Theorem dchrelbas4 24968

Description: A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of CC, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
dchrmhm.g	|- G = (DChr ` N )
dchrmhm.z	|- Z = (ℤ/nℤ ` N )
dchrmhm.b	|- D = ( Base ` G )
dchrelbas4.l	|- L = ( ZRHom ` Z )

Assertion

Ref	Expression
dchrelbas4	|- ( X e. D <-> ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) )

Distinct variable groups:   x, L   x, N   x, X   x, Z   x, D

Allowed substitution hint:   G( x)

Proof of Theorem dchrelbas4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrmhm.g	. . . 4 |- G = (DChr ` N )
2		dchrmhm.b	. . . 4 |- D = ( Base ` G )
3	1, 2	dchrrcl 24965	. . 3 |- ( X e. D -> N e. NN )
4		dchrmhm.z	. . . . 5 |- Z = (ℤ/nℤ ` N )
5		eqid 2622	. . . . 5 |- ( Base ` Z ) = ( Base ` Z )
6		eqid 2622	. . . . 5 |- (Unit ` Z ) = (Unit ` Z )
7		id 22	. . . . 5 |- ( N e. NN -> N e. NN )
8	1, 4, 5, 6, 7, 2	dchrelbas2 24962	. . . 4 |- ( N e. NN -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. y e. ( Base ` Z ) ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) ) ) )
9		nnnn0 11299	. . . . . . . 8 |- ( N e. NN -> N e. NN0 )
10	9	adantr 481	. . . . . . 7 |- ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> N e. NN0 )
11		dchrelbas4.l	. . . . . . . 8 |- L = ( ZRHom ` Z )
12	4, 5, 11	znzrhfo 19896	. . . . . . 7 |- ( N e. NN0 -> L : ZZ -onto-> ( Base ` Z ) )
13		fveq2 6191	. . . . . . . . . 10 |- ( ( L ` x ) = y -> ( X ` ( L ` x ) ) = ( X ` y ) )
14	13	neeq1d 2853	. . . . . . . . 9 |- ( ( L ` x ) = y -> ( ( X ` ( L ` x ) ) =/= 0 <-> ( X ` y ) =/= 0 ) )
15		eleq1 2689	. . . . . . . . 9 |- ( ( L ` x ) = y -> ( ( L ` x ) e. (Unit ` Z ) <-> y e. (Unit ` Z ) ) )
16	14, 15	imbi12d 334	. . . . . . . 8 |- ( ( L ` x ) = y -> ( ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) ) )
17	16	cbvfo 6544	. . . . . . 7 |- ( L : ZZ -onto-> ( Base ` Z ) -> ( A. x e. ZZ ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> A. y e. ( Base ` Z ) ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) ) )
18	10, 12, 17	3syl 18	. . . . . 6 |- ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( A. x e. ZZ ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> A. y e. ( Base ` Z ) ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) ) )
19		df-ne 2795	. . . . . . . . . 10 |- ( ( X ` ( L ` x ) ) =/= 0 <-> -. ( X ` ( L ` x ) ) = 0 )
20	19	a1i 11	. . . . . . . . 9 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( X ` ( L ` x ) ) =/= 0 <-> -. ( X ` ( L ` x ) ) = 0 ) )
21	4, 6, 11	znunit 19912	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ x e. ZZ ) -> ( ( L ` x ) e. (Unit ` Z ) <-> ( x gcd N ) = 1 ) )
22	10, 21	sylan 488	. . . . . . . . . 10 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( L ` x ) e. (Unit ` Z ) <-> ( x gcd N ) = 1 ) )
23		1red 10055	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> 1 e. RR )
24		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> x e. ZZ )
25		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> N e. NN )
26	25	nnzd 11481	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> N e. ZZ )
27		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( N e. NN -> N =/= 0 )
28		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( x = 0 /\ N = 0 ) -> N = 0 )
29	28	necon3ai 2819	. . . . . . . . . . . . . . 15 |- ( N =/= 0 -> -. ( x = 0 /\ N = 0 ) )
30	25, 27, 29	3syl 18	. . . . . . . . . . . . . 14 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> -. ( x = 0 /\ N = 0 ) )
31		gcdn0cl 15224	. . . . . . . . . . . . . 14 |- ( ( ( x e. ZZ /\ N e. ZZ ) /\ -. ( x = 0 /\ N = 0 ) ) -> ( x gcd N ) e. NN )
32	24, 26, 30, 31	syl21anc 1325	. . . . . . . . . . . . 13 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( x gcd N ) e. NN )
33	32	nnred 11035	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( x gcd N ) e. RR )
34	32	nnge1d 11063	. . . . . . . . . . . 12 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> 1 <_ ( x gcd N ) )
35	23, 33, 34	leltned 10190	. . . . . . . . . . 11 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( 1 < ( x gcd N ) <-> ( x gcd N ) =/= 1 ) )
36	35	necon2bbid 2837	. . . . . . . . . 10 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( x gcd N ) = 1 <-> -. 1 < ( x gcd N ) ) )
37	22, 36	bitrd 268	. . . . . . . . 9 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( L ` x ) e. (Unit ` Z ) <-> -. 1 < ( x gcd N ) ) )
38	20, 37	imbi12d 334	. . . . . . . 8 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> ( -. ( X ` ( L ` x ) ) = 0 -> -. 1 < ( x gcd N ) ) ) )
39		con34b 306	. . . . . . . 8 |- ( ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) <-> ( -. ( X ` ( L ` x ) ) = 0 -> -. 1 < ( x gcd N ) ) )
40	38, 39	syl6bbr 278	. . . . . . 7 |- ( ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) /\ x e. ZZ ) -> ( ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) )
41	40	ralbidva 2985	. . . . . 6 |- ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( A. x e. ZZ ( ( X ` ( L ` x ) ) =/= 0 -> ( L ` x ) e. (Unit ` Z ) ) <-> A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) )
42	18, 41	bitr3d 270	. . . . 5 |- ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) ) -> ( A. y e. ( Base ` Z ) ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) <-> A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) )
43	42	pm5.32da 673	. . . 4 |- ( N e. NN -> ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. y e. ( Base ` Z ) ( ( X ` y ) =/= 0 -> y e. (Unit ` Z ) ) ) <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) ) )
44	8, 43	bitrd 268	. . 3 |- ( N e. NN -> ( X e. D <-> ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) ) )
45	3, 44	biadan2 674	. 2 |- ( X e. D <-> ( N e. NN /\ ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) ) )
46		3anass 1042	. 2 |- ( ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) <-> ( N e. NN /\ ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) ) )
47	45, 46	bitr4i 267	1 |- ( X e. D <-> ( N e. NN /\ X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ A. x e. ZZ ( 1 < ( x gcd N ) -> ( X ` ( L ` x ) ) = 0 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   < clt 10074   NNcn 11020   NN0cn0 11292   ZZcz 11377   gcd cgcd 15216   Basecbs 15857   MndHom cmhm 17333  mulGrpcmgp 18489  Unitcui 18639  ℂ_fldccnfld 19746   ZRHomczrh 19848  ℤ/nℤczn 19851  DChrcdchr 24957

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-inf2 8538  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-pre-sup 10014  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-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-uni 4437  df-int 4476  df-iun 4522  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-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-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-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  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-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  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-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855  df-dchr 24958

This theorem is referenced by: (None)