Theorem psgneu 17926

Description: A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)

Hypotheses

Ref	Expression
psgnval.g	|- G = ( SymGrp ` D )
psgnval.t	|- T = ran (pmTrsp ` D )
psgnval.n	|- N = (pmSgn ` D )

Assertion

Ref	Expression
psgneu	|- ( P e. dom N -> E! s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )

Distinct variable groups:   w, s, G   N, s, w   P, s, w   T, s, w   D, s, w

Proof of Theorem psgneu

Dummy variables t x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnval.g	. . . . . . . . 9 |- G = ( SymGrp ` D )
2		psgnval.n	. . . . . . . . 9 |- N = (pmSgn ` D )
3		eqid 2622	. . . . . . . . 9 |- ( Base ` G ) = ( Base ` G )
4	1, 2, 3	psgneldm 17923	. . . . . . . 8 |- ( P e. dom N <-> ( P e. ( Base ` G ) /\ dom ( P \ _I ) e. Fin ) )
5	4	simplbi 476	. . . . . . 7 |- ( P e. dom N -> P e. ( Base ` G ) )
6	1, 3	elbasfv 15920	. . . . . . 7 |- ( P e. ( Base ` G ) -> D e. _V )
7	5, 6	syl 17	. . . . . 6 |- ( P e. dom N -> D e. _V )
8		psgnval.t	. . . . . . 7 |- T = ran (pmTrsp ` D )
9	1, 8, 2	psgneldm2 17924	. . . . . 6 |- ( D e. _V -> ( P e. dom N <-> E. w e. Word T P = ( G gsumg w ) ) )
10	7, 9	syl 17	. . . . 5 |- ( P e. dom N -> ( P e. dom N <-> E. w e. Word T P = ( G gsumg w ) ) )
11	10	ibi 256	. . . 4 |- ( P e. dom N -> E. w e. Word T P = ( G gsumg w ) )
12		simpr 477	. . . . . . 7 |- ( ( ( P e. dom N /\ w e. Word T ) /\ P = ( G gsumg w ) ) -> P = ( G gsumg w ) )
13		eqid 2622	. . . . . . 7 |- ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` w ) )
14		ovex 6678	. . . . . . . 8 |- ( -u 1 ^ ( # ` w ) ) e. _V
15		eqeq1 2626	. . . . . . . . 9 |- ( s = ( -u 1 ^ ( # ` w ) ) -> ( s = ( -u 1 ^ ( # ` w ) ) <-> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` w ) ) ) )
16	15	anbi2d 740	. . . . . . . 8 |- ( s = ( -u 1 ^ ( # ` w ) ) -> ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( P = ( G gsumg w ) /\ ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` w ) ) ) ) )
17	14, 16	spcev 3300	. . . . . . 7 |- ( ( P = ( G gsumg w ) /\ ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` w ) ) ) -> E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
18	12, 13, 17	sylancl 694	. . . . . 6 |- ( ( ( P e. dom N /\ w e. Word T ) /\ P = ( G gsumg w ) ) -> E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
19	18	ex 450	. . . . 5 |- ( ( P e. dom N /\ w e. Word T ) -> ( P = ( G gsumg w ) -> E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
20	19	reximdva 3017	. . . 4 |- ( P e. dom N -> ( E. w e. Word T P = ( G gsumg w ) -> E. w e. Word T E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) ) )
21	11, 20	mpd 15	. . 3 |- ( P e. dom N -> E. w e. Word T E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
22		rexcom4 3225	. . 3 |- ( E. w e. Word T E. s ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
23	21, 22	sylib 208	. 2 |- ( P e. dom N -> E. s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )
24		reeanv 3107	. . . 4 |- ( E. w e. Word T E. x e. Word T ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) <-> ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) )
25	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> D e. _V )
26		simplrl 800	. . . . . . . 8 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> w e. Word T )
27		simplrr 801	. . . . . . . 8 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> x e. Word T )
28		simprll 802	. . . . . . . . 9 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> P = ( G gsumg w ) )
29		simprrl 804	. . . . . . . . 9 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> P = ( G gsumg x ) )
30	28, 29	eqtr3d 2658	. . . . . . . 8 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> ( G gsumg w ) = ( G gsumg x ) )
31	1, 8, 25, 26, 27, 30	psgnuni 17919	. . . . . . 7 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` x ) ) )
32		simprlr 803	. . . . . . 7 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> s = ( -u 1 ^ ( # ` w ) ) )
33		simprrr 805	. . . . . . 7 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> t = ( -u 1 ^ ( # ` x ) ) )
34	31, 32, 33	3eqtr4d 2666	. . . . . 6 |- ( ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) /\ ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) ) -> s = t )
35	34	ex 450	. . . . 5 |- ( ( P e. dom N /\ ( w e. Word T /\ x e. Word T ) ) -> ( ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) -> s = t ) )
36	35	rexlimdvva 3038	. . . 4 |- ( P e. dom N -> ( E. w e. Word T E. x e. Word T ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) -> s = t ) )
37	24, 36	syl5bir 233	. . 3 |- ( P e. dom N -> ( ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) -> s = t ) )
38	37	alrimivv 1856	. 2 |- ( P e. dom N -> A. s A. t ( ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) -> s = t ) )
39		eqeq1 2626	. . . . . 6 |- ( s = t -> ( s = ( -u 1 ^ ( # ` w ) ) <-> t = ( -u 1 ^ ( # ` w ) ) ) )
40	39	anbi2d 740	. . . . 5 |- ( s = t -> ( ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( P = ( G gsumg w ) /\ t = ( -u 1 ^ ( # ` w ) ) ) ) )
41	40	rexbidv 3052	. . . 4 |- ( s = t -> ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. w e. Word T ( P = ( G gsumg w ) /\ t = ( -u 1 ^ ( # ` w ) ) ) ) )
42		oveq2 6658	. . . . . . 7 |- ( w = x -> ( G gsumg w ) = ( G gsumg x ) )
43	42	eqeq2d 2632	. . . . . 6 |- ( w = x -> ( P = ( G gsumg w ) <-> P = ( G gsumg x ) ) )
44		fveq2 6191	. . . . . . . 8 |- ( w = x -> ( # ` w ) = ( # ` x ) )
45	44	oveq2d 6666	. . . . . . 7 |- ( w = x -> ( -u 1 ^ ( # ` w ) ) = ( -u 1 ^ ( # ` x ) ) )
46	45	eqeq2d 2632	. . . . . 6 |- ( w = x -> ( t = ( -u 1 ^ ( # ` w ) ) <-> t = ( -u 1 ^ ( # ` x ) ) ) )
47	43, 46	anbi12d 747	. . . . 5 |- ( w = x -> ( ( P = ( G gsumg w ) /\ t = ( -u 1 ^ ( # ` w ) ) ) <-> ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) )
48	47	cbvrexv 3172	. . . 4 |- ( E. w e. Word T ( P = ( G gsumg w ) /\ t = ( -u 1 ^ ( # ` w ) ) ) <-> E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) )
49	41, 48	syl6bb 276	. . 3 |- ( s = t -> ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) )
50	49	eu4 2518	. 2 |- ( E! s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) <-> ( E. s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ A. s A. t ( ( E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) /\ E. x e. Word T ( P = ( G gsumg x ) /\ t = ( -u 1 ^ ( # ` x ) ) ) ) -> s = t ) ) )
51	23, 38, 50	sylanbrc 698	1 |- ( P e. dom N -> E! s E. w e. Word T ( P = ( G gsumg w ) /\ s = ( -u 1 ^ ( # ` w ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E.wrex 2913   _Vcvv 3200   \ cdif 3571   _I cid 5023   dom cdm 5114   ran crn 5115   ` cfv 5888  (class class class)co 6650   Fincfn 7955   1c1 9937   -ucneg 10267   ^cexp 12860   #chash 13117  Word cword 13291   Basecbs 15857   gsum_g cgsu 16101   SymGrpcsymg 17797  pmTrspcpmtr 17861  pmSgncpsgn 17909

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-iin 4523  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-mre 16246  df-mrc 16247  df-acs 16249  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  df-psgn 17911

This theorem is referenced by:  psgnvali  17928  psgnvalii  17929  psgnfieu  17938