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Theorem 4sqlem13 13254
Description: Lemma for 4sq 13261. (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 2389 . . 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 2389 . . 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 13253 . 2  |-  ( ph  ->  E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ [ _i ]  ( ( ( abs `  u
) ^ 2 )  +  1 )  =  ( k  x.  P
) )
8 simplrl 737 . . . . . . . 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 11014 . . . . . . . 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 448 . . . . . . . 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 12031 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1312oveq1i 6032 . . . . . . . . . . 11  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
14 sq1 11405 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
1513, 14eqtri 2409 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  1
1615oveq2i 6033 . . . . . . . . 9  |-  ( ( ( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( ( ( abs `  u ) ^ 2 )  +  1 )
17 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  u  e.  ZZ [ _i ] )
18 1z 10245 . . . . . . . . . . 11  |-  1  e.  ZZ
19 zgz 13230 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  1  e.  ZZ [ _i ]
2114sqlem4a 13248 . . . . . . . . . 10  |-  ( ( u  e.  ZZ [
_i ]  /\  1  e.  ZZ [ _i ]
)  ->  ( (
( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
2217, 20, 21sylancl 644 . . . . . . . . 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 2474 . . . . . . . 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 2464 . . . . . . 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 6029 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  x.  P )  =  ( k  x.  P ) )
2625eleq1d 2455 . . . . . . . 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 3039 . . . . . . 7  |-  ( k  e.  T  <->  ( k  e.  NN  /\  ( k  x.  P )  e.  S ) )
2910, 24, 28sylanbrc 646 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  T )
30 ne0i 3579 . . . . . 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 3373 . . . . . . . . 9  |-  { i  e.  NN  |  ( i  x.  P )  e.  S }  C_  NN
3327, 32eqsstri 3323 . . . . . . . 8  |-  T  C_  NN
34 4sq.7 . . . . . . . . 9  |-  M  =  sup ( T ,  RR ,  `'  <  )
35 nnuz 10455 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3633, 35sseqtri 3325 . . . . . . . . . 10  |-  T  C_  ( ZZ>= `  1 )
37 infmssuzcl 10493 . . . . . . . . . 10  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
3836, 31, 37sylancr 645 . . . . . . . . 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 2473 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  T )
4033, 39sseldi 3291 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  NN )
4140nnred 9949 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  RR )
4210nnred 9949 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  RR )
434ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  Prime )
44 prmnn 13011 . . . . . . . 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 9949 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  RR )
47 infmssuzle 10492 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  k  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  k
)
4836, 29, 47sylancr 645 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  sup ( T ,  RR ,  `'  <  )  <_ 
k )
4934, 48syl5eqbr 4188 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <_  k )
50 prmz 13012 . . . . . . . . . 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 11048 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  P  e.  ZZ )  ->  ( k  e.  ( 1 ... ( P  -  1 ) )  <-> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) ) )
5318, 51, 52sylancr 645 . . . . . . . 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 202 . . . . . . 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 971 . . . . . 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 9162 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <  P )
5731, 56jca 519 . . . 4  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( T  =/=  (/)  /\  M  <  P ) )
5857ex 424 . . 3  |-  ( (
ph  /\  ( k  e.  ( 1 ... ( P  -  1 ) )  /\  u  e.  ZZ [ _i ]
) )  ->  (
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P )  ->  ( T  =/=  (/)  /\  M  < 
P ) ) )
5958rexlimdvva 2782 . 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2375    =/= wne 2552   E.wrex 2652   {crab 2655    C_ wss 3265   (/)c0 3573   class class class wbr 4155    e. cmpt 4209   `'ccnv 4819   ` cfv 5396  (class class class)co 6022   supcsup 7382   RRcr 8924   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    < clt 9055    <_ cle 9056    - cmin 9225   NNcn 9934   2c2 9983   ZZcz 10216   ZZ>=cuz 10422   ...cfz 10977    mod cmo 11179   ^cexp 11311   abscabs 11968   Primecprime 13008   ZZ [ _i ]cgz 13226
This theorem is referenced by:  4sqlem14  13255  4sqlem17  13258  4sqlem18  13259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-n0 10156  df-z 10217  df-uz 10423  df-rp 10547  df-fz 10978  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-dvds 12782  df-gcd 12936  df-prm 13009  df-gz 13227
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