Theorem psgnuni 17919

Description: If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Hypotheses

Ref	Expression
psgnuni.g	|- G = ( SymGrp ` D )
psgnuni.t	|- T = ran (pmTrsp ` D )
psgnuni.d	|- ( ph -> D e. V )
psgnuni.w	|- ( ph -> W e. Word T )
psgnuni.x	|- ( ph -> X e. Word T )
psgnuni.e	|- ( ph -> ( G gsumg W ) = ( G gsumg X ) )

Assertion

Ref	Expression
psgnuni	|- ( ph -> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` X ) ) )

Proof of Theorem psgnuni

Step	Hyp	Ref	Expression
1		psgnuni.w	. . . . . 6 |- ( ph -> W e. Word T )
2		lencl 13324	. . . . . 6 |- ( W e. Word T -> ( # ` W ) e. NN0 )
3	1, 2	syl 17	. . . . 5 |- ( ph -> ( # ` W ) e. NN0 )
4	3	nn0zd 11480	. . . 4 |- ( ph -> ( # ` W ) e. ZZ )
5		m1expcl 12883	. . . 4 |- ( ( # ` W ) e. ZZ -> ( -u 1 ^ ( # ` W ) ) e. ZZ )
6	4, 5	syl 17	. . 3 |- ( ph -> ( -u 1 ^ ( # ` W ) ) e. ZZ )
7	6	zcnd 11483	. 2 |- ( ph -> ( -u 1 ^ ( # ` W ) ) e. CC )
8		psgnuni.x	. . . . . 6 |- ( ph -> X e. Word T )
9		lencl 13324	. . . . . 6 |- ( X e. Word T -> ( # ` X ) e. NN0 )
10	8, 9	syl 17	. . . . 5 |- ( ph -> ( # ` X ) e. NN0 )
11	10	nn0zd 11480	. . . 4 |- ( ph -> ( # ` X ) e. ZZ )
12		m1expcl 12883	. . . 4 |- ( ( # ` X ) e. ZZ -> ( -u 1 ^ ( # ` X ) ) e. ZZ )
13	11, 12	syl 17	. . 3 |- ( ph -> ( -u 1 ^ ( # ` X ) ) e. ZZ )
14	13	zcnd 11483	. 2 |- ( ph -> ( -u 1 ^ ( # ` X ) ) e. CC )
15		neg1cn 11124	. . . 4 |- -u 1 e. CC
16		neg1ne0 11126	. . . 4 |- -u 1 =/= 0
17		expne0i 12892	. . . 4 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( # ` X ) ) =/= 0 )
18	15, 16, 17	mp3an12 1414	. . 3 |- ( ( # ` X ) e. ZZ -> ( -u 1 ^ ( # ` X ) ) =/= 0 )
19	11, 18	syl 17	. 2 |- ( ph -> ( -u 1 ^ ( # ` X ) ) =/= 0 )
20		m1expaddsub 17918	. . . . 5 |- ( ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) )
21	4, 11, 20	syl2anc 693	. . . 4 |- ( ph -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) )
22		expsub 12908	. . . . . 6 |- ( ( ( -u 1 e. CC /\ -u 1 =/= 0 ) /\ ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) )
23	15, 16, 22	mpanl12 718	. . . . 5 |- ( ( ( # ` W ) e. ZZ /\ ( # ` X ) e. ZZ ) -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) )
24	4, 11, 23	syl2anc 693	. . . 4 |- ( ph -> ( -u 1 ^ ( ( # ` W ) - ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) )
25	21, 24	eqtr3d 2658	. . 3 |- ( ph -> ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) = ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) )
26		revcl 13510	. . . . . . . 8 |- ( X e. Word T -> (reverse ` X ) e. Word T )
27	8, 26	syl 17	. . . . . . 7 |- ( ph -> (reverse ` X ) e. Word T )
28		ccatlen 13360	. . . . . . 7 |- ( ( W e. Word T /\ (reverse ` X ) e. Word T ) -> ( # ` ( W ++ (reverse ` X ) ) ) = ( ( # ` W ) + ( # ` (reverse ` X ) ) ) )
29	1, 27, 28	syl2anc 693	. . . . . 6 |- ( ph -> ( # ` ( W ++ (reverse ` X ) ) ) = ( ( # ` W ) + ( # ` (reverse ` X ) ) ) )
30		revlen 13511	. . . . . . . 8 |- ( X e. Word T -> ( # ` (reverse ` X ) ) = ( # ` X ) )
31	8, 30	syl 17	. . . . . . 7 |- ( ph -> ( # ` (reverse ` X ) ) = ( # ` X ) )
32	31	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( # ` W ) + ( # ` (reverse ` X ) ) ) = ( ( # ` W ) + ( # ` X ) ) )
33	29, 32	eqtrd 2656	. . . . 5 |- ( ph -> ( # ` ( W ++ (reverse ` X ) ) ) = ( ( # ` W ) + ( # ` X ) ) )
34	33	oveq2d 6666	. . . 4 |- ( ph -> ( -u 1 ^ ( # ` ( W ++ (reverse ` X ) ) ) ) = ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) )
35		psgnuni.g	. . . . 5 |- G = ( SymGrp ` D )
36		psgnuni.t	. . . . 5 |- T = ran (pmTrsp ` D )
37		psgnuni.d	. . . . 5 |- ( ph -> D e. V )
38		ccatcl 13359	. . . . . 6 |- ( ( W e. Word T /\ (reverse ` X ) e. Word T ) -> ( W ++ (reverse ` X ) ) e. Word T )
39	1, 27, 38	syl2anc 693	. . . . 5 |- ( ph -> ( W ++ (reverse ` X ) ) e. Word T )
40		psgnuni.e	. . . . . . . . . 10 |- ( ph -> ( G gsumg W ) = ( G gsumg X ) )
41	40	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( ( invg ` G ) ` ( G gsumg W ) ) = ( ( invg ` G ) ` ( G gsumg X ) ) )
42		eqid 2622	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
43	36, 35, 42	symgtrinv 17892	. . . . . . . . . 10 |- ( ( D e. V /\ X e. Word T ) -> ( ( invg ` G ) ` ( G gsumg X ) ) = ( G gsumg (reverse ` X ) ) )
44	37, 8, 43	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( invg ` G ) ` ( G gsumg X ) ) = ( G gsumg (reverse ` X ) ) )
45	41, 44	eqtr2d 2657	. . . . . . . 8 |- ( ph -> ( G gsumg (reverse ` X ) ) = ( ( invg ` G ) ` ( G gsumg W ) ) )
46	45	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) ( +g ` G ) ( G gsumg (reverse ` X ) ) ) = ( ( G gsumg W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsumg W ) ) ) )
47	35	symggrp 17820	. . . . . . . . 9 |- ( D e. V -> G e. Grp )
48	37, 47	syl 17	. . . . . . . 8 |- ( ph -> G e. Grp )
49		grpmnd 17429	. . . . . . . . . 10 |- ( G e. Grp -> G e. Mnd )
50	48, 49	syl 17	. . . . . . . . 9 |- ( ph -> G e. Mnd )
51		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` G ) = ( Base ` G )
52	36, 35, 51	symgtrf 17889	. . . . . . . . . . 11 |- T C_ ( Base ` G )
53		sswrd 13313	. . . . . . . . . . 11 |- ( T C_ ( Base ` G ) -> Word T C_ Word ( Base ` G ) )
54	52, 53	ax-mp 5	. . . . . . . . . 10 |- Word T C_ Word ( Base ` G )
55	54, 1	sseldi 3601	. . . . . . . . 9 |- ( ph -> W e. Word ( Base ` G ) )
56	51	gsumwcl 17377	. . . . . . . . 9 |- ( ( G e. Mnd /\ W e. Word ( Base ` G ) ) -> ( G gsumg W ) e. ( Base ` G ) )
57	50, 55, 56	syl2anc 693	. . . . . . . 8 |- ( ph -> ( G gsumg W ) e. ( Base ` G ) )
58		eqid 2622	. . . . . . . . 9 |- ( +g ` G ) = ( +g ` G )
59		eqid 2622	. . . . . . . . 9 |- ( 0g ` G ) = ( 0g ` G )
60	51, 58, 59, 42	grprinv 17469	. . . . . . . 8 |- ( ( G e. Grp /\ ( G gsumg W ) e. ( Base ` G ) ) -> ( ( G gsumg W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsumg W ) ) ) = ( 0g ` G ) )
61	48, 57, 60	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( G gsumg W ) ( +g ` G ) ( ( invg ` G ) ` ( G gsumg W ) ) ) = ( 0g ` G ) )
62	46, 61	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( G gsumg W ) ( +g ` G ) ( G gsumg (reverse ` X ) ) ) = ( 0g ` G ) )
63	54, 27	sseldi 3601	. . . . . . 7 |- ( ph -> (reverse ` X ) e. Word ( Base ` G ) )
64	51, 58	gsumccat 17378	. . . . . . 7 |- ( ( G e. Mnd /\ W e. Word ( Base ` G ) /\ (reverse ` X ) e. Word ( Base ` G ) ) -> ( G gsumg ( W ++ (reverse ` X ) ) ) = ( ( G gsumg W ) ( +g ` G ) ( G gsumg (reverse ` X ) ) ) )
65	50, 55, 63, 64	syl3anc 1326	. . . . . 6 |- ( ph -> ( G gsumg ( W ++ (reverse ` X ) ) ) = ( ( G gsumg W ) ( +g ` G ) ( G gsumg (reverse ` X ) ) ) )
66	35	symgid 17821	. . . . . . 7 |- ( D e. V -> ( _I |` D ) = ( 0g ` G ) )
67	37, 66	syl 17	. . . . . 6 |- ( ph -> ( _I |` D ) = ( 0g ` G ) )
68	62, 65, 67	3eqtr4d 2666	. . . . 5 |- ( ph -> ( G gsumg ( W ++ (reverse ` X ) ) ) = ( _I |` D ) )
69	35, 36, 37, 39, 68	psgnunilem4 17917	. . . 4 |- ( ph -> ( -u 1 ^ ( # ` ( W ++ (reverse ` X ) ) ) ) = 1 )
70	34, 69	eqtr3d 2658	. . 3 |- ( ph -> ( -u 1 ^ ( ( # ` W ) + ( # ` X ) ) ) = 1 )
71	25, 70	eqtr3d 2658	. 2 |- ( ph -> ( ( -u 1 ^ ( # ` W ) ) / ( -u 1 ^ ( # ` X ) ) ) = 1 )
72	7, 14, 19, 71	diveq1d 10809	1 |- ( ph -> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   _I cid 5023   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ^cexp 12860   #chash 13117  Word cword 13291   ++ cconcat 13293  reversecreverse 13297   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422   invgcminusg 17423   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-xor 1465  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-ot 4186  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-se 5074  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-isom 5897  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-div 10685  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-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  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-subg 17591  df-ghm 17658  df-gim 17701  df-oppg 17776  df-symg 17798  df-pmtr 17862

This theorem is referenced by:  psgneu  17926  psgndiflemA  19947