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Theorem lgsquad2 23836
Description: Extend lgsquad 23833 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 463 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  NN )
74adantr 463 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  N )
8 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
9 eldifi 3612 . . . . . 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 14307 . . . . 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 4142 . . . . . . . 8  |-  ( m  e.  ( Prime  \  {
2 } )  ->  m  =/=  2 )
148, 13syl 16 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  =/=  2
)
1514necomd 2725 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  2  =/=  m
)
1615neneqd 2656 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  =  m )
17 2z 10892 . . . . . . 7  |-  2  e.  ZZ
18 uzid 11096 . . . . . . 7  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1917, 18ax-mp 5 . . . . . 6  |-  2  e.  ( ZZ>= `  2 )
20 dvdsprm 14327 . . . . . 6  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  m  e.  Prime )  ->  (
2  ||  m  <->  2  =  m ) )
2119, 10, 20sylancr 661 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( 2  ||  m 
<->  2  =  m ) )
2216, 21mtbird 299 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  -.  2  ||  m )
236nnzd 10964 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  ZZ )
2412nnzd 10964 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  ZZ )
25 gcdcom 14245 . . . . . 6  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  gcd  m
)  =  ( m  gcd  N ) )
2623, 24, 25syl2anc 659 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  ( m  gcd  N ) )
27 simprr 755 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  gcd  N )  =  1 )
2826, 27eqtrd 2495 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  gcd  m )  =  1 )
29 simprl 754 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  e.  ( Prime  \  { 2 } ) )
308adantr 463 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  m  e.  ( Prime  \  { 2 } ) )
31 eldifi 3612 . . . . . . . 8  |-  ( n  e.  ( Prime  \  {
2 } )  ->  n  e.  Prime )
32 prmrp 14329 . . . . . . . 8  |-  ( ( n  e.  Prime  /\  m  e.  Prime )  ->  (
( n  gcd  m
)  =  1  <->  n  =/=  m ) )
3331, 10, 32syl2anr 476 . . . . . . 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 617 . . . . 5  |-  ( ( ( ph  /\  (
m  e.  ( Prime  \  { 2 } )  /\  ( m  gcd  N )  =  1 ) )  /\  ( n  e.  ( Prime  \  {
2 } )  /\  ( n  gcd  m )  =  1 ) )  ->  n  =/=  m
)
36 lgsquad 23833 . . . . 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 1226 . . . 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 23835 . . 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 23786 . . . . 5  |-  ( ( m  e.  ZZ  /\  N  e.  ZZ )  ->  ( m  /L
N )  e.  ZZ )
4124, 23, 40syl2anc 659 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  /L N )  e.  ZZ )
42 lgscl 23786 . . . . 5  |-  ( ( N  e.  ZZ  /\  m  e.  ZZ )  ->  ( N  /L
m )  e.  ZZ )
4323, 24, 42syl2anc 659 . . . 4  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  /L m )  e.  ZZ )
44 zcn 10865 . . . . 5  |-  ( ( m  /L N )  e.  ZZ  ->  ( m  /L N )  e.  CC )
45 zcn 10865 . . . . 5  |-  ( ( N  /L m )  e.  ZZ  ->  ( N  /L m )  e.  CC )
46 mulcom 9567 . . . . 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 475 . . . 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 659 . . 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 10547 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  m  e.  CC )
50 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
51 subcl 9810 . . . . . . 7  |-  ( ( m  e.  CC  /\  1  e.  CC )  ->  ( m  -  1 )  e.  CC )
5249, 50, 51sylancl 660 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( m  - 
1 )  e.  CC )
5352halfcld 10779 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( m  -  1 )  / 
2 )  e.  CC )
546nncnd 10547 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  N  e.  CC )
55 subcl 9810 . . . . . . 7  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  -  1 )  e.  CC )
5654, 50, 55sylancl 660 . . . . . 6  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( N  - 
1 )  e.  CC )
5756halfcld 10779 . . . . 5  |-  ( (
ph  /\  ( m  e.  ( Prime  \  { 2 } )  /\  (
m  gcd  N )  =  1 ) )  ->  ( ( N  -  1 )  / 
2 )  e.  CC )
5853, 57mulcomd 9606 . . . 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 6286 . . 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 2505 . 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 23835 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458   {csn 4016   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   1c1 9482    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   2c2 10581   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675   ^cexp 12151    || cdvds 14073    gcd cgcd 14231   Primecprime 14304    /Lclgs 23770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-dvds 14074  df-gcd 14232  df-prm 14305  df-phi 14383  df-pc 14448  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-0g 14934  df-gsum 14935  df-imas 15000  df-qus 15001  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-submnd 16169  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mulg 16262  df-subg 16400  df-nsg 16401  df-eqg 16402  df-ghm 16467  df-cntz 16557  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-cring 17399  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-dvr 17530  df-rnghom 17562  df-drng 17596  df-field 17597  df-subrg 17625  df-lmod 17712  df-lss 17777  df-lsp 17816  df-sra 18016  df-rgmod 18017  df-lidl 18018  df-rsp 18019  df-2idl 18078  df-nzr 18104  df-rlreg 18129  df-domn 18130  df-idom 18131  df-cnfld 18619  df-zring 18687  df-zrh 18719  df-zn 18722  df-lgs 23771
This theorem is referenced by:  lgsquad3  23837
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