Theorem psgninv 19150

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 2451	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
4	1, 2, 3	psgnghm2 19149	. . . 4 |- ( D e. Fin -> N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
5		psgninv.p	. . . . 5 |- P = ( Base ` S )
6		eqid 2451	. . . . 5 |- ( invg ` S ) = ( invg ` S )
7		eqid 2451	. . . . 5 |- ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
8	5, 6, 7	ghminv 16890	. . . 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 474	. . 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 17043	. . . . 5 |- ( F e. P -> ( ( invg ` S ) ` F ) = `' F )
11	10	adantl 468	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` S ) ` F ) = `' F )
12	11	fveq2d 5869	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( N ` `' F ) )
13		eqid 2451	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 19145	. . . . 5 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15	3	cnmsgnbas 19146	. . . . . . . 8 |- { 1 , -u 1 } = ( Base ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
16	5, 15	ghmf 16887	. . . . . . 7 |- ( N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) -> N : P --> { 1 , -u 1 } )
17	4, 16	syl 17	. . . . . 6 |- ( D e. Fin -> N : P --> { 1 , -u 1 } )
18	17	ffvelrnda 6022	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. { 1 , -u 1 } )
19		cnex 9620	. . . . . . . . 9 |- CC e. _V
20		difss 3560	. . . . . . . . 9 |- ( CC \ { 0 } ) C_ CC
21	19, 20	ssexi 4548	. . . . . . . 8 |- ( CC \ { 0 } ) e. _V
22		ax-1cn 9597	. . . . . . . . . 10 |- 1 e. CC
23		ax-1ne0 9608	. . . . . . . . . 10 |- 1 =/= 0
24		eldifsn 4097	. . . . . . . . . 10 |- ( 1 e. ( CC \ { 0 } ) <-> ( 1 e. CC /\ 1 =/= 0 ) )
25	22, 23, 24	mpbir2an 931	. . . . . . . . 9 |- 1 e. ( CC \ { 0 } )
26		neg1cn 10713	. . . . . . . . . 10 |- -u 1 e. CC
27		neg1ne0 10715	. . . . . . . . . 10 |- -u 1 =/= 0
28		eldifsn 4097	. . . . . . . . . 10 |- ( -u 1 e. ( CC \ { 0 } ) <-> ( -u 1 e. CC /\ -u 1 =/= 0 ) )
29	26, 27, 28	mpbir2an 931	. . . . . . . . 9 |- -u 1 e. ( CC \ { 0 } )
30		prssi 4128	. . . . . . . . 9 |- ( ( 1 e. ( CC \ { 0 } ) /\ -u 1 e. ( CC \ { 0 } ) ) -> { 1 , -u 1 } C_ ( CC \ { 0 } ) )
31	25, 29, 30	mp2an 678	. . . . . . . 8 |- { 1 , -u 1 } C_ ( CC \ { 0 } )
32		ressabs 15188	. . . . . . . 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 678	. . . . . . 7 |- ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
34	33	eqcomi 2460	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } )
35		cnfldbas 18974	. . . . . . . 8 |- CC = ( Base ` ℂfld )
36		cnfld0 18992	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
37		cndrng 18997	. . . . . . . 8 |- ℂfld e. DivRing
38	35, 36, 37	drngui 17981	. . . . . . 7 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
39		eqid 2451	. . . . . . 7 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
40	38, 13, 39	invrfval 17901	. . . . . 6 |- ( invr ` ℂfld ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
41	34, 40, 7	subginv 16824	. . . . 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 669	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
43	31, 18	sseldi 3430	. . . . . 6 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. ( CC \ { 0 } ) )
44		eldifsn 4097	. . . . . 6 |- ( ( N ` F ) e. ( CC \ { 0 } ) <-> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
45	43, 44	sylib 200	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
46		cnfldinv 18999	. . . . 5 |- ( ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
47	45, 46	syl 17	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
48	42, 47	eqtr3d 2487	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
49	9, 12, 48	3eqtr3d 2493	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( 1 / ( N ` F ) ) )
50		fvex 5875	. . . . 5 |- ( N ` F ) e. _V
51	50	elpr 3986	. . . 4 |- ( ( N ` F ) e. { 1 , -u 1 } <-> ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) )
52		1div1e1 10300	. . . . . 6 |- ( 1 / 1 ) = 1
53		oveq2 6298	. . . . . 6 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( 1 / 1 ) )
54		id 22	. . . . . 6 |- ( ( N ` F ) = 1 -> ( N ` F ) = 1 )
55	52, 53, 54	3eqtr4a 2511	. . . . 5 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
56		divneg2 10331	. . . . . . . 8 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
57	22, 22, 23, 56	mp3an 1364	. . . . . . 7 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
58	52	negeqi 9868	. . . . . . 7 |- -u ( 1 / 1 ) = -u 1
59	57, 58	eqtr3i 2475	. . . . . 6 |- ( 1 / -u 1 ) = -u 1
60		oveq2 6298	. . . . . 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 2511	. . . . 5 |- ( ( N ` F ) = -u 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
63	55, 62	jaoi 381	. . . 4 |- ( ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) -> ( 1 / ( N ` F ) ) = ( N ` F ) )
64	51, 63	sylbi 199	. . 3 |- ( ( N ` F ) e. { 1 , -u 1 } -> ( 1 / ( N ` F ) ) = ( N ` F ) )
65	18, 64	syl 17	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( 1 / ( N ` F ) ) = ( N ` F ) )
66	49, 65	eqtrd 2485	1 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( N ` F ) )

Syntax hints:   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   _Vcvv 3045   \ cdif 3401   C_ wss 3404   {csn 3968   {cpr 3970   `'ccnv 4833   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   CCcc 9537   0cc0 9539   1c1 9540   -ucneg 9861   / cdiv 10269   Basecbs 15121   ↾_s cress 15122   invgcminusg 16670  SubGrpcsubg 16811   GrpHom cghm 16880   SymGrpcsymg 17018  pmSgncpsgn 17130  mulGrpcmgp 17723   invrcinvr 17899  ℂ_fldccnfld 18970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-xor 1406  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-ot 3977  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-tpos 6973  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-word 12664  df-lsw 12665  df-concat 12666  df-s1 12667  df-substr 12668  df-splice 12669  df-reverse 12670  df-s2 12944  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-0g 15340  df-gsum 15341  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-mhm 16582  df-submnd 16583  df-grp 16673  df-minusg 16674  df-subg 16814  df-ghm 16881  df-gim 16923  df-oppg 16997  df-symg 17019  df-pmtr 17083  df-psgn 17132  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-cring 17783  df-oppr 17851  df-dvdsr 17869  df-unit 17870  df-invr 17900  df-dvr 17911  df-drng 17977  df-cnfld 18971

This theorem is referenced by:  zrhpsgninv  19153  evpmodpmf1o  19164  madjusmdetlem4  28656