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Theorem lgsquad2 20615
Description: Extend lgsquad 20612 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 3311 . . . . . 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 12777 . . . . 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 3763 . . . . . . . 8  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  =/=  2 )
148, 13syl 15 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  =/=  2
)
1514necomd 2542 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  2  =/=  m
)
1615neneqd 2475 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  =  m )
17 2z 10070 . . . . . . 7  |-  2  e.  ZZ
18 uzid 10258 . . . . . . 7  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1917, 18ax-mp 8 . . . . . 6  |-  2  e.  ( ZZ>= `  2 )
20 dvdsprm 12794 . . . . . 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 10132 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  ZZ )
2412nnzd 10132 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ZZ )
25 gcdcom 12715 . . . . . 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 2328 . . . 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 3311 . . . . . . . 8  |-  ( n  e.  ( Prime  \  {
2 } )  ->  n  e.  Prime )
32 prmrp 12796 . . . . . . . 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 20612 . . . . 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 20614 . . 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 20565 . . . . 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 20565 . . . . 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 10045 . . . . 5  |-  ( ( m  / L N
)  e.  ZZ  ->  ( m  / L N
)  e.  CC )
45 zcn 10045 . . . . 5  |-  ( ( N  / L m )  e.  ZZ  ->  ( N  / L m )  e.  CC )
46 mulcom 8839 . . . . 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 9778 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  CC )
50 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
51 subcl 9067 . . . . . . 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 9972 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  -  1 )  / 
2 )  e.  CC )
546nncnd 9778 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  CC )
55 subcl 9067 . . . . . . 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 9972 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  -  1 )  / 
2 )  e.  CC )
5853, 57mulcomd 8872 . . . 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 5890 . . 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 2338 . 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 20614 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162   {csn 3653   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774    / Lclgs 20549
This theorem is referenced by:  lgsquad3  20616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-phi 12850  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-gsum 13421  df-imas 13427  df-divs 13428  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-nsg 14635  df-eqg 14636  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-rnghom 15512  df-drng 15530  df-field 15531  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-2idl 16000  df-nzr 16026  df-rlreg 16040  df-domn 16041  df-idom 16042  df-cnfld 16394  df-zrh 16471  df-zn 16474  df-lgs 20550
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