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Theorem 4sqlem13 13004
Description: Lemma for 4sq 13011. (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 2283 . . 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 2283 . . 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 13003 . 2  |-  ( ph  ->  E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ [ _i ]  ( ( ( abs `  u
) ^ 2 )  +  1 )  =  ( k  x.  P
) )
8 simplrl 736 . . . . . . . 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 10819 . . . . . . . 8  |-  ( k  e.  ( 1 ... ( P  -  1 ) )  ->  k  e.  NN )
108, 9syl 15 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  NN )
11 simpr 447 . . . . . . . 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 11782 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1312oveq1i 5868 . . . . . . . . . . 11  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
14 sq1 11198 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
1513, 14eqtri 2303 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  1
1615oveq2i 5869 . . . . . . . . 9  |-  ( ( ( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( ( ( abs `  u ) ^ 2 )  +  1 )
17 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  u  e.  ZZ [ _i ] )
18 1z 10053 . . . . . . . . . . 11  |-  1  e.  ZZ
19 zgz 12980 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  1  e.  ZZ [ _i ]
2114sqlem4a 12998 . . . . . . . . . 10  |-  ( ( u  e.  ZZ [
_i ]  /\  1  e.  ZZ [ _i ]
)  ->  ( (
( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
2217, 20, 21sylancl 643 . . . . . . . . 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 2368 . . . . . . . 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 2358 . . . . . . 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 5865 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  x.  P )  =  ( k  x.  P ) )
2625eleq1d 2349 . . . . . . . 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 2925 . . . . . . 7  |-  ( k  e.  T  <->  ( k  e.  NN  /\  ( k  x.  P )  e.  S ) )
2910, 24, 28sylanbrc 645 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  T )
30 ne0i 3461 . . . . . 6  |-  ( k  e.  T  ->  T  =/=  (/) )
3129, 30syl 15 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  T  =/=  (/) )
32 ssrab2 3258 . . . . . . . . 9  |-  { i  e.  NN  |  ( i  x.  P )  e.  S }  C_  NN
3327, 32eqsstri 3208 . . . . . . . 8  |-  T  C_  NN
34 4sq.7 . . . . . . . . 9  |-  M  =  sup ( T ,  RR ,  `'  <  )
35 nnuz 10263 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3633, 35sseqtri 3210 . . . . . . . . . 10  |-  T  C_  ( ZZ>= `  1 )
37 infmssuzcl 10301 . . . . . . . . . 10  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
3836, 31, 37sylancr 644 . . . . . . . . 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 2367 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  T )
4033, 39sseldi 3178 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  NN )
4140nnred 9761 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  RR )
4210nnred 9761 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  RR )
434ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  Prime )
44 prmnn 12761 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  NN )
4543, 44syl 15 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  NN )
4645nnred 9761 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  RR )
47 infmssuzle 10300 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  k  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  k
)
4836, 29, 47sylancr 644 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  sup ( T ,  RR ,  `'  <  )  <_ 
k )
4934, 48syl5eqbr 4056 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <_  k )
50 prmz 12762 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  ZZ )
5143, 50syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  ZZ )
52 elfzm11 10853 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  P  e.  ZZ )  ->  ( k  e.  ( 1 ... ( P  -  1 ) )  <-> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) ) )
5318, 51, 52sylancr 644 . . . . . . . 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 201 . . . . . . 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 969 . . . . . 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 8974 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <  P )
5731, 56jca 518 . . . 4  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( T  =/=  (/)  /\  M  <  P ) )
5857ex 423 . . 3  |-  ( (
ph  /\  ( k  e.  ( 1 ... ( P  -  1 ) )  /\  u  e.  ZZ [ _i ]
) )  ->  (
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P )  ->  ( T  =/=  (/)  /\  M  < 
P ) ) )
5958rexlimdvva 2674 . 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 14 1  |-  ( ph  ->  ( T  =/=  (/)  /\  M  <  P ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   2c2 9795   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    mod cmo 10973   ^cexp 11104   abscabs 11719   Primecprime 12758   ZZ [ _i ]cgz 12976
This theorem is referenced by:  4sqlem14  13005  4sqlem17  13008  4sqlem18  13009
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-rep 4131  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-card 7572  df-cda 7794  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-4 9806  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-hash 11338  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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