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Theorem 2sqlem9 22717
Description: Lemma for 2sq 22720. (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 2449 . . . . . . . 8  |-  ( z  =  N  ->  (
z  =  ( ( x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) )
32anbi2d 703 . . . . . . 7  |-  ( z  =  N  ->  (
( ( x  gcd  y )  =  1  /\  z  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^
2 )  +  ( y ^ 2 ) ) ) ) )
432rexbidv 2763 . . . . . 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 6103 . . . . . . . . 9  |-  ( x  =  u  ->  (
x  gcd  y )  =  ( u  gcd  y ) )
65eqeq1d 2451 . . . . . . . 8  |-  ( x  =  u  ->  (
( x  gcd  y
)  =  1  <->  (
u  gcd  y )  =  1 ) )
7 oveq1 6103 . . . . . . . . . 10  |-  ( x  =  u  ->  (
x ^ 2 )  =  ( u ^
2 ) )
87oveq1d 6111 . . . . . . . . 9  |-  ( x  =  u  ->  (
( x ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )
98eqeq2d 2454 . . . . . . . 8  |-  ( x  =  u  ->  ( N  =  ( (
x ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) )
106, 9anbi12d 710 . . . . . . 7  |-  ( x  =  u  ->  (
( ( x  gcd  y )  =  1  /\  N  =  ( ( x ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^
2 )  +  ( y ^ 2 ) ) ) ) )
11 oveq2 6104 . . . . . . . . 9  |-  ( y  =  v  ->  (
u  gcd  y )  =  ( u  gcd  v ) )
1211eqeq1d 2451 . . . . . . . 8  |-  ( y  =  v  ->  (
( u  gcd  y
)  =  1  <->  (
u  gcd  v )  =  1 ) )
13 oveq1 6103 . . . . . . . . . 10  |-  ( y  =  v  ->  (
y ^ 2 )  =  ( v ^
2 ) )
1413oveq2d 6112 . . . . . . . . 9  |-  ( y  =  v  ->  (
( u ^ 2 )  +  ( y ^ 2 ) )  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) )
1514eqeq2d 2454 . . . . . . . 8  |-  ( y  =  v  ->  ( N  =  ( (
u ^ 2 )  +  ( y ^
2 ) )  <->  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
1612, 15anbi12d 710 . . . . . . 7  |-  ( y  =  v  ->  (
( ( u  gcd  y )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( y ^ 2 ) ) )  <->  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) ) )
1710, 16cbvrex2v 2961 . . . . . 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 261 . . . . 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 3113 . . . 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 241 . . 3  |-  ( N  e.  Y  ->  E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) ) )
221, 21syl 16 . 2  |-  ( ph  ->  E. u  e.  ZZ  E. v  e.  ZZ  (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) ) )
23 simpr 461 . . . . . 6  |-  ( ( ( ( ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  /\  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  /\  M  =  1 )  ->  M  = 
1 )
24 1z 10681 . . . . . . . . 9  |-  1  e.  ZZ
25 zgz 13999 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2624, 25ax-mp 5 . . . . . . . 8  |-  1  e.  ZZ[_i]
27 sq1 11965 . . . . . . . . 9  |-  ( 1 ^ 2 )  =  1
2827eqcomi 2447 . . . . . . . 8  |-  1  =  ( 1 ^ 2 )
29 fveq2 5696 . . . . . . . . . . . 12  |-  ( x  =  1  ->  ( abs `  x )  =  ( abs `  1
) )
30 abs1 12791 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
3129, 30syl6eq 2491 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( abs `  x )  =  1 )
3231oveq1d 6111 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( abs `  x
) ^ 2 )  =  ( 1 ^ 2 ) )
3332eqeq2d 2454 . . . . . . . . 9  |-  ( x  =  1  ->  (
1  =  ( ( abs `  x ) ^ 2 )  <->  1  =  ( 1 ^ 2 ) ) )
3433rspcev 3078 . . . . . . . 8  |-  ( ( 1  e.  ZZ[_i]  /\  1  =  ( 1 ^ 2 ) )  ->  E. x  e.  ZZ[_i]  1  =  ( ( abs `  x ) ^ 2 ) )
3526, 28, 34mp2an 672 . . . . . . 7  |-  E. x  e.  ZZ[_i] 
1  =  ( ( abs `  x ) ^ 2 )
36 2sq.1 . . . . . . . 8  |-  S  =  ran  ( w  e.  ZZ[_i]  |->  ( ( abs `  w
) ^ 2 ) )
37362sqlem1 22707 . . . . . . 7  |-  ( 1  e.  S  <->  E. x  e.  ZZ[_i] 
1  =  ( ( abs `  x ) ^ 2 ) )
3835, 37mpbir 209 . . . . . 6  |-  1  e.  S
3923, 38syl6eqel 2531 . . . . 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 725 . . . . . . 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 725 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  ||  N )
4436, 192sqlem7 22714 . . . . . . . . . 10  |-  Y  C_  ( S  i^i  NN )
45 inss2 3576 . . . . . . . . . 10  |-  ( S  i^i  NN )  C_  NN
4644, 45sstri 3370 . . . . . . . . 9  |-  Y  C_  NN
4746, 1sseldi 3359 . . . . . . . 8  |-  ( ph  ->  N  e.  NN )
4847ad2antrr 725 . . . . . . 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 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  NN )
51 simprr 756 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  =/=  1 )
52 eluz2b3 10933 . . . . . . . 8  |-  ( M  e.  ( ZZ>= `  2
)  <->  ( M  e.  NN  /\  M  =/=  1 ) )
5350, 51, 52sylanbrc 664 . . . . . . 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 759 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  u  e.  ZZ )
55 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  v  e.  ZZ )
56 simprll 761 . . . . . . 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 762 . . . . . . 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 2443 . . . . . . 7  |-  ( ( ( u  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( u  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
59 eqid 2443 . . . . . . 7  |-  ( ( ( v  +  ( M  /  2 ) )  mod  M )  -  ( M  / 
2 ) )  =  ( ( ( v  +  ( M  / 
2 ) )  mod 
M )  -  ( M  /  2 ) )
60 eqid 2443 . . . . . . 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 2443 . . . . . . 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 22716 . . . . . 6  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
( u  gcd  v
)  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^
2 ) ) )  /\  M  =/=  1
) )  ->  M  e.  S )
6362anassrs 648 . . . . 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 2694 . . . 4  |-  ( ( ( ph  /\  (
u  e.  ZZ  /\  v  e.  ZZ )
)  /\  ( (
u  gcd  v )  =  1  /\  N  =  ( ( u ^ 2 )  +  ( v ^ 2 ) ) ) )  ->  M  e.  S
)
6564ex 434 . . 3  |-  ( (
ph  /\  ( u  e.  ZZ  /\  v  e.  ZZ ) )  -> 
( ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
6665rexlimdvva 2853 . 2  |-  ( ph  ->  ( E. u  e.  ZZ  E. v  e.  ZZ  ( ( u  gcd  v )  =  1  /\  N  =  ( ( u ^
2 )  +  ( v ^ 2 ) ) )  ->  M  e.  S ) )
6722, 66mpd 15 1  |-  ( ph  ->  M  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2611   A.wral 2720   E.wrex 2721    i^i cin 3332   class class class wbr 4297    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096   1c1 9288    + caddc 9290    - cmin 9600    / cdiv 9998   NNcn 10327   2c2 10376   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442    mod cmo 11713   ^cexp 11870   abscabs 12728    || cdivides 13540    gcd cgcd 13695   ZZ[_i]cgz 13995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-dvds 13541  df-gcd 13696  df-prm 13769  df-gz 13996
This theorem is referenced by:  2sqlem10  22718
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