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Theorem 4sqlem13 13327
Description: Lemma for 4sq 13334. (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 2438 . . 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 2438 . . 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 13326 . 2  |-  ( ph  ->  E. k  e.  ( 1 ... ( P  -  1 ) ) E. u  e.  ZZ [ _i ]  ( ( ( abs `  u
) ^ 2 )  +  1 )  =  ( k  x.  P
) )
8 simplrl 738 . . . . . . . 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 11082 . . . . . . . 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 449 . . . . . . . 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 12104 . . . . . . . . . . . 12  |-  ( abs `  1 )  =  1
1312oveq1i 6093 . . . . . . . . . . 11  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
14 sq1 11478 . . . . . . . . . . 11  |-  ( 1 ^ 2 )  =  1
1513, 14eqtri 2458 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  1
1615oveq2i 6094 . . . . . . . . 9  |-  ( ( ( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( ( ( abs `  u ) ^ 2 )  +  1 )
17 simplrr 739 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  u  e.  ZZ [ _i ] )
18 1z 10313 . . . . . . . . . . 11  |-  1  e.  ZZ
19 zgz 13303 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  1  e.  ZZ [ _i ]
2114sqlem4a 13321 . . . . . . . . . 10  |-  ( ( u  e.  ZZ [
_i ]  /\  1  e.  ZZ [ _i ]
)  ->  ( (
( abs `  u
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
2217, 20, 21sylancl 645 . . . . . . . . 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 2523 . . . . . . . 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 2513 . . . . . . 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 6090 . . . . . . . . 9  |-  ( i  =  k  ->  (
i  x.  P )  =  ( k  x.  P ) )
2625eleq1d 2504 . . . . . . . 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 3096 . . . . . . 7  |-  ( k  e.  T  <->  ( k  e.  NN  /\  ( k  x.  P )  e.  S ) )
2910, 24, 28sylanbrc 647 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  T )
30 ne0i 3636 . . . . . 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 3430 . . . . . . . . 9  |-  { i  e.  NN  |  ( i  x.  P )  e.  S }  C_  NN
3327, 32eqsstri 3380 . . . . . . . 8  |-  T  C_  NN
34 4sq.7 . . . . . . . . 9  |-  M  =  sup ( T ,  RR ,  `'  <  )
35 nnuz 10523 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
3633, 35sseqtri 3382 . . . . . . . . . 10  |-  T  C_  ( ZZ>= `  1 )
37 infmssuzcl 10561 . . . . . . . . . 10  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  T  =/=  (/) )  ->  sup ( T ,  RR ,  `'  <  )  e.  T
)
3836, 31, 37sylancr 646 . . . . . . . . 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 2522 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  T )
4033, 39sseldi 3348 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  NN )
4140nnred 10017 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  e.  RR )
4210nnred 10017 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
k  e.  RR )
434ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  Prime )
44 prmnn 13084 . . . . . . . 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 10017 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  P  e.  RR )
47 infmssuzle 10560 . . . . . . . 8  |-  ( ( T  C_  ( ZZ>= ` 
1 )  /\  k  e.  T )  ->  sup ( T ,  RR ,  `'  <  )  <_  k
)
4836, 29, 47sylancr 646 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  sup ( T ,  RR ,  `'  <  )  <_ 
k )
4934, 48syl5eqbr 4247 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <_  k )
50 prmz 13085 . . . . . . . . . 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 11118 . . . . . . . . 9  |-  ( ( 1  e.  ZZ  /\  P  e.  ZZ )  ->  ( k  e.  ( 1 ... ( P  -  1 ) )  <-> 
( k  e.  ZZ  /\  1  <_  k  /\  k  <  P ) ) )
5318, 51, 52sylancr 646 . . . . . . . 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 203 . . . . . . 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 972 . . . . . 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 9230 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  ->  M  <  P )
5731, 56jca 520 . . . 4  |-  ( ( ( ph  /\  (
k  e.  ( 1 ... ( P  - 
1 ) )  /\  u  e.  ZZ [ _i ] ) )  /\  ( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P ) )  -> 
( T  =/=  (/)  /\  M  <  P ) )
5857ex 425 . . 3  |-  ( (
ph  /\  ( k  e.  ( 1 ... ( P  -  1 ) )  /\  u  e.  ZZ [ _i ]
) )  ->  (
( ( ( abs `  u ) ^ 2 )  +  1 )  =  ( k  x.  P )  ->  ( T  =/=  (/)  /\  M  < 
P ) ) )
5958rexlimdvva 2839 . 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {cab 2424    =/= wne 2601   E.wrex 2708   {crab 2711    C_ wss 3322   (/)c0 3630   class class class wbr 4214    e. cmpt 4268   `'ccnv 4879   ` cfv 5456  (class class class)co 6083   supcsup 7447   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    x. cmul 8997    < clt 9122    <_ cle 9123    - cmin 9293   NNcn 10002   2c2 10051   ZZcz 10284   ZZ>=cuz 10490   ...cfz 11045    mod cmo 11252   ^cexp 11384   abscabs 12041   Primecprime 13081   ZZ [ _i ]cgz 13299
This theorem is referenced by:  4sqlem14  13328  4sqlem17  13331  4sqlem18  13332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-prm 13082  df-gz 13300
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