Theorem symgtrinv 17892

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.

Step	Hyp	Ref	Expression
1		symgtrinv.g	. . . . 5 |- G = ( SymGrp ` D )
2	1	symggrp 17820	. . . 4 |- ( D e. V -> G e. Grp )
3		eqid 2622	. . . . 5 |- (oppg ` G ) = (oppg ` G )
4		symgtrinv.i	. . . . 5 |- I = ( invg ` G )
5	3, 4	invoppggim 17790	. . . 4 |- ( G e. Grp -> I e. ( G GrpIso (oppg ` G ) ) )
6		gimghm 17706	. . . 4 |- ( I e. ( G GrpIso (oppg ` G ) ) -> I e. ( G GrpHom (oppg ` G ) ) )
7		ghmmhm 17670	. . . 4 |- ( I e. ( G GrpHom (oppg ` G ) ) -> I e. ( G MndHom (oppg ` G ) ) )
8	2, 5, 6, 7	4syl 19	. . 3 |- ( D e. V -> I e. ( G MndHom (oppg ` G ) ) )
9		symgtrinv.t	. . . . . 6 |- T = ran (pmTrsp ` D )
10		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
11	9, 1, 10	symgtrf 17889	. . . . 5 |- T C_ ( Base ` G )
12		sswrd 13313	. . . . 5 |- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) )
13	11, 12	ax-mp 5	. . . 4 |- Word T C_ Word ( Base ` G )
14	13	sseli 3599	. . 3 |- ( W e. Word T -> W e. Word ( Base ` G ) )
15	10	gsumwmhm 17382	. . 3 |- ( ( I e. ( G MndHom (oppg ` G ) ) /\ W e. Word ( Base ` G ) ) -> ( I ` ( G gsumg W ) ) = ( (oppg ` G ) gsumg ( I o. W ) ) )
16	8, 14, 15	syl2an 494	. 2 |- ( ( D e. V /\ W e. Word T ) -> ( I ` ( G gsumg W ) ) = ( (oppg ` G ) gsumg ( I o. W ) ) )
17	10, 4	grpinvf 17466	. . . . . . . 8 |- ( G e. Grp -> I : ( Base ` G ) --> ( Base ` G ) )
18	2, 17	syl 17	. . . . . . 7 |- ( D e. V -> I : ( Base ` G ) --> ( Base ` G ) )
19	18	adantr 481	. . . . . 6 |- ( ( D e. V /\ W e. Word T ) -> I : ( Base ` G ) --> ( Base ` G ) )
20		wrdf 13310	. . . . . . . 8 |- ( W e. Word T -> W : ( 0..^ ( # ` W ) ) --> T )
21	20	adantl 482	. . . . . . 7 |- ( ( D e. V /\ W e. Word T ) -> W : ( 0..^ ( # ` W ) ) --> T )
22		fss 6056	. . . . . . 7 |- ( ( W : ( 0..^ ( # ` W ) ) --> T /\ T C_ ( Base ` G ) ) -> W : ( 0..^ ( # ` W ) ) --> ( Base ` G ) )
23	21, 11, 22	sylancl 694	. . . . . 6 |- ( ( D e. V /\ W e. Word T ) -> W : ( 0..^ ( # ` W ) ) --> ( Base ` G ) )
24		fco 6058	. . . . . 6 |- ( ( I : ( Base ` G ) --> ( Base ` G ) /\ W : ( 0..^ ( # ` W ) ) --> ( Base ` G ) ) -> ( I o. W ) : ( 0..^ ( # ` W ) ) --> ( Base ` G ) )
25	19, 23, 24	syl2anc 693	. . . . 5 |- ( ( D e. V /\ W e. Word T ) -> ( I o. W ) : ( 0..^ ( # ` W ) ) --> ( Base ` G ) )
26		ffn 6045	. . . . 5 |- ( ( I o. W ) : ( 0..^ ( # ` W ) ) --> ( Base ` G ) -> ( I o. W ) Fn ( 0..^ ( # ` W ) ) )
27	25, 26	syl 17	. . . 4 |- ( ( D e. V /\ W e. Word T ) -> ( I o. W ) Fn ( 0..^ ( # ` W ) ) )
28		ffn 6045	. . . . 5 |- ( W : ( 0..^ ( # ` W ) ) --> T -> W Fn ( 0..^ ( # ` W ) ) )
29	21, 28	syl 17	. . . 4 |- ( ( D e. V /\ W e. Word T ) -> W Fn ( 0..^ ( # ` W ) ) )
30		fvco2 6273	. . . . . 6 |- ( ( W Fn ( 0..^ ( # ` W ) ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( I o. W ) ` x ) = ( I ` ( W ` x ) ) )
31	29, 30	sylan 488	. . . . 5 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( I o. W ) ` x ) = ( I ` ( W ` x ) ) )
32	21	ffvelrnda 6359	. . . . . . 7 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( W ` x ) e. T )
33	11, 32	sseldi 3601	. . . . . 6 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( W ` x ) e. ( Base ` G ) )
34	1, 10, 4	symginv 17822	. . . . . 6 |- ( ( W ` x ) e. ( Base ` G ) -> ( I ` ( W ` x ) ) = `' ( W ` x ) )
35	33, 34	syl 17	. . . . 5 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( I ` ( W ` x ) ) = `' ( W ` x ) )
36		eqid 2622	. . . . . . 7 |- (pmTrsp ` D ) = (pmTrsp ` D )
37	36, 9	pmtrfcnv 17884	. . . . . 6 |- ( ( W ` x ) e. T -> `' ( W ` x ) = ( W ` x ) )
38	32, 37	syl 17	. . . . 5 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> `' ( W ` x ) = ( W ` x ) )
39	31, 35, 38	3eqtrd 2660	. . . 4 |- ( ( ( D e. V /\ W e. Word T ) /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( I o. W ) ` x ) = ( W ` x ) )
40	27, 29, 39	eqfnfvd 6314	. . 3 |- ( ( D e. V /\ W e. Word T ) -> ( I o. W ) = W )
41	40	oveq2d 6666	. 2 |- ( ( D e. V /\ W e. Word T ) -> ( (oppg ` G ) gsumg ( I o. W ) ) = ( (oppg ` G ) gsumg W ) )
42		grpmnd 17429	. . . 4 |- ( G e. Grp -> G e. Mnd )
43	2, 42	syl 17	. . 3 |- ( D e. V -> G e. Mnd )
44	10, 3	gsumwrev 17796	. . 3 |- ( ( G e. Mnd /\ W e. Word ( Base ` G ) ) -> ( (oppg ` G ) gsumg W ) = ( G gsumg (reverse ` W ) ) )
45	43, 14, 44	syl2an 494	. 2 |- ( ( D e. V /\ W e. Word T ) -> ( (oppg ` G ) gsumg W ) = ( G gsumg (reverse ` W ) ) )
46	16, 41, 45	3eqtrd 2660	1 |- ( ( D e. V /\ W e. Word T ) -> ( I ` ( G gsumg W ) ) = ( G gsumg (reverse ` W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   `'ccnv 5113   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936  ..^cfzo 12465   #chash 13117  Word cword 13291  reversecreverse 13297   Basecbs 15857   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333   Grpcgrp 17422   invgcminusg 17423   GrpHom cghm 17657   GrpIso cgim 17699  opp_gcoppg 17775   SymGrpcsymg 17797  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-reverse 13305  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-ghm 17658  df-gim 17701  df-oppg 17776  df-symg 17798  df-pmtr 17862

This theorem is referenced by:  psgnuni  17919