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Theorem 4sqlem19 14492
Description: Lemma for 4sq 14493. The proof is by strong induction - we show that if all the integers less than  k are in  S, then  k is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 14491. If  k is  0 ,  1 ,  2, we show  k  e.  S directly; otherwise if  k is composite,  k is the product of two numbers less than it (and hence in  S by assumption), so by mul4sq 14483  k  e.  S. (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
Hypothesis
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 ) ) ) }
Assertion
Ref Expression
4sqlem19  |-  NN0  =  S
Distinct variable groups:    w, n, x, y, z    S, n
Allowed substitution hints:    S( x, y, z, w)

Proof of Theorem 4sqlem19
Dummy variables  j 
k  i  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 10818 . . . 4  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2 eleq1 2529 . . . . . 6  |-  ( j  =  1  ->  (
j  e.  S  <->  1  e.  S ) )
3 eleq1 2529 . . . . . 6  |-  ( j  =  m  ->  (
j  e.  S  <->  m  e.  S ) )
4 eleq1 2529 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  S  <->  i  e.  S ) )
5 eleq1 2529 . . . . . 6  |-  ( j  =  ( m  x.  i )  ->  (
j  e.  S  <->  ( m  x.  i )  e.  S
) )
6 eleq1 2529 . . . . . 6  |-  ( j  =  k  ->  (
j  e.  S  <->  k  e.  S ) )
7 abs1 13141 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
87oveq1i 6306 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
9 sq1 12264 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
108, 9eqtri 2486 . . . . . . . . 9  |-  ( ( abs `  1 ) ^ 2 )  =  1
11 abs0 13129 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1211oveq1i 6306 . . . . . . . . . 10  |-  ( ( abs `  0 ) ^ 2 )  =  ( 0 ^ 2 )
13 sq0 12261 . . . . . . . . . 10  |-  ( 0 ^ 2 )  =  0
1412, 13eqtri 2486 . . . . . . . . 9  |-  ( ( abs `  0 ) ^ 2 )  =  0
1510, 14oveq12i 6308 . . . . . . . 8  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 1  +  0 )
16 1p0e1 10669 . . . . . . . 8  |-  ( 1  +  0 )  =  1
1715, 16eqtri 2486 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  1
18 1z 10915 . . . . . . . . 9  |-  1  e.  ZZ
19 zgz 14462 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ[_i]
)
2018, 19ax-mp 5 . . . . . . . 8  |-  1  e.  ZZ[_i]
21 0z 10896 . . . . . . . . 9  |-  0  e.  ZZ
22 zgz 14462 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  ZZ[_i]
)
2321, 22ax-mp 5 . . . . . . . 8  |-  0  e.  ZZ[_i]
24 4sq.1 . . . . . . . . 9  |-  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 ) ) ) }
25244sqlem4a 14480 . . . . . . . 8  |-  ( ( 1  e.  ZZ[_i]  /\  0  e.  ZZ[_i]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
2620, 23, 25mp2an 672 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
2717, 26eqeltrri 2542 . . . . . 6  |-  1  e.  S
28 eleq1 2529 . . . . . . 7  |-  ( j  =  2  ->  (
j  e.  S  <->  2  e.  S ) )
29 eldifsn 4157 . . . . . . . . 9  |-  ( j  e.  ( Prime  \  {
2 } )  <->  ( j  e.  Prime  /\  j  =/=  2 ) )
30 oddprm 14350 . . . . . . . . . . 11  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
( ( j  - 
1 )  /  2
)  e.  NN )
3130adantr 465 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  /  2 )  e.  NN )
32 eldifi 3622 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
j  e.  Prime )
3332adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  Prime )
34 prmnn 14231 . . . . . . . . . . . . . . 15  |-  ( j  e.  Prime  ->  j  e.  NN )
35 nncn 10564 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  j  e.  CC )
3633, 34, 353syl 20 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  CC )
37 ax-1cn 9567 . . . . . . . . . . . . . 14  |-  1  e.  CC
38 subcl 9838 . . . . . . . . . . . . . 14  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  -  1 )  e.  CC )
3936, 37, 38sylancl 662 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  CC )
40 2cnd 10629 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  e.  CC )
41 2ne0 10649 . . . . . . . . . . . . . 14  |-  2  =/=  0
4241a1i 11 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  =/=  0 )
4339, 40, 42divcan2d 10343 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 2  x.  ( ( j  -  1 )  / 
2 ) )  =  ( j  -  1 ) )
4443oveq1d 6311 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
2  x.  ( ( j  -  1 )  /  2 ) )  +  1 )  =  ( ( j  - 
1 )  +  1 ) )
45 npcan 9848 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  - 
1 )  +  1 )  =  j )
4636, 37, 45sylancl 662 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  +  1 )  =  j )
4744, 46eqtr2d 2499 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  =  ( ( 2  x.  ( ( j  - 
1 )  /  2
) )  +  1 ) )
4843oveq2d 6312 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  =  ( 0 ... (
j  -  1 ) ) )
49 nnm1nn0 10858 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  (
j  -  1 )  e.  NN0 )
5033, 34, 493syl 20 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e. 
NN0 )
51 elnn0uz 11143 . . . . . . . . . . . . . 14  |-  ( ( j  -  1 )  e.  NN0  <->  ( j  - 
1 )  e.  (
ZZ>= `  0 ) )
5250, 51sylib 196 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  ( ZZ>= `  0 )
)
53 eluzfz1 11718 . . . . . . . . . . . . 13  |-  ( ( j  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... (
j  -  1 ) ) )
54 fzsplit 11736 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 ... ( j  -  1 ) )  ->  (
0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5552, 53, 543syl 20 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5648, 55eqtrd 2498 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
57 fzsn 11750 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5821, 57ax-mp 5 . . . . . . . . . . . . . 14  |-  ( 0 ... 0 )  =  { 0 }
5914, 14oveq12i 6308 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 0  +  0 )
60 00id 9772 . . . . . . . . . . . . . . . . 17  |-  ( 0  +  0 )  =  0
6159, 60eqtri 2486 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  0
62244sqlem4a 14480 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ[_i]  /\  0  e.  ZZ[_i]
)  ->  ( (
( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
6323, 23, 62mp2an 672 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
6461, 63eqeltrri 2542 . . . . . . . . . . . . . . 15  |-  0  e.  S
65 snssi 4176 . . . . . . . . . . . . . . 15  |-  ( 0  e.  S  ->  { 0 }  C_  S )
6664, 65ax-mp 5 . . . . . . . . . . . . . 14  |-  { 0 }  C_  S
6758, 66eqsstri 3529 . . . . . . . . . . . . 13  |-  ( 0 ... 0 )  C_  S
6867a1i 11 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... 0 )  C_  S )
69 0p1e1 10668 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7069oveq1i 6306 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... ( j  - 
1 ) )  =  ( 1 ... (
j  -  1 ) )
71 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
72 dfss3 3489 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( j  -  1 ) ) 
C_  S  <->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
7371, 72sylibr 212 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 1 ... ( j  - 
1 ) )  C_  S )
7470, 73syl5eqss 3543 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0  +  1 ) ... ( j  - 
1 ) )  C_  S )
7568, 74unssd 3676 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0 ... 0 )  u.  ( ( 0  +  1 ) ... (
j  -  1 ) ) )  C_  S
)
7656, 75eqsstrd 3533 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  C_  S )
77 oveq1 6303 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  x.  j )  =  ( i  x.  j ) )
7877eleq1d 2526 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( k  x.  j
)  e.  S  <->  ( i  x.  j )  e.  S
) )
7978cbvrabv 3108 . . . . . . . . . 10  |-  { k  e.  NN  |  ( k  x.  j )  e.  S }  =  { i  e.  NN  |  ( i  x.  j )  e.  S }
80 eqid 2457 . . . . . . . . . 10  |-  sup ( { k  e.  NN  |  ( k  x.  j )  e.  S } ,  RR ,  `'  <  )  =  sup ( { k  e.  NN  |  ( k  x.  j )  e.  S } ,  RR ,  `'  <  )
8124, 31, 47, 33, 76, 79, 804sqlem18 14491 . . . . . . . . 9  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8229, 81sylanbr 473 . . . . . . . 8  |-  ( ( ( j  e.  Prime  /\  j  =/=  2 )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8382an32s 804 . . . . . . 7  |-  ( ( ( j  e.  Prime  /\ 
A. m  e.  ( 1 ... ( j  -  1 ) ) m  e.  S )  /\  j  =/=  2
)  ->  j  e.  S )
8410, 10oveq12i 6308 . . . . . . . . . 10  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( 1  +  1 )
85 df-2 10615 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
8684, 85eqtr4i 2489 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  2
87244sqlem4a 14480 . . . . . . . . . 10  |-  ( ( 1  e.  ZZ[_i]  /\  1  e.  ZZ[_i]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
8820, 20, 87mp2an 672 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S
8986, 88eqeltrri 2542 . . . . . . . 8  |-  2  e.  S
9089a1i 11 . . . . . . 7  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
2  e.  S )
9128, 83, 90pm2.61ne 2772 . . . . . 6  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
j  e.  S )
9224mul4sq 14483 . . . . . . 7  |-  ( ( m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S )
9392a1i 11 . . . . . 6  |-  ( ( m  e.  ( ZZ>= ` 
2 )  /\  i  e.  ( ZZ>= `  2 )
)  ->  ( (
m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S ) )
942, 3, 4, 5, 6, 27, 91, 93prmind2 14239 . . . . 5  |-  ( k  e.  NN  ->  k  e.  S )
95 id 22 . . . . . 6  |-  ( k  =  0  ->  k  =  0 )
9695, 64syl6eqel 2553 . . . . 5  |-  ( k  =  0  ->  k  e.  S )
9794, 96jaoi 379 . . . 4  |-  ( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  S
)
981, 97sylbi 195 . . 3  |-  ( k  e.  NN0  ->  k  e.  S )
9998ssriv 3503 . 2  |-  NN0  C_  S
100244sqlem1 14477 . 2  |-  S  C_  NN0
10199, 100eqssi 3515 1  |-  NN0  =  S
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    \ cdif 3468    u. cun 3469    C_ wss 3471   {csn 4032   `'ccnv 5007   ` cfv 5594  (class class class)co 6296   supcsup 7918   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    < clt 9645    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   ^cexp 12168   abscabs 13078   Primecprime 14228   ZZ[_i]cgz 14458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fl 11931  df-mod 11999  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-dvds 13998  df-gcd 14156  df-prm 14229  df-gz 14459
This theorem is referenced by:  4sq  14493
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