Theorem dchrinvcl 24978

Description: Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
dchrmhm.g	|- G = (DChr ` N )
dchrmhm.z	|- Z = (ℤ/nℤ ` N )
dchrmhm.b	|- D = ( Base ` G )
dchrn0.b	|- B = ( Base ` Z )
dchrn0.u	|- U = (Unit ` Z )
dchr1cl.o	|- .1. = ( k e. B |-> if ( k e. U , 1 , 0 ) )
dchrmulid2.t	|- .x. = ( +g ` G )
dchrmulid2.x	|- ( ph -> X e. D )
dchrinvcl.n	|- K = ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) )

Assertion

Ref	Expression
dchrinvcl	|- ( ph -> ( K e. D /\ ( K .x. X ) = .1. ) )

Distinct variable groups:   B, k   U, k   k, N   ph, k   k, X   k, Z

Allowed substitution hints:   D( k)   .x. ( k)   .1. ( k)   G( k)   K( k)

Proof of Theorem dchrinvcl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrinvcl.n	. . 3 |- K = ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) )
2		dchrmhm.g	. . . 4 |- G = (DChr ` N )
3		dchrmhm.z	. . . 4 |- Z = (ℤ/nℤ ` N )
4		dchrn0.b	. . . 4 |- B = ( Base ` Z )
5		dchrn0.u	. . . 4 |- U = (Unit ` Z )
6		dchrmulid2.x	. . . . 5 |- ( ph -> X e. D )
7		dchrmhm.b	. . . . . 6 |- D = ( Base ` G )
8	2, 7	dchrrcl 24965	. . . . 5 |- ( X e. D -> N e. NN )
9	6, 8	syl 17	. . . 4 |- ( ph -> N e. NN )
10		fveq2 6191	. . . . 5 |- ( k = x -> ( X ` k ) = ( X ` x ) )
11	10	oveq2d 6666	. . . 4 |- ( k = x -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` x ) ) )
12		fveq2 6191	. . . . 5 |- ( k = y -> ( X ` k ) = ( X ` y ) )
13	12	oveq2d 6666	. . . 4 |- ( k = y -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` y ) ) )
14		fveq2 6191	. . . . 5 |- ( k = ( x ( .r ` Z ) y ) -> ( X ` k ) = ( X ` ( x ( .r ` Z ) y ) ) )
15	14	oveq2d 6666	. . . 4 |- ( k = ( x ( .r ` Z ) y ) -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) )
16		fveq2 6191	. . . . 5 |- ( k = ( 1r ` Z ) -> ( X ` k ) = ( X ` ( 1r ` Z ) ) )
17	16	oveq2d 6666	. . . 4 |- ( k = ( 1r ` Z ) -> ( 1 / ( X ` k ) ) = ( 1 / ( X ` ( 1r ` Z ) ) ) )
18	2, 3, 7, 4, 6	dchrf 24967	. . . . . 6 |- ( ph -> X : B --> CC )
19	4, 5	unitss 18660	. . . . . . 7 |- U C_ B
20	19	sseli 3599	. . . . . 6 |- ( k e. U -> k e. B )
21		ffvelrn 6357	. . . . . 6 |- ( ( X : B --> CC /\ k e. B ) -> ( X ` k ) e. CC )
22	18, 20, 21	syl2an 494	. . . . 5 |- ( ( ph /\ k e. U ) -> ( X ` k ) e. CC )
23		simpr 477	. . . . . 6 |- ( ( ph /\ k e. U ) -> k e. U )
24	6	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. U ) -> X e. D )
25	20	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. U ) -> k e. B )
26	2, 3, 7, 4, 5, 24, 25	dchrn0 24975	. . . . . 6 |- ( ( ph /\ k e. U ) -> ( ( X ` k ) =/= 0 <-> k e. U ) )
27	23, 26	mpbird 247	. . . . 5 |- ( ( ph /\ k e. U ) -> ( X ` k ) =/= 0 )
28	22, 27	reccld 10794	. . . 4 |- ( ( ph /\ k e. U ) -> ( 1 / ( X ` k ) ) e. CC )
29		1t1e1 11175	. . . . . . . 8 |- ( 1 x. 1 ) = 1
30	29	eqcomi 2631	. . . . . . 7 |- 1 = ( 1 x. 1 )
31	30	a1i 11	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> 1 = ( 1 x. 1 ) )
32	2, 3, 7	dchrmhm 24966	. . . . . . . 8 |- D C_ ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) )
33	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X e. D )
34	32, 33	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
35		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> x e. U )
36	19, 35	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> x e. B )
37		simprr 796	. . . . . . . 8 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> y e. U )
38	19, 37	sseldi 3601	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> y e. B )
39		eqid 2622	. . . . . . . . 9 |- (mulGrp ` Z ) = (mulGrp ` Z )
40	39, 4	mgpbas 18495	. . . . . . . 8 |- B = ( Base ` (mulGrp ` Z ) )
41		eqid 2622	. . . . . . . . 9 |- ( .r ` Z ) = ( .r ` Z )
42	39, 41	mgpplusg 18493	. . . . . . . 8 |- ( .r ` Z ) = ( +g ` (mulGrp ` Z ) )
43		eqid 2622	. . . . . . . . 9 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
44		cnfldmul 19752	. . . . . . . . 9 |- x. = ( .r ` ℂfld )
45	43, 44	mgpplusg 18493	. . . . . . . 8 |- x. = ( +g ` (mulGrp ` ℂfld ) )
46	40, 42, 45	mhmlin 17342	. . . . . . 7 |- ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ x e. B /\ y e. B ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
47	34, 36, 38, 46	syl3anc 1326	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` ( x ( .r ` Z ) y ) ) = ( ( X ` x ) x. ( X ` y ) ) )
48	31, 47	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) = ( ( 1 x. 1 ) / ( ( X ` x ) x. ( X ` y ) ) ) )
49		1cnd 10056	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> 1 e. CC )
50	18	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> X : B --> CC )
51	50, 36	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` x ) e. CC )
52	50, 38	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` y ) e. CC )
53	2, 3, 7, 4, 5, 33, 36	dchrn0 24975	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( X ` x ) =/= 0 <-> x e. U ) )
54	35, 53	mpbird 247	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` x ) =/= 0 )
55	2, 3, 7, 4, 5, 33, 38	dchrn0 24975	. . . . . . 7 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( X ` y ) =/= 0 <-> y e. U ) )
56	37, 55	mpbird 247	. . . . . 6 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( X ` y ) =/= 0 )
57	49, 51, 49, 52, 54, 56	divmuldivd 10842	. . . . 5 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( ( 1 / ( X ` x ) ) x. ( 1 / ( X ` y ) ) ) = ( ( 1 x. 1 ) / ( ( X ` x ) x. ( X ` y ) ) ) )
58	48, 57	eqtr4d 2659	. . . 4 |- ( ( ph /\ ( x e. U /\ y e. U ) ) -> ( 1 / ( X ` ( x ( .r ` Z ) y ) ) ) = ( ( 1 / ( X ` x ) ) x. ( 1 / ( X ` y ) ) ) )
59	32, 6	sseldi 3601	. . . . . . 7 |- ( ph -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
60		eqid 2622	. . . . . . . . 9 |- ( 1r ` Z ) = ( 1r ` Z )
61	39, 60	ringidval 18503	. . . . . . . 8 |- ( 1r ` Z ) = ( 0g ` (mulGrp ` Z ) )
62		cnfld1 19771	. . . . . . . . 9 |- 1 = ( 1r ` ℂfld )
63	43, 62	ringidval 18503	. . . . . . . 8 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
64	61, 63	mhm0 17343	. . . . . . 7 |- ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) -> ( X ` ( 1r ` Z ) ) = 1 )
65	59, 64	syl 17	. . . . . 6 |- ( ph -> ( X ` ( 1r ` Z ) ) = 1 )
66	65	oveq2d 6666	. . . . 5 |- ( ph -> ( 1 / ( X ` ( 1r ` Z ) ) ) = ( 1 / 1 ) )
67		1div1e1 10717	. . . . 5 |- ( 1 / 1 ) = 1
68	66, 67	syl6eq 2672	. . . 4 |- ( ph -> ( 1 / ( X ` ( 1r ` Z ) ) ) = 1 )
69	2, 3, 4, 5, 9, 7, 11, 13, 15, 17, 28, 58, 68	dchrelbasd 24964	. . 3 |- ( ph -> ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) ) e. D )
70	1, 69	syl5eqel 2705	. 2 |- ( ph -> K e. D )
71		dchrmulid2.t	. . . 4 |- .x. = ( +g ` G )
72	2, 3, 7, 71, 70, 6	dchrmul 24973	. . 3 |- ( ph -> ( K .x. X ) = ( K oF x. X ) )
73		fvex 6201	. . . . . . 7 |- ( Base ` Z ) e. _V
74	4, 73	eqeltri 2697	. . . . . 6 |- B e. _V
75	74	a1i 11	. . . . 5 |- ( ph -> B e. _V )
76		ovex 6678	. . . . . . 7 |- ( 1 / ( X ` k ) ) e. _V
77		c0ex 10034	. . . . . . 7 |- 0 e. _V
78	76, 77	ifex 4156	. . . . . 6 |- if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) e. _V
79	78	a1i 11	. . . . 5 |- ( ( ph /\ k e. B ) -> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) e. _V )
80	18	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ k e. B ) -> ( X ` k ) e. CC )
81	1	a1i 11	. . . . 5 |- ( ph -> K = ( k e. B |-> if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) ) )
82	18	feqmptd 6249	. . . . 5 |- ( ph -> X = ( k e. B |-> ( X ` k ) ) )
83	75, 79, 80, 81, 82	offval2 6914	. . . 4 |- ( ph -> ( K oF x. X ) = ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) )
84		ovif 6737	. . . . . . 7 |- ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) = if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) )
85	80	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( X ` k ) e. CC )
86	6	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ k e. B ) -> X e. D )
87		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ k e. B ) -> k e. B )
88	2, 3, 7, 4, 5, 86, 87	dchrn0 24975	. . . . . . . . . . 11 |- ( ( ph /\ k e. B ) -> ( ( X ` k ) =/= 0 <-> k e. U ) )
89	88	biimpar 502	. . . . . . . . . 10 |- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( X ` k ) =/= 0 )
90	85, 89	recid2d 10797	. . . . . . . . 9 |- ( ( ( ph /\ k e. B ) /\ k e. U ) -> ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) = 1 )
91	90	ifeq1da 4116	. . . . . . . 8 |- ( ( ph /\ k e. B ) -> if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , ( 0 x. ( X ` k ) ) ) )
92	80	mul02d 10234	. . . . . . . . 9 |- ( ( ph /\ k e. B ) -> ( 0 x. ( X ` k ) ) = 0 )
93	92	ifeq2d 4105	. . . . . . . 8 |- ( ( ph /\ k e. B ) -> if ( k e. U , 1 , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , 0 ) )
94	91, 93	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ k e. B ) -> if ( k e. U , ( ( 1 / ( X ` k ) ) x. ( X ` k ) ) , ( 0 x. ( X ` k ) ) ) = if ( k e. U , 1 , 0 ) )
95	84, 94	syl5eq 2668	. . . . . 6 |- ( ( ph /\ k e. B ) -> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) = if ( k e. U , 1 , 0 ) )
96	95	mpteq2dva 4744	. . . . 5 |- ( ph -> ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) = ( k e. B |-> if ( k e. U , 1 , 0 ) ) )
97		dchr1cl.o	. . . . 5 |- .1. = ( k e. B |-> if ( k e. U , 1 , 0 ) )
98	96, 97	syl6reqr 2675	. . . 4 |- ( ph -> .1. = ( k e. B |-> ( if ( k e. U , ( 1 / ( X ` k ) ) , 0 ) x. ( X ` k ) ) ) )
99	83, 98	eqtr4d 2659	. . 3 |- ( ph -> ( K oF x. X ) = .1. )
100	72, 99	eqtrd 2656	. 2 |- ( ph -> ( K .x. X ) = .1. )
101	70, 100	jca 554	1 |- ( ph -> ( K e. D /\ ( K .x. X ) = .1. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   NNcn 11020   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   MndHom cmhm 17333  mulGrpcmgp 18489   1rcur 18501  Unitcui 18639  ℂ_fldccnfld 19746  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-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-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-of 6897  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-fz 12327  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-subg 17591  df-nsg 17592  df-eqg 17593  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-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-zn 19855  df-dchr 24958

This theorem is referenced by:  dchrabl  24979