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Theorem 4sqlem19 13286
Description: Lemma for 4sq 13287. 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 13285. 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 13277  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 10179 . . . 4  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2 eleq1 2464 . . . . . 6  |-  ( j  =  1  ->  (
j  e.  S  <->  1  e.  S ) )
3 eleq1 2464 . . . . . 6  |-  ( j  =  m  ->  (
j  e.  S  <->  m  e.  S ) )
4 eleq1 2464 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  S  <->  i  e.  S ) )
5 eleq1 2464 . . . . . 6  |-  ( j  =  ( m  x.  i )  ->  (
j  e.  S  <->  ( m  x.  i )  e.  S
) )
6 eleq1 2464 . . . . . 6  |-  ( j  =  k  ->  (
j  e.  S  <->  k  e.  S ) )
7 abs1 12057 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
87oveq1i 6050 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
9 sq1 11431 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
108, 9eqtri 2424 . . . . . . . . 9  |-  ( ( abs `  1 ) ^ 2 )  =  1
11 abs0 12045 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1211oveq1i 6050 . . . . . . . . . 10  |-  ( ( abs `  0 ) ^ 2 )  =  ( 0 ^ 2 )
13 sq0 11428 . . . . . . . . . 10  |-  ( 0 ^ 2 )  =  0
1412, 13eqtri 2424 . . . . . . . . 9  |-  ( ( abs `  0 ) ^ 2 )  =  0
1510, 14oveq12i 6052 . . . . . . . 8  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 1  +  0 )
16 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
1716addid1i 9209 . . . . . . . 8  |-  ( 1  +  0 )  =  1
1815, 17eqtri 2424 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  1
19 1z 10267 . . . . . . . . 9  |-  1  e.  ZZ
20 zgz 13256 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2119, 20ax-mp 8 . . . . . . . 8  |-  1  e.  ZZ [ _i ]
22 0z 10249 . . . . . . . . 9  |-  0  e.  ZZ
23 zgz 13256 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  ZZ [ _i ]
)
2422, 23ax-mp 8 . . . . . . . 8  |-  0  e.  ZZ [ _i ]
25 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 ) ) ) }
26254sqlem4a 13274 . . . . . . . 8  |-  ( ( 1  e.  ZZ [
_i ]  /\  0  e.  ZZ [ _i ]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
2721, 24, 26mp2an 654 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
2818, 27eqeltrri 2475 . . . . . 6  |-  1  e.  S
29 eleq1 2464 . . . . . . 7  |-  ( j  =  2  ->  (
j  e.  S  <->  2  e.  S ) )
30 eldifsn 3887 . . . . . . . . 9  |-  ( j  e.  ( Prime  \  {
2 } )  <->  ( j  e.  Prime  /\  j  =/=  2 ) )
31 oddprm 13144 . . . . . . . . . . 11  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
( ( j  - 
1 )  /  2
)  e.  NN )
3231adantr 452 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  /  2 )  e.  NN )
33 eldifi 3429 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
j  e.  Prime )
3433adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  Prime )
35 prmnn 13037 . . . . . . . . . . . . . . 15  |-  ( j  e.  Prime  ->  j  e.  NN )
36 nncn 9964 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  j  e.  CC )
3734, 35, 363syl 19 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  CC )
38 subcl 9261 . . . . . . . . . . . . . 14  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  -  1 )  e.  CC )
3937, 16, 38sylancl 644 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  CC )
40 2cn 10026 . . . . . . . . . . . . . 14  |-  2  e.  CC
4140a1i 11 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  e.  CC )
42 2ne0 10039 . . . . . . . . . . . . . 14  |-  2  =/=  0
4342a1i 11 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  =/=  0 )
4439, 41, 43divcan2d 9748 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 2  x.  ( ( j  -  1 )  / 
2 ) )  =  ( j  -  1 ) )
4544oveq1d 6055 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
2  x.  ( ( j  -  1 )  /  2 ) )  +  1 )  =  ( ( j  - 
1 )  +  1 ) )
46 npcan 9270 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  - 
1 )  +  1 )  =  j )
4737, 16, 46sylancl 644 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  +  1 )  =  j )
4845, 47eqtr2d 2437 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  =  ( ( 2  x.  ( ( j  - 
1 )  /  2
) )  +  1 ) )
4944oveq2d 6056 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  =  ( 0 ... (
j  -  1 ) ) )
50 nnm1nn0 10217 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  (
j  -  1 )  e.  NN0 )
5134, 35, 503syl 19 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e. 
NN0 )
52 elnn0uz 10479 . . . . . . . . . . . . . 14  |-  ( ( j  -  1 )  e.  NN0  <->  ( j  - 
1 )  e.  (
ZZ>= `  0 ) )
5351, 52sylib 189 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  ( ZZ>= `  0 )
)
54 eluzfz1 11020 . . . . . . . . . . . . 13  |-  ( ( j  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... (
j  -  1 ) ) )
55 fzsplit 11033 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 ... ( j  -  1 ) )  ->  (
0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5653, 54, 553syl 19 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5749, 56eqtrd 2436 . . . . . . . . . . 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 ) ) ) )
58 fzsn 11050 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5922, 58ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 0 ... 0 )  =  { 0 }
6014, 14oveq12i 6052 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 0  +  0 )
61 00id 9197 . . . . . . . . . . . . . . . . 17  |-  ( 0  +  0 )  =  0
6260, 61eqtri 2424 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  0
63254sqlem4a 13274 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ [
_i ]  /\  0  e.  ZZ [ _i ]
)  ->  ( (
( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
6424, 24, 63mp2an 654 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
6562, 64eqeltrri 2475 . . . . . . . . . . . . . . 15  |-  0  e.  S
66 snssi 3902 . . . . . . . . . . . . . . 15  |-  ( 0  e.  S  ->  { 0 }  C_  S )
6765, 66ax-mp 8 . . . . . . . . . . . . . 14  |-  { 0 }  C_  S
6859, 67eqsstri 3338 . . . . . . . . . . . . 13  |-  ( 0 ... 0 )  C_  S
6968a1i 11 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... 0 )  C_  S )
70 0p1e1 10049 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7170oveq1i 6050 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... ( j  - 
1 ) )  =  ( 1 ... (
j  -  1 ) )
72 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
73 dfss3 3298 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( j  -  1 ) ) 
C_  S  <->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
7472, 73sylibr 204 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 1 ... ( j  - 
1 ) )  C_  S )
7571, 74syl5eqss 3352 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0  +  1 ) ... ( j  - 
1 ) )  C_  S )
7669, 75unssd 3483 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0 ... 0 )  u.  ( ( 0  +  1 ) ... (
j  -  1 ) ) )  C_  S
)
7757, 76eqsstrd 3342 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  C_  S )
78 oveq1 6047 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  x.  j )  =  ( i  x.  j ) )
7978eleq1d 2470 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( k  x.  j
)  e.  S  <->  ( i  x.  j )  e.  S
) )
8079cbvrabv 2915 . . . . . . . . . 10  |-  { k  e.  NN  |  ( k  x.  j )  e.  S }  =  { i  e.  NN  |  ( i  x.  j )  e.  S }
81 eqid 2404 . . . . . . . . . 10  |-  sup ( { k  e.  NN  |  ( k  x.  j )  e.  S } ,  RR ,  `'  <  )  =  sup ( { k  e.  NN  |  ( k  x.  j )  e.  S } ,  RR ,  `'  <  )
8225, 32, 48, 34, 77, 80, 814sqlem18 13285 . . . . . . . . 9  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8330, 82sylanbr 460 . . . . . . . 8  |-  ( ( ( j  e.  Prime  /\  j  =/=  2 )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8483an32s 780 . . . . . . 7  |-  ( ( ( j  e.  Prime  /\ 
A. m  e.  ( 1 ... ( j  -  1 ) ) m  e.  S )  /\  j  =/=  2
)  ->  j  e.  S )
8510, 10oveq12i 6052 . . . . . . . . . 10  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( 1  +  1 )
86 df-2 10014 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
8785, 86eqtr4i 2427 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  2
88254sqlem4a 13274 . . . . . . . . . 10  |-  ( ( 1  e.  ZZ [
_i ]  /\  1  e.  ZZ [ _i ]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
8921, 21, 88mp2an 654 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S
9087, 89eqeltrri 2475 . . . . . . . 8  |-  2  e.  S
9190a1i 11 . . . . . . 7  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
2  e.  S )
9229, 84, 91pm2.61ne 2642 . . . . . 6  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
j  e.  S )
9325mul4sq 13277 . . . . . . 7  |-  ( ( m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S )
9493a1i 11 . . . . . 6  |-  ( ( m  e.  ( ZZ>= ` 
2 )  /\  i  e.  ( ZZ>= `  2 )
)  ->  ( (
m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S ) )
952, 3, 4, 5, 6, 28, 92, 94prmind2 13045 . . . . 5  |-  ( k  e.  NN  ->  k  e.  S )
96 id 20 . . . . . 6  |-  ( k  =  0  ->  k  =  0 )
9796, 65syl6eqel 2492 . . . . 5  |-  ( k  =  0  ->  k  e.  S )
9895, 97jaoi 369 . . . 4  |-  ( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  S
)
991, 98sylbi 188 . . 3  |-  ( k  e.  NN0  ->  k  e.  S )
10099ssriv 3312 . 2  |-  NN0  C_  S
101254sqlem1 13271 . 2  |-  S  C_  NN0
102100, 101eqssi 3324 1  |-  NN0  =  S
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    \ cdif 3277    u. cun 3278    C_ wss 3280   {csn 3774   `'ccnv 4836   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   ^cexp 11337   abscabs 11994   Primecprime 13034   ZZ [ _i ]cgz 13252
This theorem is referenced by:  4sq  13287
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-dvds 12808  df-gcd 12962  df-prm 13035  df-gz 13253
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