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Theorem 4sqlem19 13331
Description: Lemma for 4sq 13332. 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 13330. 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 13322  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 10223 . . . 4  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2 eleq1 2496 . . . . . 6  |-  ( j  =  1  ->  (
j  e.  S  <->  1  e.  S ) )
3 eleq1 2496 . . . . . 6  |-  ( j  =  m  ->  (
j  e.  S  <->  m  e.  S ) )
4 eleq1 2496 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  S  <->  i  e.  S ) )
5 eleq1 2496 . . . . . 6  |-  ( j  =  ( m  x.  i )  ->  (
j  e.  S  <->  ( m  x.  i )  e.  S
) )
6 eleq1 2496 . . . . . 6  |-  ( j  =  k  ->  (
j  e.  S  <->  k  e.  S ) )
7 abs1 12102 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
87oveq1i 6091 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
9 sq1 11476 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
108, 9eqtri 2456 . . . . . . . . 9  |-  ( ( abs `  1 ) ^ 2 )  =  1
11 abs0 12090 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1211oveq1i 6091 . . . . . . . . . 10  |-  ( ( abs `  0 ) ^ 2 )  =  ( 0 ^ 2 )
13 sq0 11473 . . . . . . . . . 10  |-  ( 0 ^ 2 )  =  0
1412, 13eqtri 2456 . . . . . . . . 9  |-  ( ( abs `  0 ) ^ 2 )  =  0
1510, 14oveq12i 6093 . . . . . . . 8  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 1  +  0 )
16 ax-1cn 9048 . . . . . . . . 9  |-  1  e.  CC
1716addid1i 9253 . . . . . . . 8  |-  ( 1  +  0 )  =  1
1815, 17eqtri 2456 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  1
19 1z 10311 . . . . . . . . 9  |-  1  e.  ZZ
20 zgz 13301 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2119, 20ax-mp 8 . . . . . . . 8  |-  1  e.  ZZ [ _i ]
22 0z 10293 . . . . . . . . 9  |-  0  e.  ZZ
23 zgz 13301 . . . . . . . . 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 13319 . . . . . . . 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 2507 . . . . . 6  |-  1  e.  S
29 eleq1 2496 . . . . . . 7  |-  ( j  =  2  ->  (
j  e.  S  <->  2  e.  S ) )
30 eldifsn 3927 . . . . . . . . 9  |-  ( j  e.  ( Prime  \  {
2 } )  <->  ( j  e.  Prime  /\  j  =/=  2 ) )
31 oddprm 13189 . . . . . . . . . . 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 3469 . . . . . . . . . . . . . . . 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 13082 . . . . . . . . . . . . . . 15  |-  ( j  e.  Prime  ->  j  e.  NN )
36 nncn 10008 . . . . . . . . . . . . . . 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 9305 . . . . . . . . . . . . . 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 10070 . . . . . . . . . . . . . 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 10083 . . . . . . . . . . . . . 14  |-  2  =/=  0
4342a1i 11 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  =/=  0 )
4439, 41, 43divcan2d 9792 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 2  x.  ( ( j  -  1 )  / 
2 ) )  =  ( j  -  1 ) )
4544oveq1d 6096 . . . . . . . . . . 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 9314 . . . . . . . . . . . 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 2469 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  =  ( ( 2  x.  ( ( j  - 
1 )  /  2
) )  +  1 ) )
4944oveq2d 6097 . . . . . . . . . . . 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 10261 . . . . . . . . . . . . . . 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 10523 . . . . . . . . . . . . . 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 11064 . . . . . . . . . . . . 13  |-  ( ( j  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... (
j  -  1 ) ) )
55 fzsplit 11077 . . . . . . . . . . . . 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 2468 . . . . . . . . . . 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 11094 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5922, 58ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 0 ... 0 )  =  { 0 }
6014, 14oveq12i 6093 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 0  +  0 )
61 00id 9241 . . . . . . . . . . . . . . . . 17  |-  ( 0  +  0 )  =  0
6260, 61eqtri 2456 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  0
63254sqlem4a 13319 . . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . 15  |-  0  e.  S
66 snssi 3942 . . . . . . . . . . . . . . 15  |-  ( 0  e.  S  ->  { 0 }  C_  S )
6765, 66ax-mp 8 . . . . . . . . . . . . . 14  |-  { 0 }  C_  S
6859, 67eqsstri 3378 . . . . . . . . . . . . 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 10093 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7170oveq1i 6091 . . . . . . . . . . . . 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 3338 . . . . . . . . . . . . . 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 3392 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0  +  1 ) ... ( j  - 
1 ) )  C_  S )
7669, 75unssd 3523 . . . . . . . . . . 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 3382 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  C_  S )
78 oveq1 6088 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  x.  j )  =  ( i  x.  j ) )
7978eleq1d 2502 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( k  x.  j
)  e.  S  <->  ( i  x.  j )  e.  S
) )
8079cbvrabv 2955 . . . . . . . . . 10  |-  { k  e.  NN  |  ( k  x.  j )  e.  S }  =  { i  e.  NN  |  ( i  x.  j )  e.  S }
81 eqid 2436 . . . . . . . . . 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 13330 . . . . . . . . 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 6093 . . . . . . . . . 10  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( 1  +  1 )
86 df-2 10058 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
8785, 86eqtr4i 2459 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  2
88254sqlem4a 13319 . . . . . . . . . 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 2507 . . . . . . . 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 2679 . . . . . 6  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
j  e.  S )
9325mul4sq 13322 . . . . . . 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 13090 . . . . 5  |-  ( k  e.  NN  ->  k  e.  S )
96 id 20 . . . . . 6  |-  ( k  =  0  ->  k  =  0 )
9796, 65syl6eqel 2524 . . . . 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 3352 . 2  |-  NN0  C_  S
101254sqlem1 13316 . 2  |-  S  C_  NN0
102100, 101eqssi 3364 1  |-  NN0  =  S
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    - cmin 9291    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382   abscabs 12039   Primecprime 13079   ZZ [ _i ]cgz 13297
This theorem is referenced by:  4sq  13332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-dvds 12853  df-gcd 13007  df-prm 13080  df-gz 13298
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