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Theorem lgsquad2 22815
Description: Extend lgsquad 22812 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 465 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  NN )
74adantr 465 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  N )
8 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
9 eldifi 3576 . . . . . 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 13868 . . . . 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 4099 . . . . . . . 8  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  =/=  2 )
148, 13syl 16 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  =/=  2
)
1514necomd 2719 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  2  =/=  m
)
1615neneqd 2651 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  =  m )
17 2z 10779 . . . . . . 7  |-  2  e.  ZZ
18 uzid 10976 . . . . . . 7  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1917, 18ax-mp 5 . . . . . 6  |-  2  e.  ( ZZ>= `  2 )
20 dvdsprm 13887 . . . . . 6  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  m  e.  Prime )  ->  (
2  ||  m  <->  2  =  m ) )
2119, 10, 20sylancr 663 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( 2  ||  m 
<->  2  =  m ) )
2216, 21mtbird 301 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  m )
236nnzd 10847 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  ZZ )
2412nnzd 10847 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ZZ )
25 gcdcom 13806 . . . . . 6  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  gcd  m
)  =  ( m  gcd  N ) )
2623, 24, 25syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  ( m  gcd  N ) )
27 simprr 756 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  gcd  N )  =  1 )
2826, 27eqtrd 2492 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  1 )
29 simprl 755 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  e.  ( Prime  \  { 2 } ) )
308adantr 465 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
31 eldifi 3576 . . . . . . . 8  |-  ( n  e.  ( Prime  \  {
2 } )  ->  n  e.  Prime )
32 prmrp 13889 . . . . . . . 8  |-  ( ( n  e.  Prime  /\  m  e.  Prime )  ->  (
( n  gcd  m
)  =  1  <->  n  =/=  m ) )
3331, 10, 32syl2anr 478 . . . . . . 7  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  <-> 
n  =/=  m ) )
3433biimpd 207 . . . . . 6  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  n  e.  ( Prime  \  { 2 } ) )  -> 
( ( n  gcd  m )  =  1  ->  n  =/=  m
) )
3534impr 619 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  =/=  m
)
36 lgsquad 22812 . . . . 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 1219 . . . 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 236 . . . 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 22814 . . 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 22765 . . . . 5  |-  ( ( m  e.  ZZ  /\  N  e.  ZZ )  ->  ( m  /L
N )  e.  ZZ )
4124, 23, 40syl2anc 661 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  /L N )  e.  ZZ )
42 lgscl 22765 . . . . 5  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  /L
m )  e.  ZZ )
4323, 24, 42syl2anc 661 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  /L m )  e.  ZZ )
44 zcn 10752 . . . . 5  |-  ( ( m  /L N )  e.  ZZ  ->  ( m  /L N )  e.  CC )
45 zcn 10752 . . . . 5  |-  ( ( N  /L m )  e.  ZZ  ->  ( N  /L m )  e.  CC )
46 mulcom 9469 . . . . 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 477 . . . 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 661 . . 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 10439 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  CC )
50 ax-1cn 9441 . . . . . . 7  |-  1  e.  CC
51 subcl 9710 . . . . . . 7  |-  ( ( m  e.  CC  /\  1  e.  CC )  ->  ( m  -  1 )  e.  CC )
5249, 50, 51sylancl 662 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  - 
1 )  e.  CC )
5352halfcld 10670 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  -  1 )  / 
2 )  e.  CC )
546nncnd 10439 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  CC )
55 subcl 9710 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
5654, 50, 55sylancl 662 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  - 
1 )  e.  CC )
5756halfcld 10670 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  -  1 )  / 
2 )  e.  CC )
5853, 57mulcomd 9508 . . . 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 6206 . . 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 2502 . 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 236 . 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 22814 1  |-  ( ph  ->  ( ( M  /L N )  x.  ( N  /L
M ) )  =  ( -u 1 ^ ( ( ( M  -  1 )  / 
2 )  x.  (
( N  -  1 )  /  2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795    \ cdif 3423   {csn 3975   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   CCcc 9381   1c1 9384    x. cmul 9388    - cmin 9696   -ucneg 9697    / cdiv 10094   NNcn 10423   2c2 10472   ZZcz 10747   ZZ>=cuz 10962   ...cfz 11538   ^cexp 11966    || cdivides 13637    gcd cgcd 13792   Primecprime 13865    /Lclgs 22749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-disj 4361  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-ec 7203  df-qs 7207  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-fz 11539  df-fzo 11650  df-fl 11743  df-mod 11810  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-dvds 13638  df-gcd 13793  df-prm 13866  df-phi 13943  df-pc 14006  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-0g 14482  df-gsum 14483  df-imas 14548  df-divs 14549  df-mnd 15517  df-mhm 15566  df-submnd 15567  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-nsg 15781  df-eqg 15782  df-ghm 15847  df-cntz 15937  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-dvr 16881  df-rnghom 16912  df-drng 16940  df-field 16941  df-subrg 16969  df-lmod 17056  df-lss 17120  df-lsp 17159  df-sra 17359  df-rgmod 17360  df-lidl 17361  df-rsp 17362  df-2idl 17420  df-nzr 17446  df-rlreg 17460  df-domn 17461  df-idom 17462  df-cnfld 17928  df-zring 17993  df-zrh 18044  df-zn 18047  df-lgs 22750
This theorem is referenced by:  lgsquad3  22816
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