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Theorem 4sqlem13 14030
Description: Lemma for 4sq 14037. (Contributed by Mario Carneiro, 16-Jul-2014.)
Hypotheses
Ref Expression
4sq.1  |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
2 )  +  ( y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
2 ) ) ) }
4sq.2  |-  ( ph  ->  N  e.  NN )
4sq.3  |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 ) )
4sq.4  |-  ( ph  ->  P  e.  Prime )
4sq.5  |-  ( ph  ->  ( 0 ... (
2  x.  N ) )  C_  S )
4sq.6  |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }
4sq.7  |-  M  =  sup ( T ,  RR ,  `'  <  )
Assertion
Ref Expression
4sqlem13  |-  ( ph  ->  ( T  =/=  (/)  /\  M  <  P ) )
Distinct variable groups:    w, n, x, y, z    i, n, M    n, N    P, i, n    ph, n    S, i, n
Allowed substitution hints:    ph( x, y, z, w, i)    P( x, y, z, w)    S( x, y, z, w)    T( x, y, z, w, i, n)    M( x, y, z, w)    N( x, y, z, w, i)

Proof of Theorem 4sqlem13
Dummy variables  k 
v  u  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 4sq.1 . . 3  |-  S  =  { n  |  E. x  e.  ZZ  E. y  e.  ZZ  E. z  e.  ZZ  E. w  e.  ZZ  n  =  ( ( ( x ^
2 )  +  ( y ^ 2 ) )  +  ( ( z ^ 2 )  +  ( w ^
2 ) ) ) }
2 4sq.2 . . 3  |-  ( ph  ->  N  e.  NN )
3 4sq.3 . . 3  |-  ( ph  ->  P  =  ( ( 2  x.  N )  +  1 ) )
4 4sq.4 . . 3  |-  ( ph  ->  P  e.  Prime )
5 eqid 2443 . . 3  |-  { u  |  E. m  e.  ( 0 ... N ) u  =  ( ( m ^ 2 )  mod  P ) }  =  { u  |  E. m  e.  ( 0 ... N ) u  =  ( ( m ^ 2 )  mod  P ) }
6 eqid 2443 . . 3  |-  ( v  e.  { u  |  E. m  e.  ( 0 ... N ) u  =  ( ( m ^ 2 )  mod  P ) } 
|->  ( ( P  - 
1 )  -  v
) )  =  ( v  e.  { u  |  E. m  e.  ( 0 ... N ) u  =  ( ( m ^ 2 )  mod  P ) } 
|->  ( ( P  - 
1 )  -  v
) )
71, 2, 3, 4, 5, 64sqlem12 14029 . 2  |-  ( ph  ->  E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ[_i]  ( ( ( abs `  u
) ^ 2 )  +  1 )  =  ( k  x.  P
) )
8 simplrl 759 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  ( 1 ... ( P  - 
1 ) ) )
9 elfznn 11490 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  k  e.  NN )
108, 9syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  NN )
11 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )
12 abs1 12798 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1312oveq1i 6113 . . . . . . . . . . 11  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
14 sq1 11972 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
1513, 14eqtri 2463 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  1
1615oveq2i 6114 . . . . . . . . 9  |-  ( ( ( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( ( ( abs `  u ) ^ 2 )  +  1 )
17 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  u  e.  ZZ[_i] )
18 1z 10688 . . . . . . . . . . 11  |-  1  e.  ZZ
19 zgz 14006 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2018, 19ax-mp 5 . . . . . . . . . 10  |-  1  e.  ZZ[_i]
2114sqlem4a 14024 . . . . . . . . . 10  |-  ( ( u  e.  ZZ[_i]  /\  1  e.  ZZ[_i]
)  ->  ( (
( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
2217, 20, 21sylancl 662 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( ( ( abs `  u ) ^ 2 )  +  ( ( abs `  1 ) ^ 2 ) )  e.  S )
2316, 22syl5eqelr 2528 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( ( ( abs `  u ) ^ 2 )  +  1 )  e.  S )
2411, 23eqeltrrd 2518 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( k  x.  P
)  e.  S )
25 oveq1 6110 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  x.  P )  =  ( k  x.  P ) )
2625eleq1d 2509 . . . . . . . 8  |-  ( i  =  k  ->  (
( i  x.  P
)  e.  S  <->  ( k  x.  P )  e.  S
) )
27 4sq.6 . . . . . . . 8  |-  T  =  { i  e.  NN  |  ( i  x.  P )  e.  S }
2826, 27elrab2 3131 . . . . . . 7  |-  ( k  e.  T  <->  ( k  e.  NN  /\  ( k  x.  P )  e.  S ) )
2910, 24, 28sylanbrc 664 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  T )
30 ne0i 3655 . . . . . 6  |-  ( k  e.  T  ->  T  =/=  (/) )
3129, 30syl 16 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  T  =/=  (/) )
32 ssrab2 3449 . . . . . . . . 9  |-  { i  e.  NN  |  ( i  x.  P )  e.  S }  C_  NN
3327, 32eqsstri 3398 . . . . . . . 8  |-  T  C_  NN
34 4sq.7 . . . . . . . . 9  |-  M  =  sup ( T ,  RR ,  `'  <  )
35 nnuz 10908 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3633, 35sseqtri 3400 . . . . . . . . . 10  |-  T  C_  ( ZZ>= `  1 )
37 infmssuzcl 10950 . . . . . . . . . 10  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
3836, 31, 37sylancr 663 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T )
3934, 38syl5eqel 2527 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  T )
4033, 39sseldi 3366 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  NN )
4140nnred 10349 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  RR )
4210nnred 10349 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  RR )
434ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  Prime )
44 prmnn 13778 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
4543, 44syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  NN )
4645nnred 10349 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  RR )
47 infmssuzle 10949 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  k  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  k
)
4836, 29, 47sylancr 663 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  sup ( T ,  RR ,  `'  <  )  <_ 
k )
4934, 48syl5eqbr 4337 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <_  k )
50 prmz 13779 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
5143, 50syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  ZZ )
52 elfzm11 11540 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  P  e.  ZZ )  ->  ( k  e.  ( 1 ... ( P  -  1 ) )  <-> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) ) )
5318, 51, 52sylancr 663 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( k  e.  ( 1 ... ( P  -  1 ) )  <-> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) ) )
548, 53mpbid 210 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) )
5554simp3d 1002 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  <  P )
5641, 42, 46, 49, 55lelttrd 9541 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <  P )
5731, 56jca 532 . . . 4  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( T  =/=  (/)  /\  M  <  P ) )
5857ex 434 . . 3  |-  ( (
ph  /\  ( k  e.  ( 1 ... ( P  -  1 ) )  /\  u  e.  ZZ[_i]
) )  ->  (
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P )  ->  ( T  =/=  (/)  /\  M  < 
P ) ) )
5958rexlimdvva 2860 . 2  |-  ( ph  ->  ( E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ[_i] 
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P )  ->  ( T  =/=  (/)  /\  M  < 
P ) ) )
607, 59mpd 15 1  |-  ( ph  ->  ( T  =/=  (/)  /\  M  <  P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429    =/= wne 2618   E.wrex 2728   {crab 2731    C_ wss 3340   (/)c0 3649   class class class wbr 4304    e. cmpt 4362   `'ccnv 4851   ` cfv 5430  (class class class)co 6103   supcsup 7702   RRcr 9293   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299    < clt 9430    <_ cle 9431    - cmin 9607   NNcn 10334   2c2 10383   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449    mod cmo 11720   ^cexp 11877   abscabs 12735   Primecprime 13775   ZZ[_i]cgz 14002
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-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fl 11654  df-mod 11721  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-gcd 13703  df-prm 13776  df-gz 14003
This theorem is referenced by:  4sqlem14  14031  4sqlem17  14034  4sqlem18  14035
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