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Theorem 2sqlem9 20612
Description: Lemma for 2sq 20615. (Contributed by Mario Carneiro, 19-Jun-2015.)
Hypotheses
Ref Expression
2sq.1  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
2sqlem7.2  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
2sqlem9.5  |-  ( ph  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S ) )
2sqlem9.7  |-  ( ph  ->  M  ||  N )
2sqlem9.6  |-  ( ph  ->  M  e.  NN )
2sqlem9.4  |-  ( ph  ->  N  e.  Y )
Assertion
Ref Expression
2sqlem9  |-  ( ph  ->  M  e.  S )
Distinct variable groups:    a, b, w, x, y, z    ph, x, y    M, a, b, x, y, z    S, a, b, x, y, z   
x, N, y, z    Y, a, b, x, y
Allowed substitution hints:    ph( z, w, a, b)    S( w)    M( w)    N( w, a, b)    Y( z, w)

Proof of Theorem 2sqlem9
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2sqlem9.4 . . 3  |-  ( ph  ->  N  e.  Y )
2 eqeq1 2289 . . . . . . . 8  |-  ( z  =  N  ->  (
z  =  ( ( x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) )
32anbi2d 684 . . . . . . 7  |-  ( z  =  N  ->  (
( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) ) )
432rexbidv 2586 . . . . . 6  |-  ( z  =  N  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  <->  E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) ) ) )
5 oveq1 5865 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  gcd  y )  =  ( u  gcd  y ) )
65eqeq1d 2291 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  gcd  y
)  =  1  <->  (
u  gcd  y )  =  1 ) )
7 oveq1 5865 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x ^ 2 )  =  ( u ^
2 ) )
87oveq1d 5873 . . . . . . . . 9  |-  ( x  =  u  ->  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )
98eqeq2d 2294 . . . . . . . 8  |-  ( x  =  u  ->  ( N  =  ( (
x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) )
106, 9anbi12d 691 . . . . . . 7  |-  ( x  =  u  ->  (
( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) ) )
11 oveq2 5866 . . . . . . . . 9  |-  ( y  =  v  ->  (
u  gcd  y )  =  ( u  gcd  v ) )
1211eqeq1d 2291 . . . . . . . 8  |-  ( y  =  v  ->  (
( u  gcd  y
)  =  1  <->  (
u  gcd  v )  =  1 ) )
13 oveq1 5865 . . . . . . . . . 10  |-  ( y  =  v  ->  (
y ^ 2 )  =  ( v ^
2 ) )
1413oveq2d 5874 . . . . . . . . 9  |-  ( y  =  v  ->  (
( u ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) )
1514eqeq2d 2294 . . . . . . . 8  |-  ( y  =  v  ->  ( N  =  ( (
u ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
1612, 15anbi12d 691 . . . . . . 7  |-  ( y  =  v  ->  (
( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) ) )
1710, 16cbvrex2v 2773 . . . . . 6  |-  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) )
184, 17syl6bb 252 . . . . 5  |-  ( z  =  N  ->  ( E. x  e.  ZZ  E. y  e.  ZZ  (
( x  gcd  y
)  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^
2 ) ) )  <->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) ) )
19 2sqlem7.2 . . . . 5  |-  Y  =  { z  |  E. x  e.  ZZ  E. y  e.  ZZ  ( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) }
2018, 19elab2g 2916 . . . 4  |-  ( N  e.  Y  ->  ( N  e.  Y  <->  E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) ) )
2120ibi 232 . . 3  |-  ( N  e.  Y  ->  E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
221, 21syl 15 . 2  |-  ( ph  ->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) )
23 simpr 447 . . . . . 6  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =  1 )  ->  M  = 
1 )
24 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
25 zgz 12980 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2624, 25ax-mp 8 . . . . . . . 8  |-  1  e.  ZZ [ _i ]
27 sq1 11198 . . . . . . . . 9  |-  ( 1 ^ 2 )  =  1
2827eqcomi 2287 . . . . . . . 8  |-  1  =  ( 1 ^ 2 )
29 fveq2 5525 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( abs `  x )  =  ( abs `  1
) )
30 abs1 11782 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
3129, 30syl6eq 2331 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( abs `  x )  =  1 )
3231oveq1d 5873 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( abs `  x
) ^ 2 )  =  ( 1 ^ 2 ) )
3332eqeq2d 2294 . . . . . . . . 9  |-  ( x  =  1  ->  (
1  =  ( ( abs `  x ) ^ 2 )  <->  1  =  ( 1 ^ 2 ) ) )
3433rspcev 2884 . . . . . . . 8  |-  ( ( 1  e.  ZZ [
_i ]  /\  1  =  ( 1 ^ 2 ) )  ->  E. x  e.  ZZ [ _i ]  1  =  ( ( abs `  x
) ^ 2 ) )
3526, 28, 34mp2an 653 . . . . . . 7  |-  E. x  e.  ZZ [ _i ] 
1  =  ( ( abs `  x ) ^ 2 )
36 2sq.1 . . . . . . . 8  |-  S  =  ran  ( w  e.  ZZ [ _i ]  |->  ( ( abs `  w
) ^ 2 ) )
37362sqlem1 20602 . . . . . . 7  |-  ( 1  e.  S  <->  E. x  e.  ZZ [ _i ] 
1  =  ( ( abs `  x ) ^ 2 ) )
3835, 37mpbir 200 . . . . . 6  |-  1  e.  S
3923, 38syl6eqel 2371 . . . . 5  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =  1 )  ->  M  e.  S )
40 2sqlem9.5 . . . . . . . 8  |-  ( ph  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S ) )
4140ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  A. b  e.  ( 1 ... ( M  -  1 ) ) A. a  e.  Y  ( b  ||  a  ->  b  e.  S
) )
42 2sqlem9.7 . . . . . . . 8  |-  ( ph  ->  M  ||  N )
4342ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  ||  N )
4436, 192sqlem7 20609 . . . . . . . . . 10  |-  Y  C_  ( S  i^i  NN )
45 inss2 3390 . . . . . . . . . 10  |-  ( S  i^i  NN )  C_  NN
4644, 45sstri 3188 . . . . . . . . 9  |-  Y  C_  NN
4746, 1sseldi 3178 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
4847ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  N  e.  NN )
49 2sqlem9.6 . . . . . . . . 9  |-  ( ph  ->  M  e.  NN )
5049ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  NN )
51 simprr 733 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  =/=  1 )
52 eluz2b3 10291 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  2
)  <->  ( M  e.  NN  /\  M  =/=  1 ) )
5350, 51, 52sylanbrc 645 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  ( ZZ>= `  2 )
)
54 simplrl 736 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  u  e.  ZZ )
55 simplrr 737 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  v  e.  ZZ )
56 simprll 738 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  (
u  gcd  v )  =  1 )
57 simprlr 739 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) )
58 eqid 2283 . . . . . . 7  |-  ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( u  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
59 eqid 2283 . . . . . . 7  |-  ( ( ( v  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
60 eqid 2283 . . . . . . 7  |-  ( ( ( ( u  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )  =  ( ( ( ( u  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )
61 eqid 2283 . . . . . . 7  |-  ( ( ( ( v  +  ( M  /  2
) )  mod  M
)  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )  =  ( ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )  /  ( ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  gcd  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) ) ) )
6236, 19, 41, 43, 48, 53, 54, 55, 56, 57, 58, 59, 60, 612sqlem8 20611 . . . . . 6  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  S )
6362anassrs 629 . . . . 5  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =/=  1
)  ->  M  e.  S )
6439, 63pm2.61dane 2524 . . . 4  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  ->  M  e.  S
)
6564ex 423 . . 3  |-  ( (
ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  -> 
( ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
6665rexlimdvva 2674 . 2  |-  ( ph  ->  ( E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
6722, 66mpd 14 1  |-  ( ph  ->  M  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    mod cmo 10973   ^cexp 11104   abscabs 11719    || cdivides 12531    gcd cgcd 12685   ZZ [ _i ]cgz 12976
This theorem is referenced by:  2sqlem10  20613
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  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-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-gz 12977
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