210, 1, 11symgtrf 27079 . . . . 5  |-  T  C_  ( Base `  G )
13 sswrd 11664 . . . . 5  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
1412, 13ax-mp 8 . . . 4  |- Word  T  C_ Word  (
Base `  G )
1514sseli 3287 . . 3  |-  ( W  e. Word  T  ->  W  e. Word  ( Base `  G
) )
1611gsumwmhm 14717 . . 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 14776 . . . . . . . 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 11660 . . . . . . . 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 5539 . . . . . . 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 5540 . . . . . 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 5531 . . . . 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 5531 . . . . 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 5737 . . . . . 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 5809 . . . . . . 7  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3412, 33sseldi 3289 . . . . . 6  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  ( Base `  G ) )
351, 11, 4symginv 15032 . . . . . 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 2387 . . . . . . 7  |-  (pmTrsp `  D )  =  (pmTrsp `  D )
3837, 10pmtrfcnv 27074 . . . . . 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 2423 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( W `
 x ) )
4128, 30, 40eqfnfvd 5769 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  =  W )
4241oveq2d 6036 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  ( I  o.  W ) )  =  ( (oppg `  G
)  gsumg  W ) )
43 grpmnd 14744 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
442, 43syl 16 . . 3  |-  ( D  e.  V  ->  G  e.  Mnd )
4511, 3gsumwrev 15089 . . 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 2423 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 1649    e. wcel 1717    C_ wss 3263   `'ccnv 4817   ran crn 4819    o. ccom 4822    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   0cc0 8923  ..^cfzo 11065   #chash 11545  Word cword 11644  reversecreverse 11649   Basecbs 13396    gsumg cgsu 13651   Mndcmnd 14611   Grpcgrp 14612   inv gcminusg 14613   MndHom cmhm 14663    GrpHom cghm 14930   GrpIso cgim 14971   SymGrpcsymg 15019  oppgcoppg 15068  pmTrspcpmtr 27053
This theorem is referenced by:  psgnuni  27091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-word 11650  df-concat 11651  df-s1 11652  df-substr 11653  df-reverse 11655  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-tset 13475  df-0g 13654  df-gsum 13655  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-ghm 14931  df-gim 14973  df-symg 15020  df-oppg 15069  df-pmtr 27054
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