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Theorem lgsdchrval 23747
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/n `  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
StepHypRef Expression
1 nnnn0 10823 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
21adantr 465 . . . . 5  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  N  e.  NN0 )
3 lgsdchr.z . . . . . 6  |-  Z  =  (ℤ/n `  N )
4 lgsdchr.b . . . . . 6  |-  B  =  ( Base `  Z
)
5 lgsdchr.l . . . . . 6  |-  L  =  ( ZRHom `  Z
)
63, 4, 5znzrhfo 18712 . . . . 5  |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
7 fof 5801 . . . . 5  |-  ( L : ZZ -onto-> B  ->  L : ZZ --> B )
82, 6, 73syl 20 . . . 4  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  L : ZZ --> B )
98ffvelrnda 6032 . . 3  |-  ( ( ( N  e.  NN  /\ 
-.  2  ||  N
)  /\  A  e.  ZZ )  ->  ( L `
 A )  e.  B )
10 eqeq1 2461 . . . . . . 7  |-  ( y  =  ( L `  A )  ->  (
y  =  ( L `
 m )  <->  ( L `  A )  =  ( L `  m ) ) )
1110anbi1d 704 . . . . . 6  |-  ( y  =  ( L `  A )  ->  (
( y  =  ( L `  m )  /\  h  =  ( m  /L N ) )  <->  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) ) ) )
1211rexbidv 2968 . . . . 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 ) ) ) )
1312iotabidv 5578 . . . 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 5574 . . . 4  |-  ( iota
h E. m  e.  ZZ  ( y  =  ( L `  m
)  /\  h  =  ( m  /L N ) ) )  e. 
_V
1613, 14, 15fvmpt3i 5960 . . 3  |-  ( ( L `  A )  e.  B  ->  ( X `  ( L `  A ) )  =  ( iota h E. m  e.  ZZ  (
( L `  A
)  =  ( L `
 m )  /\  h  =  ( m  /L N ) ) ) )
179, 16syl 16 . 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 6324 . . 3  |-  ( A  /L N )  e.  _V
19 simprr 757 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( L `  A )  =  ( L `  m ) )
20 simplll 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  N  e.  NN )
2120, 1syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  N  e.  NN0 )
22 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  A  e.  ZZ )
23 simprl 756 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  m  e.  ZZ )
243, 5zndvds 18714 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  m  e.  ZZ )  ->  (
( L `  A
)  =  ( L `
 m )  <->  N  ||  ( A  -  m )
) )
2521, 22, 23, 24syl3anc 1228 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( ( L `
 A )  =  ( L `  m
)  <->  N  ||  ( A  -  m ) ) )
2619, 25mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  N  ||  ( A  -  m )
)
27 moddvds 14004 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  m  e.  ZZ )  ->  (
( A  mod  N
)  =  ( m  mod  N )  <->  N  ||  ( A  -  m )
) )
2820, 22, 23, 27syl3anc 1228 . . . . . . . . . . . . . 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 ) ) )
2926, 28mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( A  mod  N )  =  ( m  mod  N ) )
3029oveq1d 6311 . . . . . . . . . . . 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 760 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  -.  2  ||  N )
32 lgsmod 23721 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( ( A  mod  N )  /L N )  =  ( A  /L N ) )
3322, 20, 31, 32syl3anc 1228 . . . . . . . . . . . 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 23721 . . . . . . . . . . . . 13  |-  ( ( m  e.  ZZ  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( ( m  mod  N )  /L N )  =  ( m  /L N ) )
3523, 20, 31, 34syl3anc 1228 . . . . . . . . . . . 12  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( ( m  mod  N )  /L N )  =  ( m  /L
N ) )
3630, 33, 353eqtr3d 2506 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( A  /L N )  =  ( m  /L
N ) )
3736eqeq2d 2471 . . . . . . . . . 10  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( h  =  ( A  /L
N )  <->  h  =  ( m  /L N ) ) )
3837biimprd 223 . . . . . . . . 9  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( h  =  ( m  /L
N )  ->  h  =  ( A  /L N ) ) )
3938anassrs 648 . . . . . . . 8  |-  ( ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  m  e.  ZZ )  /\  ( L `  A )  =  ( L `  m ) )  ->  ( h  =  ( m  /L N )  ->  h  =  ( A  /L N ) ) )
4039expimpd 603 . . . . . . 7  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  m  e.  ZZ )  ->  (
( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) )  ->  h  =  ( A  /L N ) ) )
4140rexlimdva 2949 . . . . . 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 5872 . . . . . . . . . . . 12  |-  ( m  =  A  ->  ( L `  m )  =  ( L `  A ) )
4342eqcomd 2465 . . . . . . . . . . 11  |-  ( m  =  A  ->  ( L `  A )  =  ( L `  m ) )
4443biantrurd 508 . . . . . . . . . 10  |-  ( m  =  A  ->  (
h  =  ( m  /L N )  <-> 
( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) ) ) )
45 oveq1 6303 . . . . . . . . . . 11  |-  ( m  =  A  ->  (
m  /L N )  =  ( A  /L N ) )
4645eqeq2d 2471 . . . . . . . . . 10  |-  ( m  =  A  ->  (
h  =  ( m  /L N )  <-> 
h  =  ( A  /L N ) ) )
4744, 46bitr3d 255 . . . . . . . . 9  |-  ( m  =  A  ->  (
( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) )  <->  h  =  ( A  /L N ) ) )
4847rspcev 3210 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  h  =  ( A  /L N ) )  ->  E. m  e.  ZZ  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) ) )
4948ex 434 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
h  =  ( A  /L N )  ->  E. m  e.  ZZ  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  /L N ) ) ) )
5049adantl 466 . . . . . 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 ) ) ) )
5141, 50impbid 191 . . . . 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 ) ) )
5251adantr 465 . . . 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 ) ) )
5352iota5 5577 . . 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 ) )
5418, 53mpan2 671 . 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 ) )
5517, 54eqtrd 2498 1  |-  ( ( ( N  e.  NN  /\ 
-.  2  ||  N
)  /\  A  e.  ZZ )  ->  ( X `
 ( L `  A ) )  =  ( A  /L
N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   iotacio 5555   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296    - cmin 9824   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885    mod cmo 11998    || cdvds 13997   Basecbs 14643   ZRHomczrh 18663  ℤ/nczn 18666  DChrcdchr 23632    /Lclgs 23694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-fz 11698  df-fzo 11821  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-dvds 13998  df-gcd 14156  df-prm 14229  df-phi 14307  df-pc 14372  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-0g 14858  df-imas 14924  df-qus 14925  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-nsg 16325  df-eqg 16326  df-ghm 16391  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-cring 17327  df-oppr 17398  df-dvdsr 17416  df-rnghom 17490  df-subrg 17553  df-lmod 17640  df-lss 17705  df-lsp 17744  df-sra 17944  df-rgmod 17945  df-lidl 17946  df-rsp 17947  df-2idl 18006  df-cnfld 18547  df-zring 18615  df-zrh 18667  df-zn 18670  df-lgs 23695
This theorem is referenced by:  lgsdchr  23748
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