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Theorem lgssq2 20575
Description: The Legendre symbol at a square is equal to  1. (Contributed by Mario Carneiro, 5-Feb-2015.)
Assertion
Ref Expression
lgssq2  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L ( N ^ 2 ) )  =  1 )

Proof of Theorem lgssq2
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  A  e.  ZZ )
2 nnz 10045 . . . 4  |-  ( N  e.  NN  ->  N  e.  ZZ )
323ad2ant2 977 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  N  e.  ZZ )
4 nnne0 9778 . . . 4  |-  ( N  e.  NN  ->  N  =/=  0 )
543ad2ant2 977 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  N  =/=  0 )
6 lgsdi 20571 . . 3  |-  ( ( ( A  e.  ZZ  /\  N  e.  ZZ  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  N  =/=  0
) )  ->  ( A  / L ( N  x.  N ) )  =  ( ( A  / L N )  x.  ( A  / L N ) ) )
71, 3, 3, 5, 5, 6syl32anc 1190 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L ( N  x.  N ) )  =  ( ( A  / L N )  x.  ( A  / L N ) ) )
8 nncn 9754 . . . . 5  |-  ( N  e.  NN  ->  N  e.  CC )
983ad2ant2 977 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  N  e.  CC )
109sqvald 11242 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( N ^ 2 )  =  ( N  x.  N
) )
1110oveq2d 5874 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L ( N ^ 2 ) )  =  ( A  / L ( N  x.  N ) ) )
12 lgscl 20549 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
131, 3, 12syl2anc 642 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  ZZ )
1413zred 10117 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  RR )
15 absresq 11787 . . . 4  |-  ( ( A  / L N
)  e.  RR  ->  ( ( abs `  ( A  / L N ) ) ^ 2 )  =  ( ( A  / L N ) ^ 2 ) )
1614, 15syl 15 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  ( ( A  / L N ) ^ 2 ) )
17 lgsabs1 20573 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  ( A  / L N ) )  =  1  <->  ( A  gcd  N )  =  1 ) )
182, 17sylan2 460 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  NN )  ->  ( ( abs `  ( A  / L N ) )  =  1  <->  ( A  gcd  N )  =  1 ) )
1918biimp3ar 1282 . . . . 5  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( abs `  ( A  / L N ) )  =  1 )
2019oveq1d 5873 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  ( 1 ^ 2 ) )
21 sq1 11198 . . . 4  |-  ( 1 ^ 2 )  =  1
2220, 21syl6eq 2331 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  1 )
2313zcnd 10118 . . . 4  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  CC )
2423sqvald 11242 . . 3  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  (
( A  / L N ) ^ 2 )  =  ( ( A  / L N
)  x.  ( A  / L N ) ) )
2516, 22, 243eqtr3d 2323 . 2  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  1  =  ( ( A  / L N )  x.  ( A  / L N ) ) )
267, 11, 253eqtr4d 2325 1  |-  ( ( A  e.  ZZ  /\  N  e.  NN  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L ( N ^ 2 ) )  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   NNcn 9746   2c2 9795   ZZcz 10024   ^cexp 11104   abscabs 11719    gcd cgcd 12685    / Lclgs 20533
This theorem is referenced by:  lgs1  20577  lgsquad2lem2  20598
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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