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Theorem lgssq 21121
Description: The Legendre symbol at a square is equal to  1. Together with lgsmod 21107 this implies that the Legendre symbol takes value  1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.)
Assertion
Ref Expression
lgssq  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ 2 )  / L N
)  =  1 )

Proof of Theorem lgssq
StepHypRef Expression
1 nnz 10305 . . . 4  |-  ( A  e.  NN  ->  A  e.  ZZ )
213ad2ant1 979 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  e.  ZZ )
3 simp2 959 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  N  e.  ZZ )
4 nnne0 10034 . . . 4  |-  ( A  e.  NN  ->  A  =/=  0 )
543ad2ant1 979 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  =/=  0 )
6 lgsdir 21116 . . 3  |-  ( ( ( A  e.  ZZ  /\  A  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  A  =/=  0
) )  ->  (
( A  x.  A
)  / L N
)  =  ( ( A  / L N
)  x.  ( A  / L N ) ) )
72, 2, 3, 5, 5, 6syl32anc 1193 . 2  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A  x.  A
)  / L N
)  =  ( ( A  / L N
)  x.  ( A  / L N ) ) )
8 nncn 10010 . . . . 5  |-  ( A  e.  NN  ->  A  e.  CC )
983ad2ant1 979 . . . 4  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  A  e.  CC )
109sqvald 11522 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A ^ 2 )  =  ( A  x.  A
) )
1110oveq1d 6098 . 2  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ 2 )  / L N
)  =  ( ( A  x.  A )  / L N ) )
12 lgscl 21096 . . . . . 6  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( A  / L N )  e.  ZZ )
132, 3, 12syl2anc 644 . . . . 5  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  ZZ )
1413zred 10377 . . . 4  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  RR )
15 absresq 12109 . . . 4  |-  ( ( A  / L N
)  e.  RR  ->  ( ( abs `  ( A  / L N ) ) ^ 2 )  =  ( ( A  / L N ) ^ 2 ) )
1614, 15syl 16 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  ( ( A  / L N ) ^ 2 ) )
17 lgsabs1 21120 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( abs `  ( A  / L N ) )  =  1  <->  ( A  gcd  N )  =  1 ) )
181, 17sylan 459 . . . . . 6  |-  ( ( A  e.  NN  /\  N  e.  ZZ )  ->  ( ( abs `  ( A  / L N ) )  =  1  <->  ( A  gcd  N )  =  1 ) )
1918biimp3ar 1285 . . . . 5  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( abs `  ( A  / L N ) )  =  1 )
2019oveq1d 6098 . . . 4  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  ( 1 ^ 2 ) )
21 sq1 11478 . . . 4  |-  ( 1 ^ 2 )  =  1
2220, 21syl6eq 2486 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( abs `  ( A  / L N ) ) ^ 2 )  =  1 )
2313zcnd 10378 . . . 4  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  ( A  / L N )  e.  CC )
2423sqvald 11522 . . 3  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A  / L N ) ^ 2 )  =  ( ( A  / L N
)  x.  ( A  / L N ) ) )
2516, 22, 243eqtr3d 2478 . 2  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  1  =  ( ( A  / L N )  x.  ( A  / L N ) ) )
267, 11, 253eqtr4d 2480 1  |-  ( ( A  e.  NN  /\  N  e.  ZZ  /\  ( A  gcd  N )  =  1 )  ->  (
( A ^ 2 )  / L N
)  =  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    x. cmul 8997   NNcn 10002   2c2 10051   ZZcz 10284   ^cexp 11384   abscabs 12041    gcd cgcd 13008    / Lclgs 21080
This theorem is referenced by:  1lgs  21123  lgsqr  21132  lgsquad2lem2  21145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-n0 10224  df-z 10285  df-uz 10491  df-q 10577  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-prm 13082  df-phi 13157  df-pc 13213  df-lgs 21081
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