Theorem psgninv 19928

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 2622	. . . . 5 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
4	1, 2, 3	psgnghm2 19927	. . . 4 |- ( D e. Fin -> N e. ( S GrpHom ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) )
5		psgninv.p	. . . . 5 |- P = ( Base ` S )
6		eqid 2622	. . . . 5 |- ( invg ` S ) = ( invg ` S )
7		eqid 2622	. . . . 5 |- ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
8	5, 6, 7	ghminv 17667	. . . 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 488	. . 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 17822	. . . . 5 |- ( F e. P -> ( ( invg ` S ) ` F ) = `' F )
11	10	adantl 482	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` S ) ` F ) = `' F )
12	11	fveq2d 6195	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( N ` ( ( invg ` S ) ` F ) ) = ( N ` `' F ) )
13		eqid 2622	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
14	13	cnmsgnsubg 19923	. . . . 5 |- { 1 , -u 1 } e. (SubGrp ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
15	3	cnmsgnbas 19924	. . . . . . . 8 |- { 1 , -u 1 } = ( Base ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) )
16	5, 15	ghmf 17664	. . . . . . 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 6359	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. { 1 , -u 1 } )
19		cnex 10017	. . . . . . . . 9 |- CC e. _V
20		difss 3737	. . . . . . . . 9 |- ( CC \ { 0 } ) C_ CC
21	19, 20	ssexi 4803	. . . . . . . 8 |- ( CC \ { 0 } ) e. _V
22		ax-1cn 9994	. . . . . . . . . 10 |- 1 e. CC
23		ax-1ne0 10005	. . . . . . . . . 10 |- 1 =/= 0
24		eldifsn 4317	. . . . . . . . . 10 |- ( 1 e. ( CC \ { 0 } ) <-> ( 1 e. CC /\ 1 =/= 0 ) )
25	22, 23, 24	mpbir2an 955	. . . . . . . . 9 |- 1 e. ( CC \ { 0 } )
26		neg1cn 11124	. . . . . . . . . 10 |- -u 1 e. CC
27		neg1ne0 11126	. . . . . . . . . 10 |- -u 1 =/= 0
28		eldifsn 4317	. . . . . . . . . 10 |- ( -u 1 e. ( CC \ { 0 } ) <-> ( -u 1 e. CC /\ -u 1 =/= 0 ) )
29	26, 27, 28	mpbir2an 955	. . . . . . . . 9 |- -u 1 e. ( CC \ { 0 } )
30		prssi 4353	. . . . . . . . 9 |- ( ( 1 e. ( CC \ { 0 } ) /\ -u 1 e. ( CC \ { 0 } ) ) -> { 1 , -u 1 } C_ ( CC \ { 0 } ) )
31	25, 29, 30	mp2an 708	. . . . . . . 8 |- { 1 , -u 1 } C_ ( CC \ { 0 } )
32		ressabs 15939	. . . . . . . 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 708	. . . . . . 7 |- ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } ) = ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } )
34	33	eqcomi 2631	. . . . . 6 |- ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) = ( ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) ↾s { 1 , -u 1 } )
35		cnfldbas 19750	. . . . . . . 8 |- CC = ( Base ` ℂfld )
36		cnfld0 19770	. . . . . . . 8 |- 0 = ( 0g ` ℂfld )
37		cndrng 19775	. . . . . . . 8 |- ℂfld e. DivRing
38	35, 36, 37	drngui 18753	. . . . . . 7 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
39		eqid 2622	. . . . . . 7 |- ( invr ` ℂfld ) = ( invr ` ℂfld )
40	38, 13, 39	invrfval 18673	. . . . . 6 |- ( invr ` ℂfld ) = ( invg ` ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) ) )
41	34, 40, 7	subginv 17601	. . . . 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 695	. . . 4 |- ( ( D e. Fin /\ F e. P ) -> ( ( invr ` ℂfld ) ` ( N ` F ) ) = ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) )
43	31, 18	sseldi 3601	. . . . . 6 |- ( ( D e. Fin /\ F e. P ) -> ( N ` F ) e. ( CC \ { 0 } ) )
44		eldifsn 4317	. . . . . 6 |- ( ( N ` F ) e. ( CC \ { 0 } ) <-> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
45	43, 44	sylib 208	. . . . 5 |- ( ( D e. Fin /\ F e. P ) -> ( ( N ` F ) e. CC /\ ( N ` F ) =/= 0 ) )
46		cnfldinv 19777	. . . . 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 2658	. . 3 |- ( ( D e. Fin /\ F e. P ) -> ( ( invg ` ( (mulGrp ` ℂfld ) ↾s { 1 , -u 1 } ) ) ` ( N ` F ) ) = ( 1 / ( N ` F ) ) )
49	9, 12, 48	3eqtr3d 2664	. 2 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( 1 / ( N ` F ) ) )
50		fvex 6201	. . . . 5 |- ( N ` F ) e. _V
51	50	elpr 4198	. . . 4 |- ( ( N ` F ) e. { 1 , -u 1 } <-> ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) )
52		1div1e1 10717	. . . . . 6 |- ( 1 / 1 ) = 1
53		oveq2 6658	. . . . . 6 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( 1 / 1 ) )
54		id 22	. . . . . 6 |- ( ( N ` F ) = 1 -> ( N ` F ) = 1 )
55	52, 53, 54	3eqtr4a 2682	. . . . 5 |- ( ( N ` F ) = 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
56		divneg2 10749	. . . . . . . 8 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
57	22, 22, 23, 56	mp3an 1424	. . . . . . 7 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
58	52	negeqi 10274	. . . . . . 7 |- -u ( 1 / 1 ) = -u 1
59	57, 58	eqtr3i 2646	. . . . . 6 |- ( 1 / -u 1 ) = -u 1
60		oveq2 6658	. . . . . 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 2682	. . . . 5 |- ( ( N ` F ) = -u 1 -> ( 1 / ( N ` F ) ) = ( N ` F ) )
63	55, 62	jaoi 394	. . . 4 |- ( ( ( N ` F ) = 1 \/ ( N ` F ) = -u 1 ) -> ( 1 / ( N ` F ) ) = ( N ` F ) )
64	51, 63	sylbi 207	. . 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 2656	1 |- ( ( D e. Fin /\ F e. P ) -> ( N ` `' F ) = ( N ` F ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   `'ccnv 5113   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   -ucneg 10267   / cdiv 10684   Basecbs 15857   ↾_s cress 15858   invgcminusg 17423  SubGrpcsubg 17588   GrpHom cghm 17657   SymGrpcsymg 17797  pmSgncpsgn 17909  mulGrpcmgp 18489   invrcinvr 18671  ℂ_fldccnfld 19746

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  ax-addf 10015  ax-mulf 10016

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-dec 11494  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-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  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  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-cnfld 19747

This theorem is referenced by:  zrhpsgninv  19931  evpmodpmf1o  19942  madjusmdetlem4  29896