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Theorem symgtrinv 16089
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 16016 . . . 4  |-  ( D  e.  V  ->  G  e.  Grp )
3 eqid 2451 . . . . 5  |-  (oppg `  G
)  =  (oppg `  G
)
4 symgtrinv.i . . . . 5  |-  I  =  ( invg `  G )
53, 4invoppggim 15986 . . . 4  |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  (oppg `  G
) ) )
6 gimghm 15903 . . . 4  |-  ( I  e.  ( G GrpIso  (oppg `  G
) )  ->  I  e.  ( G  GrpHom  (oppg `  G
) ) )
7 ghmmhm 15868 . . . 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 2451 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
119, 1, 10symgtrf 16086 . . . . 5  |-  T  C_  ( Base `  G )
12 sswrd 12353 . . . . 5  |-  ( T 
C_  ( Base `  G
)  -> Word  T  C_ Word  ( Base `  G ) )
1311, 12ax-mp 5 . . . 4  |- Word  T  C_ Word  (
Base `  G )
1413sseli 3453 . . 3  |-  ( W  e. Word  T  ->  W  e. Word  ( Base `  G
) )
1510gsumwmhm 15634 . . 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 477 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I `  ( G  gsumg  W ) )  =  ( (oppg
`  G )  gsumg  ( I  o.  W ) ) )
1710, 4grpinvf 15693 . . . . . . . 8  |-  ( G  e.  Grp  ->  I : ( Base `  G
) --> ( Base `  G
) )
182, 17syl 16 . . . . . . 7  |-  ( D  e.  V  ->  I : ( Base `  G
) --> ( Base `  G
) )
1918adantr 465 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  I : ( Base `  G ) --> ( Base `  G ) )
20 wrdf 12351 . . . . . . . 8  |-  ( W  e. Word  T  ->  W : ( 0..^ (
# `  W )
) --> T )
2120adantl 466 . . . . . . 7  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> T )
22 fss 5668 . . . . . . 7  |-  ( ( W : ( 0..^ ( # `  W
) ) --> T  /\  T  C_  ( Base `  G
) )  ->  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2321, 11, 22sylancl 662 . . . . . 6  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
24 fco 5669 . . . . . 6  |-  ( ( I : ( Base `  G ) --> ( Base `  G )  /\  W : ( 0..^ (
# `  W )
) --> ( Base `  G
) )  ->  (
I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
) )
2519, 23, 24syl2anc 661 . . . . 5  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
) : ( 0..^ ( # `  W
) ) --> ( Base `  G ) )
26 ffn 5660 . . . . 5  |-  ( ( I  o.  W ) : ( 0..^ (
# `  W )
) --> ( Base `  G
)  ->  ( I  o.  W )  Fn  (
0..^ ( # `  W
) ) )
2725, 26syl 16 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  Fn  ( 0..^ ( # `  W
) ) )
28 ffn 5660 . . . . 5  |-  ( W : ( 0..^ (
# `  W )
) --> T  ->  W  Fn  ( 0..^ ( # `  W ) ) )
2921, 28syl 16 . . . 4  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  W  Fn  ( 0..^ ( # `  W
) ) )
30 fvco2 5868 . . . . . 6  |-  ( ( W  Fn  ( 0..^ ( # `  W
) )  /\  x  e.  ( 0..^ ( # `  W ) ) )  ->  ( ( I  o.  W ) `  x )  =  ( I `  ( W `
 x ) ) )
3129, 30sylan 471 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( I `
 ( W `  x ) ) )
3221ffvelrnda 5945 . . . . . . 7  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  T )
3311, 32sseldi 3455 . . . . . 6  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( W `  x
)  e.  ( Base `  G ) )
341, 10, 4symginv 16018 . . . . . 6  |-  ( ( W `  x )  e.  ( Base `  G
)  ->  ( I `  ( W `  x
) )  =  `' ( W `  x ) )
3533, 34syl 16 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( I `  ( W `  x )
)  =  `' ( W `  x ) )
36 eqid 2451 . . . . . . 7  |-  (pmTrsp `  D )  =  (pmTrsp `  D )
3736, 9pmtrfcnv 16081 . . . . . 6  |-  ( ( W `  x )  e.  T  ->  `' ( W `  x )  =  ( W `  x ) )
3832, 37syl 16 . . . . 5  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  ->  `' ( W `  x )  =  ( W `  x ) )
3931, 35, 383eqtrd 2496 . . . 4  |-  ( ( ( D  e.  V  /\  W  e. Word  T )  /\  x  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( I  o.  W ) `  x
)  =  ( W `
 x ) )
4027, 29, 39eqfnfvd 5902 . . 3  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( I  o.  W
)  =  W )
4140oveq2d 6209 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  ( I  o.  W ) )  =  ( (oppg `  G
)  gsumg  W ) )
42 grpmnd 15661 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
432, 42syl 16 . . 3  |-  ( D  e.  V  ->  G  e.  Mnd )
4410, 3gsumwrev 15992 . . 3  |-  ( ( G  e.  Mnd  /\  W  e. Word  ( Base `  G
) )  ->  (
(oppg `  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W )
) )
4543, 14, 44syl2an 477 . 2  |-  ( ( D  e.  V  /\  W  e. Word  T )  ->  ( (oppg
`  G )  gsumg  W )  =  ( G  gsumg  (reverse `  W
) ) )
4616, 41, 453eqtrd 2496 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 369    = wceq 1370    e. wcel 1758    C_ wss 3429   `'ccnv 4940   ran crn 4942    o. ccom 4945    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   0cc0 9386  ..^cfzo 11658   #chash 12213  Word cword 12332  reversecreverse 12338   Basecbs 14285    gsumg cgsu 14490   Mndcmnd 15520   Grpcgrp 15521   invgcminusg 15522   MndHom cmhm 15573    GrpHom cghm 15855   GrpIso cgim 15896  oppgcoppg 15971   SymGrpcsymg 15993  pmTrspcpmtr 16058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-seq 11917  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-substr 12344  df-reverse 12346  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-tset 14368  df-0g 14491  df-gsum 14492  df-mnd 15526  df-mhm 15575  df-submnd 15576  df-grp 15656  df-minusg 15657  df-ghm 15856  df-gim 15898  df-oppg 15972  df-symg 15994  df-pmtr 16059
This theorem is referenced by:  psgnuni  16116
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