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Theorem lgsquad2 21136
Description: Extend lgsquad 21133 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 452 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  NN )
74adantr 452 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  N )
8 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
9 eldifi 3461 . . . . . 6  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  e.  Prime )
108, 9syl 16 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  Prime )
11 prmnn 13074 . . . . 5  |-  ( m  e.  Prime  ->  m  e.  NN )
1210, 11syl 16 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  NN )
13 eldifsni 3920 . . . . . . . 8  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  =/=  2 )
148, 13syl 16 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  =/=  2
)
1514necomd 2681 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  2  =/=  m
)
1615neneqd 2614 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  =  m )
17 2z 10304 . . . . . . 7  |-  2  e.  ZZ
18 uzid 10492 . . . . . . 7  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1917, 18ax-mp 8 . . . . . 6  |-  2  e.  ( ZZ>= `  2 )
20 dvdsprm 13091 . . . . . 6  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  m  e.  Prime )  ->  (
2  ||  m  <->  2  =  m ) )
2119, 10, 20sylancr 645 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( 2  ||  m 
<->  2  =  m ) )
2216, 21mtbird 293 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  m )
236nnzd 10366 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  ZZ )
2412nnzd 10366 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ZZ )
25 gcdcom 13012 . . . . . 6  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  gcd  m
)  =  ( m  gcd  N ) )
2623, 24, 25syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  ( m  gcd  N ) )
27 simprr 734 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  gcd  N )  =  1 )
2826, 27eqtrd 2467 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  1 )
29 simprl 733 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  e.  ( Prime  \  { 2 } ) )
308adantr 452 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
31 eldifi 3461 . . . . . . . 8  |-  ( n  e.  ( Prime  \  {
2 } )  ->  n  e.  Prime )
32 prmrp 13093 . . . . . . . 8  |-  ( ( n  e.  Prime  /\  m  e.  Prime )  ->  (
( n  gcd  m
)  =  1  <->  n  =/=  m ) )
3331, 10, 32syl2anr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  <-> 
n  =/=  m ) )
3433biimpd 199 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  ->  n  =/=  m
) )
3534impr 603 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  =/=  m
)
36 lgsquad 21133 . . . . 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 1184 . . . 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 228 . . . 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 21135 . . 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 21086 . . . . 5  |-  ( ( m  e.  ZZ  /\  N  e.  ZZ )  ->  ( m  / L N )  e.  ZZ )
4124, 23, 40syl2anc 643 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  / L N )  e.  ZZ )
42 lgscl 21086 . . . . 5  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  / L
m )  e.  ZZ )
4323, 24, 42syl2anc 643 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  / L m )  e.  ZZ )
44 zcn 10279 . . . . 5  |-  ( ( m  / L N
)  e.  ZZ  ->  ( m  / L N
)  e.  CC )
45 zcn 10279 . . . . 5  |-  ( ( N  / L m )  e.  ZZ  ->  ( N  / L m )  e.  CC )
46 mulcom 9068 . . . . 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 464 . . . 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 643 . . 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 10008 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  CC )
50 ax-1cn 9040 . . . . . . 7  |-  1  e.  CC
51 subcl 9297 . . . . . . 7  |-  ( ( m  e.  CC  /\  1  e.  CC )  ->  ( m  -  1 )  e.  CC )
5249, 50, 51sylancl 644 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  - 
1 )  e.  CC )
5352halfcld 10204 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  -  1 )  / 
2 )  e.  CC )
546nncnd 10008 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  CC )
55 subcl 9297 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
5654, 50, 55sylancl 644 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  - 
1 )  e.  CC )
5756halfcld 10204 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  -  1 )  / 
2 )  e.  CC )
5853, 57mulcomd 9101 . . . 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 6089 . . 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 2477 . 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 228 . 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 21135 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    \ cdif 3309   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   NNcn 9992   2c2 10041   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035   ^cexp 11374    || cdivides 12844    gcd cgcd 12998   Primecprime 13071    / Lclgs 21070
This theorem is referenced by:  lgsquad3  21137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845  df-gcd 12999  df-prm 13072  df-phi 13147  df-pc 13203  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-gsum 13720  df-imas 13726  df-divs 13727  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-nsg 14934  df-eqg 14935  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-rnghom 15811  df-drng 15829  df-field 15830  df-subrg 15858  df-lmod 15944  df-lss 16001  df-lsp 16040  df-sra 16236  df-rgmod 16237  df-lidl 16238  df-rsp 16239  df-2idl 16295  df-nzr 16321  df-rlreg 16335  df-domn 16336  df-idom 16337  df-cnfld 16696  df-zrh 16774  df-zn 16777  df-lgs 21071
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