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Theorem lgsquad2 20599
Description: Extend lgsquad 20596 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypotheses
Ref Expression
lgsquad2.1  |-  ( ph  ->  M  e.  NN )
lgsquad2.2  |-  ( ph  ->  -.  2  ||  M
)
lgsquad2.3  |-  ( ph  ->  N  e.  NN )
lgsquad2.4  |-  ( ph  ->  -.  2  ||  N
)
lgsquad2.5  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
Assertion
Ref Expression
lgsquad2  |-  ( ph  ->  ( ( M  / L N )  x.  ( N  / L M ) )  =  ( -u
1 ^ ( ( ( M  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) ) )

Proof of Theorem lgsquad2
Dummy variables  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lgsquad2.1 . 2  |-  ( ph  ->  M  e.  NN )
2 lgsquad2.2 . 2  |-  ( ph  ->  -.  2  ||  M
)
3 lgsquad2.3 . 2  |-  ( ph  ->  N  e.  NN )
4 lgsquad2.4 . 2  |-  ( ph  ->  -.  2  ||  N
)
5 lgsquad2.5 . 2  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
63adantr 451 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  NN )
74adantr 451 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  N )
8 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
9 eldifi 3298 . . . . . 6  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  e.  Prime )
108, 9syl 15 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  Prime )
11 prmnn 12761 . . . . 5  |-  ( m  e.  Prime  ->  m  e.  NN )
1210, 11syl 15 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  NN )
13 eldifsni 3750 . . . . . . . 8  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  =/=  2 )
148, 13syl 15 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  =/=  2
)
1514necomd 2529 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  2  =/=  m
)
1615neneqd 2462 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  =  m )
17 2z 10054 . . . . . . 7  |-  2  e.  ZZ
18 uzid 10242 . . . . . . 7  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1917, 18ax-mp 8 . . . . . 6  |-  2  e.  ( ZZ>= `  2 )
20 dvdsprm 12778 . . . . . 6  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  m  e.  Prime )  ->  (
2  ||  m  <->  2  =  m ) )
2119, 10, 20sylancr 644 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( 2  ||  m 
<->  2  =  m ) )
2216, 21mtbird 292 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  m )
236nnzd 10116 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  ZZ )
2412nnzd 10116 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ZZ )
25 gcdcom 12699 . . . . . 6  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  gcd  m
)  =  ( m  gcd  N ) )
2623, 24, 25syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  ( m  gcd  N ) )
27 simprr 733 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  gcd  N )  =  1 )
2826, 27eqtrd 2315 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  1 )
29 simprl 732 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  e.  ( Prime  \  { 2 } ) )
308adantr 451 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
31 eldifi 3298 . . . . . . . 8  |-  ( n  e.  ( Prime  \  {
2 } )  ->  n  e.  Prime )
32 prmrp 12780 . . . . . . . 8  |-  ( ( n  e.  Prime  /\  m  e.  Prime )  ->  (
( n  gcd  m
)  =  1  <->  n  =/=  m ) )
3331, 10, 32syl2anr 464 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  <-> 
n  =/=  m ) )
3433biimpd 198 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  ->  n  =/=  m
) )
3534impr 602 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  =/=  m
)
36 lgsquad 20596 . . . . 5  |-  ( ( n  e.  ( Prime  \  { 2 } )  /\  m  e.  ( Prime  \  { 2 } )  /\  n  =/=  m )  ->  (
( n  / L
m )  x.  (
m  / L n ) )  =  (
-u 1 ^ (
( ( n  - 
1 )  /  2
)  x.  ( ( m  -  1 )  /  2 ) ) ) )
3729, 30, 35, 36syl3anc 1182 . . . 4  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  ( ( n  / L m )  x.  ( m  / L n ) )  =  ( -u 1 ^ ( ( ( n  -  1 )  /  2 )  x.  ( ( m  - 
1 )  /  2
) ) ) )
38 biid 227 . . . 4  |-  ( A. x  e.  ( 1 ... y ) ( ( x  gcd  (
2  x.  m ) )  =  1  -> 
( ( x  / L m )  x.  ( m  / L
x ) )  =  ( -u 1 ^ ( ( ( x  -  1 )  / 
2 )  x.  (
( m  -  1 )  /  2 ) ) ) )  <->  A. x  e.  ( 1 ... y
) ( ( x  gcd  ( 2  x.  m ) )  =  1  ->  ( (
x  / L m )  x.  ( m  / L x ) )  =  ( -u
1 ^ ( ( ( x  -  1 )  /  2 )  x.  ( ( m  -  1 )  / 
2 ) ) ) ) )
396, 7, 12, 22, 28, 37, 38lgsquad2lem2 20598 . . 3  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  / L m )  x.  ( m  / L N ) )  =  ( -u 1 ^ ( ( ( N  -  1 )  / 
2 )  x.  (
( m  -  1 )  /  2 ) ) ) )
40 lgscl 20549 . . . . 5  |-  ( ( m  e.  ZZ  /\  N  e.  ZZ )  ->  ( m  / L N )  e.  ZZ )
4124, 23, 40syl2anc 642 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  / L N )  e.  ZZ )
42 lgscl 20549 . . . . 5  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  / L
m )  e.  ZZ )
4323, 24, 42syl2anc 642 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  / L m )  e.  ZZ )
44 zcn 10029 . . . . 5  |-  ( ( m  / L N
)  e.  ZZ  ->  ( m  / L N
)  e.  CC )
45 zcn 10029 . . . . 5  |-  ( ( N  / L m )  e.  ZZ  ->  ( N  / L m )  e.  CC )
46 mulcom 8823 . . . . 5  |-  ( ( ( m  / L N )  e.  CC  /\  ( N  / L
m )  e.  CC )  ->  ( ( m  / L N )  x.  ( N  / L m ) )  =  ( ( N  / L m )  x.  ( m  / L N ) ) )
4744, 45, 46syl2an 463 . . . 4  |-  ( ( ( m  / L N )  e.  ZZ  /\  ( N  / L
m )  e.  ZZ )  ->  ( ( m  / L N )  x.  ( N  / L m ) )  =  ( ( N  / L m )  x.  ( m  / L N ) ) )
4841, 43, 47syl2anc 642 . . 3  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  / L N )  x.  ( N  / L m ) )  =  ( ( N  / L m )  x.  ( m  / L N ) ) )
4912nncnd 9762 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  CC )
50 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
51 subcl 9051 . . . . . . 7  |-  ( ( m  e.  CC  /\  1  e.  CC )  ->  ( m  -  1 )  e.  CC )
5249, 50, 51sylancl 643 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  - 
1 )  e.  CC )
5352halfcld 9956 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  -  1 )  / 
2 )  e.  CC )
546nncnd 9762 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  CC )
55 subcl 9051 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
5654, 50, 55sylancl 643 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  - 
1 )  e.  CC )
5756halfcld 9956 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  -  1 )  / 
2 )  e.  CC )
5853, 57mulcomd 8856 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( ( m  -  1 )  /  2 )  x.  ( ( N  - 
1 )  /  2
) )  =  ( ( ( N  - 
1 )  /  2
)  x.  ( ( m  -  1 )  /  2 ) ) )
5958oveq2d 5874 . . 3  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( -u 1 ^ ( ( ( m  -  1 )  /  2 )  x.  ( ( N  - 
1 )  /  2
) ) )  =  ( -u 1 ^ ( ( ( N  -  1 )  / 
2 )  x.  (
( m  -  1 )  /  2 ) ) ) )
6039, 48, 593eqtr4d 2325 . 2  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  / L N )  x.  ( N  / L m ) )  =  ( -u 1 ^ ( ( ( m  -  1 )  /  2 )  x.  ( ( N  - 
1 )  /  2
) ) ) )
61 biid 227 . 2  |-  ( A. x  e.  ( 1 ... y ) ( ( x  gcd  (
2  x.  N ) )  =  1  -> 
( ( x  / L N )  x.  ( N  / L x ) )  =  ( -u
1 ^ ( ( ( x  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) ) )  <->  A. x  e.  ( 1 ... y ) ( ( x  gcd  ( 2  x.  N
) )  =  1  ->  ( ( x  / L N )  x.  ( N  / L x ) )  =  ( -u 1 ^ ( ( ( x  -  1 )  /  2 )  x.  ( ( N  - 
1 )  /  2
) ) ) ) )
621, 2, 3, 4, 5, 60, 61lgsquad2lem2 20598 1  |-  ( ph  ->  ( ( M  / L N )  x.  ( N  / L M ) )  =  ( -u
1 ^ ( ( ( M  -  1 )  /  2 )  x.  ( ( N  -  1 )  / 
2 ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104    || cdivides 12531    gcd cgcd 12685   Primecprime 12758    / Lclgs 20533
This theorem is referenced by:  lgsquad3  20600
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-inf2 7342  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  ax-addf 8816  ax-mulf 8817
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-disj 3994  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  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-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  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-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-phi 12834  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-gsum 13405  df-imas 13411  df-divs 13412  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-nsg 14619  df-eqg 14620  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-rnghom 15496  df-drng 15514  df-field 15515  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-sra 15925  df-rgmod 15926  df-lidl 15927  df-rsp 15928  df-2idl 15984  df-nzr 16010  df-rlreg 16024  df-domn 16025  df-idom 16026  df-cnfld 16378  df-zrh 16455  df-zn 16458  df-lgs 20534
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