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Theorem lgsdchrval 21084
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 10184 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
21adantr 452 . . . . 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 16783 . . . . 5  |-  ( N  e.  NN0  ->  L : ZZ -onto-> B )
7 fof 5612 . . . . 5  |-  ( L : ZZ -onto-> B  ->  L : ZZ --> B )
82, 6, 73syl 19 . . . 4  |-  ( ( N  e.  NN  /\  -.  2  ||  N )  ->  L : ZZ --> B )
98ffvelrnda 5829 . . 3  |-  ( ( ( N  e.  NN  /\ 
-.  2  ||  N
)  /\  A  e.  ZZ )  ->  ( L `
 A )  e.  B )
10 eqeq1 2410 . . . . . . 7  |-  ( y  =  ( L `  A )  ->  (
y  =  ( L `
 m )  <->  ( L `  A )  =  ( L `  m ) ) )
1110anbi1d 686 . . . . . 6  |-  ( y  =  ( L `  A )  ->  (
( y  =  ( L `  m )  /\  h  =  ( m  / L N
) )  <->  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  / L N ) ) ) )
1211rexbidv 2687 . . . . 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 5398 . . . 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 5394 . . . 4  |-  ( iota
h E. m  e.  ZZ  ( y  =  ( L `  m
)  /\  h  =  ( m  / L N
) ) )  e. 
_V
1613, 14, 15fvmpt3i 5768 . . 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 6065 . . 3  |-  ( A  / L N )  e.  _V
19 simprr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( L `  A )  =  ( L `  m ) )
20 simplll 735 . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  A  e.  ZZ )
23 simprl 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  m  e.  ZZ )
243, 5zndvds 16785 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  m  e.  ZZ )  ->  (
( L `  A
)  =  ( L `
 m )  <->  N  ||  ( A  -  m )
) )
2521, 22, 23, 24syl3anc 1184 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 14  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  N  ||  ( A  -  m )
)
27 moddvds 12814 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  A  e.  ZZ  /\  m  e.  ZZ )  ->  (
( A  mod  N
)  =  ( m  mod  N )  <->  N  ||  ( A  -  m )
) )
2820, 22, 23, 27syl3anc 1184 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( A  mod  N )  =  ( m  mod  N ) )
3029oveq1d 6055 . . . . . . . . . . . 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 736 . . . . . . . . . . . . 13  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  -.  2  ||  N )
32 lgsmod 21058 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( ( A  mod  N )  / L N
)  =  ( A  / L N ) )
3322, 20, 31, 32syl3anc 1184 . . . . . . . . . . . 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 21058 . . . . . . . . . . . . 13  |-  ( ( m  e.  ZZ  /\  N  e.  NN  /\  -.  2  ||  N )  -> 
( ( m  mod  N )  / L N
)  =  ( m  / L N ) )
3523, 20, 31, 34syl3anc 1184 . . . . . . . . . . . 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 2444 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  NN  /\  -.  2  ||  N )  /\  A  e.  ZZ )  /\  (
m  e.  ZZ  /\  ( L `  A )  =  ( L `  m ) ) )  ->  ( A  / L N )  =  ( m  / L N
) )
3736eqeq2d 2415 . . . . . . . . . 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 215 . . . . . . . . 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 630 . . . . . . . 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 587 . . . . . . 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 2790 . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( m  =  A  ->  ( L `  m )  =  ( L `  A ) )
4342eqcomd 2409 . . . . . . . . . . 11  |-  ( m  =  A  ->  ( L `  A )  =  ( L `  m ) )
4443biantrurd 495 . . . . . . . . . 10  |-  ( m  =  A  ->  (
h  =  ( m  / L N )  <-> 
( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  / L N
) ) ) )
45 oveq1 6047 . . . . . . . . . . 11  |-  ( m  =  A  ->  (
m  / L N
)  =  ( A  / L N ) )
4645eqeq2d 2415 . . . . . . . . . 10  |-  ( m  =  A  ->  (
h  =  ( m  / L N )  <-> 
h  =  ( A  / L N ) ) )
4744, 46bitr3d 247 . . . . . . . . 9  |-  ( m  =  A  ->  (
( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  / L N
) )  <->  h  =  ( A  / L N
) ) )
4847rspcev 3012 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  h  =  ( A  / L N ) )  ->  E. m  e.  ZZ  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  / L N
) ) )
4948ex 424 . . . . . . 7  |-  ( A  e.  ZZ  ->  (
h  =  ( A  / L N )  ->  E. m  e.  ZZ  ( ( L `  A )  =  ( L `  m )  /\  h  =  ( m  / L N
) ) ) )
5049adantl 453 . . . . . 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 184 . . . . 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 452 . . . 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 5397 . . 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 653 . 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 2436 1  |-  ( ( ( N  e.  NN  /\ 
-.  2  ||  N
)  /\  A  e.  ZZ )  ->  ( X `
 ( L `  A ) )  =  ( A  / L N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   iotacio 5375   -->wf 5409   -onto->wfo 5411   ` cfv 5413  (class class class)co 6040    - cmin 9247   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238    mod cmo 11205    || cdivides 12807   Basecbs 13424   ZRHomczrh 16733  ℤ/nczn 16736  DChrcdchr 20969    / Lclgs 21031
This theorem is referenced by:  lgsdchr  21085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-phi 13110  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-cnfld 16659  df-zrh 16737  df-zn 16740  df-lgs 21032
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