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Theorem lgsvalmod 20554
Description: The Legendre symbol is equivalent to  a ^ (
( p  -  1 )  /  2 ),  mod  p. (Contributed by Mario Carneiro, 4-Feb-2015.)
Assertion
Ref Expression
lgsvalmod  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  mod  P )  =  ( ( A ^
( ( P  - 
1 )  /  2
) )  mod  P
) )

Proof of Theorem lgsvalmod
StepHypRef Expression
1 eldifi 3298 . . . . . . . 8  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
21adantl 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  Prime )
3 prmz 12762 . . . . . . 7  |-  ( P  e.  Prime  ->  P  e.  ZZ )
42, 3syl 15 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  ZZ )
5 lgscl 20549 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ZZ )  ->  ( A  / L P )  e.  ZZ )
64, 5syldan 456 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  e.  ZZ )
76zred 10117 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  e.  RR )
8 peano2re 8985 . . . 4  |-  ( ( A  / L P
)  e.  RR  ->  ( ( A  / L P )  +  1 )  e.  RR )
97, 8syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  +  1 )  e.  RR )
10 oddprm 12868 . . . . . . . 8  |-  ( P  e.  ( Prime  \  {
2 } )  -> 
( ( P  - 
1 )  /  2
)  e.  NN )
1110adantl 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  -  1 )  /  2 )  e.  NN )
1211nnnn0d 10018 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( P  -  1 )  /  2 )  e. 
NN0 )
13 zexpcl 11118 . . . . . 6  |-  ( ( A  e.  ZZ  /\  ( ( P  - 
1 )  /  2
)  e.  NN0 )  ->  ( A ^ (
( P  -  1 )  /  2 ) )  e.  ZZ )
1412, 13syldan 456 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  ZZ )
1514zred 10117 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  RR )
16 peano2re 8985 . . . 4  |-  ( ( A ^ ( ( P  -  1 )  /  2 ) )  e.  RR  ->  (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  e.  RR )
1715, 16syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  e.  RR )
18 1re 8837 . . . . 5  |-  1  e.  RR
1918renegcli 9108 . . . 4  |-  -u 1  e.  RR
2019a1i 10 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  -u 1  e.  RR )
21 prmnn 12761 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
222, 21syl 15 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  NN )
2322nnrpd 10389 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  P  e.  RR+ )
24 lgsval3 20553 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  =  ( ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  -  1 ) )
2524eqcomd 2288 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  -  1 )  =  ( A  / L P ) )
2617, 23modcld 10977 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e.  RR )
2726recnd 8861 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  e.  CC )
28 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
2928a1i 10 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  1  e.  CC )
307recnd 8861 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A  / L P )  e.  CC )
3127, 29, 30subadd2d 9176 . . . . . 6  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
)  -  1 )  =  ( A  / L P )  <->  ( ( A  / L P )  +  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
) ) )
3225, 31mpbid 201 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  +  1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
) )
3332oveq1d 5873 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A  / L P )  +  1 )  mod  P )  =  ( ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P )  mod 
P ) )
34 modabs2 10998 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  e.  RR  /\  P  e.  RR+ )  -> 
( ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P )  mod  P
)  =  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  mod  P ) )
3517, 23, 34syl2anc 642 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P )  mod  P )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  mod  P
) )
3633, 35eqtrd 2315 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A  / L P )  +  1 )  mod  P )  =  ( ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  mod 
P ) )
37 modadd1 11001 . . 3  |-  ( ( ( ( ( A  / L P )  +  1 )  e.  RR  /\  ( ( A ^ ( ( P  -  1 )  /  2 ) )  +  1 )  e.  RR )  /\  ( -u 1  e.  RR  /\  P  e.  RR+ )  /\  ( ( ( A  / L P )  +  1 )  mod 
P )  =  ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  mod  P ) )  ->  ( (
( ( A  / L P )  +  1 )  +  -u 1
)  mod  P )  =  ( ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  +  -u 1 )  mod 
P ) )
389, 17, 20, 23, 36, 37syl221anc 1193 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A  / L P )  +  1 )  +  -u 1
)  mod  P )  =  ( ( ( ( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  +  -u 1 )  mod 
P ) )
399recnd 8861 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  +  1 )  e.  CC )
40 negsub 9095 . . . . 5  |-  ( ( ( ( A  / L P )  +  1 )  e.  CC  /\  1  e.  CC )  ->  ( ( ( A  / L P )  +  1 )  + 
-u 1 )  =  ( ( ( A  / L P )  +  1 )  - 
1 ) )
4139, 28, 40sylancl 643 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A  / L P )  +  1 )  +  -u 1
)  =  ( ( ( A  / L P )  +  1 )  -  1 ) )
42 pncan 9057 . . . . 5  |-  ( ( ( A  / L P )  e.  CC  /\  1  e.  CC )  ->  ( ( ( A  / L P
)  +  1 )  -  1 )  =  ( A  / L P ) )
4330, 28, 42sylancl 643 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A  / L P )  +  1 )  -  1 )  =  ( A  / L P ) )
4441, 43eqtrd 2315 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A  / L P )  +  1 )  +  -u 1
)  =  ( A  / L P ) )
4544oveq1d 5873 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A  / L P )  +  1 )  +  -u 1
)  mod  P )  =  ( ( A  / L P )  mod  P ) )
4617recnd 8861 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  e.  CC )
47 negsub 9095 . . . . 5  |-  ( ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  e.  CC  /\  1  e.  CC )  ->  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  +  -u
1 )  =  ( ( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  -  1 ) )
4846, 28, 47sylancl 643 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  +  -u 1 )  =  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  -  1 ) )
4915recnd 8861 . . . . 5  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( A ^ ( ( P  -  1 )  / 
2 ) )  e.  CC )
50 pncan 9057 . . . . 5  |-  ( ( ( A ^ (
( P  -  1 )  /  2 ) )  e.  CC  /\  1  e.  CC )  ->  ( ( ( A ^ ( ( P  -  1 )  / 
2 ) )  +  1 )  -  1 )  =  ( A ^ ( ( P  -  1 )  / 
2 ) ) )
5149, 28, 50sylancl 643 . . . 4  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  -  1 )  =  ( A ^ (
( P  -  1 )  /  2 ) ) )
5248, 51eqtrd 2315 . . 3  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( A ^ (
( P  -  1 )  /  2 ) )  +  1 )  +  -u 1 )  =  ( A ^ (
( P  -  1 )  /  2 ) ) )
5352oveq1d 5873 . 2  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
( ( A ^
( ( P  - 
1 )  /  2
) )  +  1 )  +  -u 1
)  mod  P )  =  ( ( A ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
5438, 45, 533eqtr3d 2323 1  |-  ( ( A  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( ( A  / L P )  mod  P )  =  ( ( A ^
( ( P  - 
1 )  /  2
) )  mod  P
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149   {csn 3640  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   RR+crp 10354    mod cmo 10973   ^cexp 11104   Primecprime 12758    / Lclgs 20533
This theorem is referenced by:  lgsdirprm  20568  lgsne0  20572  lgsqrlem3  20582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-lgs 20534
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