Theorem lgsdchrval 25079

Description: The Legendre symbol function X ( m ) = ( m /L N ), where N is an odd positive number, is a Dirichlet character modulo N. (Contributed by Mario Carneiro, 28-Apr-2016.)

Hypotheses

Ref	Expression
lgsdchr.g	|- G = (DChr ` N )
lgsdchr.z	|- Z = (ℤ/nℤ ` N )
lgsdchr.d	|- D = ( Base ` G )
lgsdchr.b	|- B = ( Base ` Z )
lgsdchr.l	|- L = ( ZRHom ` Z )
lgsdchr.x	|- X = ( y e. B |-> ( iota h E. m e. ZZ ( y = ( L ` m ) /\ h = ( m /L N ) ) ) )

Assertion

Ref	Expression
lgsdchrval	|- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( X ` ( L ` A ) ) = ( A /L N ) )

Distinct variable groups:   y, B   h, m, y, L   h, N, m, y   y, X   A, h, m, y   y, Z

Allowed substitution hints:   B( h, m)   D( y, h, m)   G( y, h, m)   X( h, m)   Z( h, m)

Proof of Theorem lgsdchrval

Step	Hyp	Ref	Expression
1		nnnn0 11299	. . . . . 6 |- ( N e. NN -> N e. NN0 )
2	1	adantr 481	. . . . 5 |- ( ( N e. NN /\ -. 2 || N ) -> N e. NN0 )
3		lgsdchr.z	. . . . . 6 |- Z = (ℤ/nℤ ` N )
4		lgsdchr.b	. . . . . 6 |- B = ( Base ` Z )
5		lgsdchr.l	. . . . . 6 |- L = ( ZRHom ` Z )
6	3, 4, 5	znzrhfo 19896	. . . . 5 |- ( N e. NN0 -> L : ZZ -onto-> B )
7		fof 6115	. . . . 5 |- ( L : ZZ -onto-> B -> L : ZZ --> B )
8	2, 6, 7	3syl 18	. . . 4 |- ( ( N e. NN /\ -. 2 || N ) -> L : ZZ --> B )
9	8	ffvelrnda 6359	. . 3 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( L ` A ) e. B )
10		eqeq1 2626	. . . . . . 7 |- ( y = ( L ` A ) -> ( y = ( L ` m ) <-> ( L ` A ) = ( L ` m ) ) )
11	10	anbi1d 741	. . . . . 6 |- ( y = ( L ` A ) -> ( ( y = ( L ` m ) /\ h = ( m /L N ) ) <-> ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
12	11	rexbidv 3052	. . . . 5 |- ( y = ( L ` A ) -> ( E. m e. ZZ ( y = ( L ` m ) /\ h = ( m /L N ) ) <-> E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
13	12	iotabidv 5872	. . . 4 |- ( y = ( L ` A ) -> ( iota h E. m e. ZZ ( y = ( L ` m ) /\ h = ( m /L N ) ) ) = ( iota h E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
14		lgsdchr.x	. . . 4 |- X = ( y e. B |-> ( iota h E. m e. ZZ ( y = ( L ` m ) /\ h = ( m /L N ) ) ) )
15		iotaex 5868	. . . 4 |- ( iota h E. m e. ZZ ( y = ( L ` m ) /\ h = ( m /L N ) ) ) e. _V
16	13, 14, 15	fvmpt3i 6287	. . 3 |- ( ( L ` A ) e. B -> ( X ` ( L ` A ) ) = ( iota h E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
17	9, 16	syl 17	. 2 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( X ` ( L ` A ) ) = ( iota h E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
18		ovex 6678	. . 3 |- ( A /L N ) e. _V
19		simprr 796	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( L ` A ) = ( L ` m ) )
20		simplll 798	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> N e. NN )
21	20, 1	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> N e. NN0 )
22		simplr 792	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> A e. ZZ )
23		simprl 794	. . . . . . . . . . . . . . . 16 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> m e. ZZ )
24	3, 5	zndvds 19898	. . . . . . . . . . . . . . . 16 |- ( ( N e. NN0 /\ A e. ZZ /\ m e. ZZ ) -> ( ( L ` A ) = ( L ` m ) <-> N || ( A - m ) ) )
25	21, 22, 23, 24	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( ( L ` A ) = ( L ` m ) <-> N || ( A - m ) ) )
26	19, 25	mpbid 222	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> N || ( A - m ) )
27		moddvds 14991	. . . . . . . . . . . . . . 15 |- ( ( N e. NN /\ A e. ZZ /\ m e. ZZ ) -> ( ( A mod N ) = ( m mod N ) <-> N || ( A - m ) ) )
28	20, 22, 23, 27	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( ( A mod N ) = ( m mod N ) <-> N || ( A - m ) ) )
29	26, 28	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( A mod N ) = ( m mod N ) )
30	29	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( ( A mod N ) /L N ) = ( ( m mod N ) /L N ) )
31		simpllr 799	. . . . . . . . . . . . 13 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> -. 2 || N )
32		lgsmod 25048	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ N e. NN /\ -. 2 || N ) -> ( ( A mod N ) /L N ) = ( A /L N ) )
33	22, 20, 31, 32	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( ( A mod N ) /L N ) = ( A /L N ) )
34		lgsmod 25048	. . . . . . . . . . . . 13 |- ( ( m e. ZZ /\ N e. NN /\ -. 2 || N ) -> ( ( m mod N ) /L N ) = ( m /L N ) )
35	23, 20, 31, 34	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( ( m mod N ) /L N ) = ( m /L N ) )
36	30, 33, 35	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( A /L N ) = ( m /L N ) )
37	36	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( h = ( A /L N ) <-> h = ( m /L N ) ) )
38	37	biimprd 238	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( m e. ZZ /\ ( L ` A ) = ( L ` m ) ) ) -> ( h = ( m /L N ) -> h = ( A /L N ) ) )
39	38	anassrs 680	. . . . . . . 8 |- ( ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ m e. ZZ ) /\ ( L ` A ) = ( L ` m ) ) -> ( h = ( m /L N ) -> h = ( A /L N ) ) )
40	39	expimpd 629	. . . . . . 7 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ m e. ZZ ) -> ( ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) -> h = ( A /L N ) ) )
41	40	rexlimdva 3031	. . . . . 6 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) -> h = ( A /L N ) ) )
42		fveq2 6191	. . . . . . . . . . . 12 |- ( m = A -> ( L ` m ) = ( L ` A ) )
43	42	eqcomd 2628	. . . . . . . . . . 11 |- ( m = A -> ( L ` A ) = ( L ` m ) )
44	43	biantrurd 529	. . . . . . . . . 10 |- ( m = A -> ( h = ( m /L N ) <-> ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
45		oveq1 6657	. . . . . . . . . . 11 |- ( m = A -> ( m /L N ) = ( A /L N ) )
46	45	eqeq2d 2632	. . . . . . . . . 10 |- ( m = A -> ( h = ( m /L N ) <-> h = ( A /L N ) ) )
47	44, 46	bitr3d 270	. . . . . . . . 9 |- ( m = A -> ( ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) <-> h = ( A /L N ) ) )
48	47	rspcev 3309	. . . . . . . 8 |- ( ( A e. ZZ /\ h = ( A /L N ) ) -> E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) )
49	48	ex 450	. . . . . . 7 |- ( A e. ZZ -> ( h = ( A /L N ) -> E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
50	49	adantl 482	. . . . . 6 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( h = ( A /L N ) -> E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) )
51	41, 50	impbid 202	. . . . 5 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) <-> h = ( A /L N ) ) )
52	51	adantr 481	. . . 4 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( A /L N ) e. _V ) -> ( E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) <-> h = ( A /L N ) ) )
53	52	iota5 5871	. . 3 |- ( ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) /\ ( A /L N ) e. _V ) -> ( iota h E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) = ( A /L N ) )
54	18, 53	mpan2 707	. 2 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( iota h E. m e. ZZ ( ( L ` A ) = ( L ` m ) /\ h = ( m /L N ) ) ) = ( A /L N ) )
55	17, 54	eqtrd 2656	1 |- ( ( ( N e. NN /\ -. 2 || N ) /\ A e. ZZ ) -> ( X ` ( L ` A ) ) = ( A /L N ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   iotacio 5849   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   mod cmo 12668   || cdvds 14983   Basecbs 15857   ZRHomczrh 19848  ℤ/nℤczn 19851  DChrcdchr 24957   /Lclgs 25019

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-2o 7561  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-card 8765  df-cda 8990  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-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-phi 15471  df-pc 15542  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-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-lgs 25020

This theorem is referenced by:  lgsdchr  25080