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Theorem 4sqlem13 14337
Description: Lemma for 4sq 14344. (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 2467 . . 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 2467 . . 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 14336 . 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 11715 . . . . . . . 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 13096 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1312oveq1i 6295 . . . . . . . . . . 11  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
14 sq1 12231 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
1513, 14eqtri 2496 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  1
1615oveq2i 6296 . . . . . . . . 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 10895 . . . . . . . . . . 11  |-  1  e.  ZZ
19 zgz 14313 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2018, 19ax-mp 5 . . . . . . . . . 10  |-  1  e.  ZZ[_i]
2114sqlem4a 14331 . . . . . . . . . 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 2560 . . . . . . . 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 2556 . . . . . . 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 6292 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  x.  P )  =  ( k  x.  P ) )
2625eleq1d 2536 . . . . . . . 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 3263 . . . . . . 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 3791 . . . . . 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 3585 . . . . . . . . 9  |-  { i  e.  NN  |  ( i  x.  P )  e.  S }  C_  NN
3327, 32eqsstri 3534 . . . . . . . 8  |-  T  C_  NN
34 4sq.7 . . . . . . . . 9  |-  M  =  sup ( T ,  RR ,  `'  <  )
35 nnuz 11118 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3633, 35sseqtri 3536 . . . . . . . . . 10  |-  T  C_  ( ZZ>= `  1 )
37 infmssuzcl 11166 . . . . . . . . . 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 2559 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  T )
4033, 39sseldi 3502 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  NN )
4140nnred 10552 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  RR )
4210nnred 10552 . . . . . 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 14082 . . . . . . . 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 10552 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  RR )
47 infmssuzle 11165 . . . . . . . 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 4480 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ[_i] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <_  k )
50 prmz 14083 . . . . . . . . . 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 11750 . . . . . . . . 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 1010 . . . . . 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 9740 . . . . 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 2962 . 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 973    = wceq 1379    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   ` cfv 5588  (class class class)co 6285   supcsup 7901   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    < clt 9629    <_ cle 9630    - cmin 9806   NNcn 10537   2c2 10586   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673    mod cmo 11965   ^cexp 12135   abscabs 13033   Primecprime 14079   ZZ[_i]cgz 14309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-card 8321  df-cda 8549  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007  df-prm 14080  df-gz 14310
This theorem is referenced by:  4sqlem14  14338  4sqlem17  14341  4sqlem18  14342
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