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Theorem symgtrinv 16713
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  =  ( invg `  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 . . . . 5  |-  G  =  ( SymGrp `  D )
21symggrp 16641 . . . 4  |-  ( D  e.  V  ->  G  e.  Grp )
3 eqid 2402 . . . . 5  |-  (oppg `  G
)  =  (oppg `  G
)
4 symgtrinv.i . . . . 5  |-  I  =  ( invg `  G )
53, 4invoppggim 16611 . . . 4  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
6 gimghm 16528 . . . 4  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
7 ghmmhm 16493 . . . 4  |-  ( I  e.  ( G  GrpHom  (oppg `  G ) )  ->  I  e.  ( G MndHom  (oppg `  G ) ) )
82, 5, 6, 74syl 21 . . 3  |-  ( D  e.  V  ->  I  e.  ( G MndHom  (oppg `  G
) ) )
9 symgtrinv.t . . . . . 6  |-  T  =  ran  (pmTrsp `  D
)
10 eqid 2402 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
119, 1, 10symgtrf 16710 . . . . 5  |-  T  C_  ( Base `  G )
12 sswrd 12513 . . . . 5  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
1311, 12ax-mp 5 . . . 4  |- Word  T  C_ Word  (
Base `  G )
1413sseli 3437 . . 3  |-  ( W  e. Word  T  ->  W  e. Word  ( Base `  G
) )
1510gsumwmhm 16229 . . 3  |-  ( ( I  e.  ( G MndHom 
(oppg `  G ) )  /\  W  e. Word  ( Base `  G
) )  ->  (
I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
168, 14, 15syl2an 475 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
1710, 4grpinvf 16310 . . . . . . . 8  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
182, 17syl 17 . . . . . . 7  |-  ( D  e.  V  ->  I : ( Base `  G
) --> ( Base `  G
) )
1918adantr 463 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  I : ( Base `  G ) --> ( Base `  G ) )
20 wrdf 12510 . . . . . . . 8  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
2120adantl 464 . . . . . . 7  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> T )
22 fss 5678 . . . . . . 7  |-  ( ( W : ( 0..^ ( # `  W
) ) --> T  /\  T  C_  ( Base `  G
) )  ->  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2321, 11, 22sylancl 660 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
24 fco 5680 . . . . . 6  |-  ( ( I : ( Base `  G ) --> ( Base `  G )  /\  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )  ->  (
I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2519, 23, 24syl2anc 659 . . . . 5  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
) : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
26 ffn 5670 . . . . 5  |-  ( ( I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
)  ->  ( I  o.  W )  Fn  (
0..^ ( # `  W
) ) )
2725, 26syl 17 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  Fn  ( 0..^ ( # `  W
) ) )
28 ffn 5670 . . . . 5  |-  ( W : ( 0..^ (
# `  W )
) --> T  ->  W  Fn  ( 0..^ ( # `  W ) ) )
2921, 28syl 17 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
30 fvco2 5880 . . . . . 6  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( I  o.  W ) `  x )  =  ( I `  ( W `
 x ) ) )
3129, 30sylan 469 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( I `
 ( W `  x ) ) )
3221ffvelrnda 5965 . . . . . . 7  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3311, 32sseldi 3439 . . . . . 6  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  ( Base `  G ) )
341, 10, 4symginv 16643 . . . . . 6  |-  ( ( W `  x )  e.  ( Base `  G
)  ->  ( I `  ( W `  x
) )  =  `' ( W `  x ) )
3533, 34syl 17 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( I `  ( W `  x )
)  =  `' ( W `  x ) )
36 eqid 2402 . . . . . . 7  |-  (pmTrsp `  D )  =  (pmTrsp `  D )
3736, 9pmtrfcnv 16705 . . . . . 6  |-  ( ( W `  x )  e.  T  ->  `' ( W `  x )  =  ( W `  x ) )
3832, 37syl 17 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  `' ( W `  x )  =  ( W `  x ) )
3931, 35, 383eqtrd 2447 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( W `
 x ) )
4027, 29, 39eqfnfvd 5918 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  =  W )
4140oveq2d 6250 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  ( I  o.  W ) )  =  ( (oppg `  G
)  gsumg  W ) )
42 grpmnd 16278 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
432, 42syl 17 . . 3  |-  ( D  e.  V  ->  G  e.  Mnd )
4410, 3gsumwrev 16617 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  (
(oppg `  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W )
) )
4543, 14, 44syl2an 475 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W
) ) )
4616, 41, 453eqtrd 2447 1  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( G  gsumg  (reverse `  W )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3413   `'ccnv 4941   ran crn 4943    o. ccom 4946    Fn wfn 5520   -->wf 5521   ` cfv 5525  (class class class)co 6234   0cc0 9442  ..^cfzo 11767   #chash 12359  Word cword 12490  reversecreverse 12496   Basecbs 14733    gsumg cgsu 14947   Mndcmnd 16135   MndHom cmhm 16180   Grpcgrp 16269   invgcminusg 16270    GrpHom cghm 16480   GrpIso cgim 16521  oppgcoppg 16596   SymGrpcsymg 16618  pmTrspcpmtr 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-tpos 6912  df-recs 6999  df-rdg 7033  df-1o 7087  df-2o 7088  df-oadd 7091  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-card 8272  df-cda 8500  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-n0 10757  df-z 10826  df-uz 11046  df-fz 11644  df-fzo 11768  df-seq 12062  df-hash 12360  df-word 12498  df-lsw 12499  df-concat 12500  df-s1 12501  df-substr 12502  df-reverse 12504  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-tset 14820  df-0g 14948  df-gsum 14949  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-mhm 16182  df-submnd 16183  df-grp 16273  df-minusg 16274  df-ghm 16481  df-gim 16523  df-oppg 16597  df-symg 16619  df-pmtr 16683
This theorem is referenced by:  psgnuni  16740
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