Theorem psgninv 18382

Description: The sign of a permutation equals the sign of the inverse of the permutation. (Contributed by SO, 9-Jul-2018.)

Hypotheses

Ref	Expression
psgninv.s	|- S = ( SymGrp ` D )
psgninv.n	|- N = (pmSgn ` D )
psgninv.p	|- P = ( Base ` S )

Assertion

Ref	Expression
psgninv	|- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( N ` F ) )

Proof of Theorem psgninv

Step	Hyp	Ref	Expression
1		psgninv.s	. . . . 5 |- S = ( SymGrp ` D )
2		psgninv.n	. . . . 5 |- N = (pmSgn ` D )
3		eqid 2467	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
4	1, 2, 3	psgnghm2 18381	. . . 4 |- ( D e. Fin -> N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
5		psgninv.p	. . . . 5 |- P = ( Base ` S )
6		eqid 2467	. . . . 5 |- ( invg ` S ) = ( invg ` S )
7		eqid 2467	. . . . 5 |- ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
8	5, 6, 7	ghminv 16066	. . . 4 |- ( ( N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
9	4, 8	sylan 471	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
10	1, 5, 6	symginv 16219	. . . . 5 |- ( F e. P -> ( ( invg ` S ) ` F ) = `' F )
11	10	adantl 466	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` S ) ` F ) = `' F )
12	11	fveq2d 5868	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( N ` `' F ) )
13		eqid 2467	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 18377	. . . . 5 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15	3	cnmsgnbas 18378	. . . . . . . 8 |- { 1 , -u 1 } = ( Base ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
16	5, 15	ghmf 16063	. . . . . . 7 |- ( N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) -> N : P --> { 1 , -u 1 } )
17	4, 16	syl 16	. . . . . 6 |- ( D e. Fin -> N : P --> { 1 , -u 1 } )
18	17	ffvelrnda 6019	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. { 1 , -u 1 } )
19		cnex 9569	. . . . . . . . 9 |- CC e. _V
20		difss 3631	. . . . . . . . 9 |- ( CC \ { 0 } ) C_ CC
21	19, 20	ssexi 4592	. . . . . . . 8 |- ( CC \ { 0 } ) e. _V
22		ax-1cn 9546	. . . . . . . . . 10 |- 1 e. CC
23		ax-1ne0 9557	. . . . . . . . . 10 |- 1 =/= 0
24		eldifsn 4152	. . . . . . . . . 10 |- ( 1 e. ( CC \ { 0 } ) <-> ( 1 e. CC /\ 1 =/= 0 ) )
25	22, 23, 24	mpbir2an 918	. . . . . . . . 9 |- 1 e. ( CC \ { 0 } )
26		neg1cn 10635	. . . . . . . . . 10 |- -u 1 e. CC
27	22, 23	negne0i 9890	. . . . . . . . . 10 |- -u 1 =/= 0
28		eldifsn 4152	. . . . . . . . . 10 |- ( -u 1 e. ( CC \ { 0 } ) <-> ( -u 1 e. CC /\ -u 1 =/= 0 ) )
29	26, 27, 28	mpbir2an 918	. . . . . . . . 9 |- -u 1 e. ( CC \ { 0 } )
30		prssi 4183	. . . . . . . . 9 |- ( ( 1 e. ( CC \ { 0 } ) /\ -u 1 e. ( CC \ { 0 } ) ) -> { 1 , -u 1 } C_ ( CC \ { 0 } ) )
31	25, 29, 30	mp2an 672	. . . . . . . 8 |- { 1 , -u 1 } C_ ( CC \ { 0 } )
32		ressabs 14546	. . . . . . . 8 |- ( ( ( CC \ { 0 } ) e. _V /\ { 1 , -u 1 } C_ ( CC \ { 0 } ) ) -> ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
33	21, 31, 32	mp2an 672	. . . . . . 7 |- ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
34	33	eqcomi 2480	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } )
35		cnfldbas 18192	. . . . . . . 8 |- CC = ( Base ` ℂfld )
36		cnfld0 18210	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
37		cndrng 18215	. . . . . . . 8 |- ℂfld e. DivRing
38	35, 36, 37	drngui 17182	. . . . . . 7 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
39		eqid 2467	. . . . . . 7 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
40	38, 13, 39	invrfval 17103	. . . . . 6 |- ( invr ` ℂfld ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
41	34, 40, 7	subginv 16000	. . . . 5 |- ( ( { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ) /\ ( N ` F ) e. { 1 , -u 1 } ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
42	14, 18, 41	sylancr 663	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
43	31, 18	sseldi 3502	. . . . . 6 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. ( CC \ { 0 } ) )
44		eldifsn 4152	. . . . . 6 |- ( ( N ` F ) e. ( CC \ { 0 } ) <-> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
45	43, 44	sylib 196	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
46		cnfldinv 18217	. . . . 5 |- ( ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
47	45, 46	syl 16	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
48	42, 47	eqtr3d 2510	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
49	9, 12, 48	3eqtr3d 2516	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( 1 / ( N ` F ) ) )
50		fvex 5874	. . . . 5 |- ( N ` F ) e. _V
51	50	elpr 4045	. . . 4 |- ( ( N ` F ) e. { 1 , -u 1 } <-> ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) )
52	22	div1i 10268	. . . . . 6 |- ( 1 / 1 ) = 1
53		oveq2 6290	. . . . . 6 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( 1 / 1 ) )
54		id 22	. . . . . 6 |- ( ( N ` F ) = 1 -> ( N ` F ) = 1 )
55	52, 53, 54	3eqtr4a 2534	. . . . 5 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
56		divneg2 10264	. . . . . . . 8 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
57	22, 22, 23, 56	mp3an 1324	. . . . . . 7 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
58	52	negeqi 9809	. . . . . . 7 |- -u ( 1 / 1 ) = -u 1
59	57, 58	eqtr3i 2498	. . . . . 6 |- ( 1 / -u 1 ) = -u 1
60		oveq2 6290	. . . . . 6 |- ( ( N ` F ) = -u 1 -> ( 1 / ( N ` F ) ) = ( 1 / -u 1 ) )
61		id 22	. . . . . 6 |- ( ( N ` F ) = -u 1 -> ( N ` F ) = -u 1 )
62	59, 60, 61	3eqtr4a 2534	. . . . 5 |- ( ( N ` F ) = -u 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
63	55, 62	jaoi 379	. . . 4 |- ( ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) -> ( 1 / ( N ` F ) ) = ( N ` F ) )
64	51, 63	sylbi 195	. . 3 |- ( ( N ` F ) e. { 1 , -u 1 } -> ( 1 / ( N ` F ) ) = ( N ` F ) )
65	18, 64	syl 16	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( 1 / ( N ` F ) ) = ( N ` F ) )
66	49, 65	eqtrd 2508	1 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( N ` F ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   _Vcvv 3113   \ cdif 3473   C_ wss 3476   {csn 4027   {cpr 4029   `'ccnv 4998   -->wf 5582   ` cfv 5586  (class class class)co 6282   Fincfn 7513   CCcc 9486   0cc0 9488   1c1 9489   -ucneg 9802   / cdiv 10202   Basecbs 14483   ↾_s cress 14484   invgcminusg 15721  SubGrpcsubg 15987   GrpHom cghm 16056   SymGrpcsymg 16194  pmSgncpsgn 16307  mulGrpcmgp 16928   invrcinvr 17101  ℂ_fldccnfld 18188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-addf 9567  ax-mulf 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-rp 11217  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-word 12502  df-concat 12504  df-s1 12505  df-substr 12506  df-splice 12507  df-reverse 12508  df-s2 12770  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-0g 14690  df-gsum 14691  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-mhm 15774  df-submnd 15775  df-grp 15855  df-minusg 15856  df-subg 15990  df-ghm 16057  df-gim 16099  df-oppg 16173  df-symg 16195  df-pmtr 16260  df-psgn 16309  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-cring 16986  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-cnfld 18189

This theorem is referenced by:  zrhpsgninv  18385  evpmodpmf1o  18396