Theorem psgninv 18012

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 2443	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
4	1, 2, 3	psgnghm2 18011	. . . 4 |- ( D e. Fin -> N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
5		psgninv.p	. . . . 5 |- P = ( Base ` S )
6		eqid 2443	. . . . 5 |- ( invg ` S ) = ( invg ` S )
7		eqid 2443	. . . . 5 |- ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
8	5, 6, 7	ghminv 15754	. . . 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 15907	. . . . 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 5695	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( N ` `' F ) )
13		eqid 2443	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 18007	. . . . 5 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15	3	cnmsgnbas 18008	. . . . . . . 8 |- { 1 , -u 1 } = ( Base ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
16	5, 15	ghmf 15751	. . . . . . 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 5843	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. { 1 , -u 1 } )
19		cnex 9363	. . . . . . . . 9 |- CC e. _V
20		difss 3483	. . . . . . . . 9 |- ( CC \ { 0 } ) C_ CC
21	19, 20	ssexi 4437	. . . . . . . 8 |- ( CC \ { 0 } ) e. _V
22		ax-1cn 9340	. . . . . . . . . 10 |- 1 e. CC
23		ax-1ne0 9351	. . . . . . . . . 10 |- 1 =/= 0
24		eldifsn 4000	. . . . . . . . . 10 |- ( 1 e. ( CC \ { 0 } ) <-> ( 1 e. CC /\ 1 =/= 0 ) )
25	22, 23, 24	mpbir2an 911	. . . . . . . . 9 |- 1 e. ( CC \ { 0 } )
26		neg1cn 10425	. . . . . . . . . 10 |- -u 1 e. CC
27	22, 23	negne0i 9683	. . . . . . . . . 10 |- -u 1 =/= 0
28		eldifsn 4000	. . . . . . . . . 10 |- ( -u 1 e. ( CC \ { 0 } ) <-> ( -u 1 e. CC /\ -u 1 =/= 0 ) )
29	26, 27, 28	mpbir2an 911	. . . . . . . . 9 |- -u 1 e. ( CC \ { 0 } )
30		prssi 4029	. . . . . . . . 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 14236	. . . . . . . 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 2447	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } )
35		cnfldbas 17822	. . . . . . . 8 |- CC = ( Base ` ℂfld )
36		cnfld0 17840	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
37		cndrng 17845	. . . . . . . 8 |- ℂfld e. DivRing
38	35, 36, 37	drngui 16838	. . . . . . 7 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
39		eqid 2443	. . . . . . 7 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
40	38, 13, 39	invrfval 16765	. . . . . 6 |- ( invr ` ℂfld ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
41	34, 40, 7	subginv 15688	. . . . 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 3354	. . . . . 6 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. ( CC \ { 0 } ) )
44		eldifsn 4000	. . . . . 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 17847	. . . . 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 2477	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
49	9, 12, 48	3eqtr3d 2483	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( 1 / ( N ` F ) ) )
50		fvex 5701	. . . . 5 |- ( N ` F ) e. _V
51	50	elpr 3895	. . . 4 |- ( ( N ` F ) e. { 1 , -u 1 } <-> ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) )
52	22	div1i 10059	. . . . . 6 |- ( 1 / 1 ) = 1
53		oveq2 6099	. . . . . 6 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( 1 / 1 ) )
54		id 22	. . . . . 6 |- ( ( N ` F ) = 1 -> ( N ` F ) = 1 )
55	52, 53, 54	3eqtr4a 2501	. . . . 5 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
56		divneg2 10055	. . . . . . . 8 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
57	22, 22, 23, 56	mp3an 1314	. . . . . . 7 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
58	52	negeqi 9603	. . . . . . 7 |- -u ( 1 / 1 ) = -u 1
59	57, 58	eqtr3i 2465	. . . . . 6 |- ( 1 / -u 1 ) = -u 1
60		oveq2 6099	. . . . . 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 2501	. . . . 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 2475	1 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( N ` F ) )

Syntax hints:   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   _Vcvv 2972   \ cdif 3325   C_ wss 3328   {csn 3877   {cpr 3879   `'ccnv 4839   -->wf 5414   ` cfv 5418  (class class class)co 6091   Fincfn 7310   CCcc 9280   0cc0 9282   1c1 9283   -ucneg 9596   / cdiv 9993   Basecbs 14174   ↾_s cress 14175   invgcminusg 15411  SubGrpcsubg 15675   GrpHom cghm 15744   SymGrpcsymg 15882  pmSgncpsgn 15995  mulGrpcmgp 16591   invrcinvr 16763  ℂ_fldccnfld 17818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-reverse 12235  df-s2 12475  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-0g 14380  df-gsum 14381  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-subg 15678  df-ghm 15745  df-gim 15787  df-oppg 15861  df-symg 15883  df-pmtr 15948  df-psgn 15997  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-cring 16648  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-cnfld 17819

This theorem is referenced by:  zrhpsgninv  18015  evpmodpmf1o  18026