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Theorem gausslemma2d 15133
Description: Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer  2: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)
Hypotheses
Ref Expression
gausslemma2d.p  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
gausslemma2d.h  |-  H  =  ( ( P  - 
1 )  /  2
)
gausslemma2d.r  |-  R  =  ( x  e.  ( 1 ... H ) 
|->  if ( ( x  x.  2 )  < 
( P  /  2
) ,  ( x  x.  2 ) ,  ( P  -  (
x  x.  2 ) ) ) )
gausslemma2d.m  |-  M  =  ( |_ `  ( P  /  4 ) )
gausslemma2d.n  |-  N  =  ( H  -  M
)
Assertion
Ref Expression
gausslemma2d  |-  ( ph  ->  ( 2  /L
P )  =  (
-u 1 ^ N
) )
Distinct variable groups:    x, H    x, P    ph, x    x, M
Allowed substitution hints:    R( x)    N( x)

Proof of Theorem gausslemma2d
StepHypRef Expression
1 gausslemma2d.p . . 3  |-  ( ph  ->  P  e.  ( Prime  \  { 2 } ) )
2 gausslemma2d.h . . 3  |-  H  =  ( ( P  - 
1 )  /  2
)
3 gausslemma2d.r . . 3  |-  R  =  ( x  e.  ( 1 ... H ) 
|->  if ( ( x  x.  2 )  < 
( P  /  2
) ,  ( x  x.  2 ) ,  ( P  -  (
x  x.  2 ) ) ) )
4 gausslemma2d.m . . 3  |-  M  =  ( |_ `  ( P  /  4 ) )
5 gausslemma2d.n . . 3  |-  N  =  ( H  -  M
)
61, 2, 3, 4, 5gausslemma2dlem7 15132 . 2  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  mod 
P )  =  1 )
71gausslemma2dlem0a 15113 . . . . . . 7  |-  ( ph  ->  P  e.  NN )
8 nnq 9688 . . . . . . 7  |-  ( P  e.  NN  ->  P  e.  QQ )
97, 8syl 14 . . . . . 6  |-  ( ph  ->  P  e.  QQ )
10 eldifi 3281 . . . . . . 7  |-  ( P  e.  ( Prime  \  {
2 } )  ->  P  e.  Prime )
11 prmgt1 12260 . . . . . . 7  |-  ( P  e.  Prime  ->  1  < 
P )
121, 10, 113syl 17 . . . . . 6  |-  ( ph  ->  1  <  P )
13 q1mod 10417 . . . . . 6  |-  ( ( P  e.  QQ  /\  1  <  P )  -> 
( 1  mod  P
)  =  1 )
149, 12, 13syl2anc 411 . . . . 5  |-  ( ph  ->  ( 1  mod  P
)  =  1 )
1514eqcomd 2199 . . . 4  |-  ( ph  ->  1  =  ( 1  mod  P ) )
1615eqeq2d 2205 . . 3  |-  ( ph  ->  ( ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  1  <->  ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) ) )
17 neg1z 9339 . . . . . . . . . 10  |-  -u 1  e.  ZZ
181, 4, 2, 5gausslemma2dlem0h 15120 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
19 zexpcl 10615 . . . . . . . . . 10  |-  ( (
-u 1  e.  ZZ  /\  N  e.  NN0 )  ->  ( -u 1 ^ N )  e.  ZZ )
2017, 18, 19sylancr 414 . . . . . . . . 9  |-  ( ph  ->  ( -u 1 ^ N )  e.  ZZ )
21 2nn 9133 . . . . . . . . . . . 12  |-  2  e.  NN
2221a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  NN )
231, 2gausslemma2dlem0b 15114 . . . . . . . . . . . 12  |-  ( ph  ->  H  e.  NN )
2423nnnn0d 9283 . . . . . . . . . . 11  |-  ( ph  ->  H  e.  NN0 )
2522, 24nnexpcld 10756 . . . . . . . . . 10  |-  ( ph  ->  ( 2 ^ H
)  e.  NN )
2625nnzd 9428 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ H
)  e.  ZZ )
2720, 26zmulcld 9435 . . . . . . . 8  |-  ( ph  ->  ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  ZZ )
28 zq 9681 . . . . . . . 8  |-  ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  ZZ  ->  ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  QQ )
2927, 28syl 14 . . . . . . 7  |-  ( ph  ->  ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  e.  QQ )
3029adantr 276 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  e.  QQ )
31 1z 9333 . . . . . . 7  |-  1  e.  ZZ
32 zq 9681 . . . . . . 7  |-  ( 1  e.  ZZ  ->  1  e.  QQ )
3331, 32mp1i 10 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  1  e.  QQ )
3420adantr 276 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  ( -u 1 ^ N )  e.  ZZ )
359adantr 276 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  P  e.  QQ )
367nngt0d 9016 . . . . . . 7  |-  ( ph  ->  0  <  P )
3736adantr 276 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  0  <  P )
38 simpr 110 . . . . . 6  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  (
( ( -u 1 ^ N )  x.  (
2 ^ H ) )  mod  P )  =  ( 1  mod 
P ) )
3930, 33, 34, 35, 37, 38modqmul1 10438 . . . . 5  |-  ( (
ph  /\  ( (
( -u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
) )  ->  (
( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( -u 1 ^ N ) )  mod 
P )  =  ( ( 1  x.  ( -u 1 ^ N ) )  mod  P ) )
4039ex 115 . . . 4  |-  ( ph  ->  ( ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
)  ->  ( (
( ( -u 1 ^ N )  x.  (
2 ^ H ) )  x.  ( -u
1 ^ N ) )  mod  P )  =  ( ( 1  x.  ( -u 1 ^ N ) )  mod 
P ) ) )
4120zcnd 9430 . . . . . . . . 9  |-  ( ph  ->  ( -u 1 ^ N )  e.  CC )
4225nncnd 8986 . . . . . . . . 9  |-  ( ph  ->  ( 2 ^ H
)  e.  CC )
4341, 42, 41mul32d 8162 . . . . . . . 8  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( -u 1 ^ N ) )  =  ( ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  x.  ( 2 ^ H
) ) )
4418nn0cnd 9285 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
45442timesd 9215 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  N
)  =  ( N  +  N ) )
4645eqcomd 2199 . . . . . . . . . . 11  |-  ( ph  ->  ( N  +  N
)  =  ( 2  x.  N ) )
4746oveq2d 5926 . . . . . . . . . 10  |-  ( ph  ->  ( -u 1 ^ ( N  +  N
) )  =  (
-u 1 ^ (
2  x.  N ) ) )
48 neg1cn 9077 . . . . . . . . . . . 12  |-  -u 1  e.  CC
4948a1i 9 . . . . . . . . . . 11  |-  ( ph  -> 
-u 1  e.  CC )
5049, 18, 18expaddd 10736 . . . . . . . . . 10  |-  ( ph  ->  ( -u 1 ^ ( N  +  N
) )  =  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) ) )
5118nn0zd 9427 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ZZ )
52 m1expeven 10647 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  ( -u 1 ^ ( 2  x.  N ) )  =  1 )
5351, 52syl 14 . . . . . . . . . 10  |-  ( ph  ->  ( -u 1 ^ ( 2  x.  N
) )  =  1 )
5447, 50, 533eqtr3d 2234 . . . . . . . . 9  |-  ( ph  ->  ( ( -u 1 ^ N )  x.  ( -u 1 ^ N ) )  =  1 )
5554oveq1d 5925 . . . . . . . 8  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( -u 1 ^ N ) )  x.  ( 2 ^ H
) )  =  ( 1  x.  ( 2 ^ H ) ) )
5642mullidd 8027 . . . . . . . 8  |-  ( ph  ->  ( 1  x.  (
2 ^ H ) )  =  ( 2 ^ H ) )
5743, 55, 563eqtrd 2230 . . . . . . 7  |-  ( ph  ->  ( ( ( -u
1 ^ N )  x.  ( 2 ^ H ) )  x.  ( -u 1 ^ N ) )  =  ( 2 ^ H
) )
5857oveq1d 5925 . . . . . 6  |-  ( ph  ->  ( ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  x.  ( -u 1 ^ N ) )  mod 
P )  =  ( ( 2 ^ H
)  mod  P )
)
5941mullidd 8027 . . . . . . 7  |-  ( ph  ->  ( 1  x.  ( -u 1 ^ N ) )  =  ( -u
1 ^ N ) )
6059oveq1d 5925 . . . . . 6  |-  ( ph  ->  ( ( 1  x.  ( -u 1 ^ N ) )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
) )
6158, 60eqeq12d 2208 . . . . 5  |-  ( ph  ->  ( ( ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  x.  ( -u
1 ^ N ) )  mod  P )  =  ( ( 1  x.  ( -u 1 ^ N ) )  mod 
P )  <->  ( (
2 ^ H )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
) ) )
622oveq2i 5921 . . . . . . . 8  |-  ( 2 ^ H )  =  ( 2 ^ (
( P  -  1 )  /  2 ) )
6362oveq1i 5920 . . . . . . 7  |-  ( ( 2 ^ H )  mod  P )  =  ( ( 2 ^ ( ( P  - 
1 )  /  2
) )  mod  P
)
6463eqeq1i 2201 . . . . . 6  |-  ( ( ( 2 ^ H
)  mod  P )  =  ( ( -u
1 ^ N )  mod  P )  <->  ( (
2 ^ ( ( P  -  1 )  /  2 ) )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
) )
65 2z 9335 . . . . . . . . . 10  |-  2  e.  ZZ
66 lgsvalmod 15083 . . . . . . . . . 10  |-  ( ( 2  e.  ZZ  /\  P  e.  ( Prime  \  { 2 } ) )  ->  ( (
2  /L P )  mod  P )  =  ( ( 2 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P ) )
6765, 1, 66sylancr 414 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  /L P )  mod 
P )  =  ( ( 2 ^ (
( P  -  1 )  /  2 ) )  mod  P ) )
6867eqcomd 2199 . . . . . . . 8  |-  ( ph  ->  ( ( 2 ^ ( ( P  - 
1 )  /  2
) )  mod  P
)  =  ( ( 2  /L P )  mod  P ) )
6968eqeq1d 2202 . . . . . . 7  |-  ( ph  ->  ( ( ( 2 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  <->  ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
) ) )
701, 4, 2, 5gausslemma2dlem0i 15121 . . . . . . 7  |-  ( ph  ->  ( ( ( 2  /L P )  mod  P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
7169, 70sylbid 150 . . . . . 6  |-  ( ph  ->  ( ( ( 2 ^ ( ( P  -  1 )  / 
2 ) )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
7264, 71biimtrid 152 . . . . 5  |-  ( ph  ->  ( ( ( 2 ^ H )  mod 
P )  =  ( ( -u 1 ^ N )  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
7361, 72sylbid 150 . . . 4  |-  ( ph  ->  ( ( ( ( ( -u 1 ^ N )  x.  (
2 ^ H ) )  x.  ( -u
1 ^ N ) )  mod  P )  =  ( ( 1  x.  ( -u 1 ^ N ) )  mod 
P )  ->  (
2  /L P )  =  ( -u
1 ^ N ) ) )
7440, 73syld 45 . . 3  |-  ( ph  ->  ( ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  ( 1  mod  P
)  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
7516, 74sylbid 150 . 2  |-  ( ph  ->  ( ( ( (
-u 1 ^ N
)  x.  ( 2 ^ H ) )  mod  P )  =  1  ->  ( 2  /L P )  =  ( -u 1 ^ N ) ) )
766, 75mpd 13 1  |-  ( ph  ->  ( 2  /L
P )  =  (
-u 1 ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164    \ cdif 3150   ifcif 3557   {csn 3618   class class class wbr 4029    |-> cmpt 4090   ` cfv 5246  (class class class)co 5910   CCcc 7860   0cc0 7862   1c1 7863    + caddc 7865    x. cmul 7867    < clt 8044    - cmin 8180   -ucneg 8181    / cdiv 8681   NNcn 8972   2c2 9023   4c4 9025   NN0cn0 9230   ZZcz 9307   QQcq 9674   ...cfz 10064   |_cfl 10327    mod cmo 10383   ^cexp 10599   Primecprime 12235    /Lclgs 15061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-mulrcl 7961  ax-addcom 7962  ax-mulcom 7963  ax-addass 7964  ax-mulass 7965  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-1rid 7969  ax-0id 7970  ax-rnegex 7971  ax-precex 7972  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978  ax-pre-mulgt0 7979  ax-pre-mulext 7980  ax-arch 7981  ax-caucvg 7982
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-po 4325  df-iso 4326  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-isom 5255  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-irdg 6414  df-frec 6435  df-1o 6460  df-2o 6461  df-oadd 6464  df-er 6578  df-en 6786  df-dom 6787  df-fin 6788  df-sup 7033  df-inf 7034  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-reap 8584  df-ap 8591  df-div 8682  df-inn 8973  df-2 9031  df-3 9032  df-4 9033  df-5 9034  df-6 9035  df-7 9036  df-8 9037  df-n0 9231  df-z 9308  df-uz 9583  df-q 9675  df-rp 9710  df-ioo 9948  df-fz 10065  df-fzo 10199  df-fl 10329  df-mod 10384  df-seqfrec 10509  df-exp 10600  df-fac 10787  df-ihash 10837  df-cj 10976  df-re 10977  df-im 10978  df-rsqrt 11132  df-abs 11133  df-clim 11412  df-proddc 11684  df-dvds 11921  df-gcd 12070  df-prm 12236  df-phi 12339  df-pc 12413  df-lgs 15062
This theorem is referenced by: (None)
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