MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  4sqlem19 Unicode version

Theorem 4sqlem19 13026
Description: Lemma for 4sq 13027. 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 13025. 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 13017  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 9983 . . . 4  |-  ( k  e.  NN0  <->  ( k  e.  NN  \/  k  =  0 ) )
2 eleq1 2356 . . . . . 6  |-  ( j  =  1  ->  (
j  e.  S  <->  1  e.  S ) )
3 eleq1 2356 . . . . . 6  |-  ( j  =  m  ->  (
j  e.  S  <->  m  e.  S ) )
4 eleq1 2356 . . . . . 6  |-  ( j  =  i  ->  (
j  e.  S  <->  i  e.  S ) )
5 eleq1 2356 . . . . . 6  |-  ( j  =  ( m  x.  i )  ->  (
j  e.  S  <->  ( m  x.  i )  e.  S
) )
6 eleq1 2356 . . . . . 6  |-  ( j  =  k  ->  (
j  e.  S  <->  k  e.  S ) )
7 abs1 11798 . . . . . . . . . . 11  |-  ( abs `  1 )  =  1
87oveq1i 5884 . . . . . . . . . 10  |-  ( ( abs `  1 ) ^ 2 )  =  ( 1 ^ 2 )
9 sq1 11214 . . . . . . . . . 10  |-  ( 1 ^ 2 )  =  1
108, 9eqtri 2316 . . . . . . . . 9  |-  ( ( abs `  1 ) ^ 2 )  =  1
11 abs0 11786 . . . . . . . . . . 11  |-  ( abs `  0 )  =  0
1211oveq1i 5884 . . . . . . . . . 10  |-  ( ( abs `  0 ) ^ 2 )  =  ( 0 ^ 2 )
13 sq0 11211 . . . . . . . . . 10  |-  ( 0 ^ 2 )  =  0
1412, 13eqtri 2316 . . . . . . . . 9  |-  ( ( abs `  0 ) ^ 2 )  =  0
1510, 14oveq12i 5886 . . . . . . . 8  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 1  +  0 )
16 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
1716addid1i 9015 . . . . . . . 8  |-  ( 1  +  0 )  =  1
1815, 17eqtri 2316 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  1
19 1z 10069 . . . . . . . . 9  |-  1  e.  ZZ
20 zgz 12996 . . . . . . . . 9  |-  ( 1  e.  ZZ  ->  1  e.  ZZ [ _i ]
)
2119, 20ax-mp 8 . . . . . . . 8  |-  1  e.  ZZ [ _i ]
22 0z 10051 . . . . . . . . 9  |-  0  e.  ZZ
23 zgz 12996 . . . . . . . . 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 13014 . . . . . . . 8  |-  ( ( 1  e.  ZZ [
_i ]  /\  0  e.  ZZ [ _i ]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
2721, 24, 26mp2an 653 . . . . . . 7  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
2818, 27eqeltrri 2367 . . . . . 6  |-  1  e.  S
29 eleq1 2356 . . . . . . 7  |-  ( j  =  2  ->  (
j  e.  S  <->  2  e.  S ) )
30 eldifsn 3762 . . . . . . . . 9  |-  ( j  e.  ( Prime  \  {
2 } )  <->  ( j  e.  Prime  /\  j  =/=  2 ) )
31 oddprm 12884 . . . . . . . . . . 11  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
( ( j  - 
1 )  /  2
)  e.  NN )
3231adantr 451 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  /  2 )  e.  NN )
33 eldifi 3311 . . . . . . . . . . . . . . . 16  |-  ( j  e.  ( Prime  \  {
2 } )  -> 
j  e.  Prime )
3433adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  Prime )
35 prmnn 12777 . . . . . . . . . . . . . . 15  |-  ( j  e.  Prime  ->  j  e.  NN )
36 nncn 9770 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  j  e.  CC )
3734, 35, 363syl 18 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  CC )
38 subcl 9067 . . . . . . . . . . . . . 14  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( j  -  1 )  e.  CC )
3937, 16, 38sylancl 643 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  CC )
40 2cn 9832 . . . . . . . . . . . . . 14  |-  2  e.  CC
4140a1i 10 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  e.  CC )
42 2ne0 9845 . . . . . . . . . . . . . 14  |-  2  =/=  0
4342a1i 10 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  2  =/=  0 )
4439, 41, 43divcan2d 9554 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 2  x.  ( ( j  -  1 )  / 
2 ) )  =  ( j  -  1 ) )
4544oveq1d 5889 . . . . . . . . . . 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 9076 . . . . . . . . . . . 12  |-  ( ( j  e.  CC  /\  1  e.  CC )  ->  ( ( j  - 
1 )  +  1 )  =  j )
4737, 16, 46sylancl 643 . . . . . . . . . . 11  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
j  -  1 )  +  1 )  =  j )
4845, 47eqtr2d 2329 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  =  ( ( 2  x.  ( ( j  - 
1 )  /  2
) )  +  1 ) )
4944oveq2d 5890 . . . . . . . . . . . 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 10021 . . . . . . . . . . . . . . 15  |-  ( j  e.  NN  ->  (
j  -  1 )  e.  NN0 )
5134, 35, 503syl 18 . . . . . . . . . . . . . 14  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e. 
NN0 )
52 elnn0uz 10281 . . . . . . . . . . . . . 14  |-  ( ( j  -  1 )  e.  NN0  <->  ( j  - 
1 )  e.  (
ZZ>= `  0 ) )
5351, 52sylib 188 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( j  -  1 )  e.  ( ZZ>= `  0 )
)
54 eluzfz1 10819 . . . . . . . . . . . . 13  |-  ( ( j  -  1 )  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... (
j  -  1 ) ) )
55 fzsplit 10832 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 0 ... ( j  -  1 ) )  ->  (
0 ... ( j  - 
1 ) )  =  ( ( 0 ... 0 )  u.  (
( 0  +  1 ) ... ( j  -  1 ) ) ) )
5653, 54, 553syl 18 . . . . . . . . . . . 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 2328 . . . . . . . . . . 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 10849 . . . . . . . . . . . . . . 15  |-  ( 0  e.  ZZ  ->  (
0 ... 0 )  =  { 0 } )
5922, 58ax-mp 8 . . . . . . . . . . . . . 14  |-  ( 0 ... 0 )  =  { 0 }
6014, 14oveq12i 5886 . . . . . . . . . . . . . . . . 17  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  ( 0  +  0 )
61 00id 9003 . . . . . . . . . . . . . . . . 17  |-  ( 0  +  0 )  =  0
6260, 61eqtri 2316 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  =  0
63254sqlem4a 13014 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  ZZ [
_i ]  /\  0  e.  ZZ [ _i ]
)  ->  ( (
( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S )
6424, 24, 63mp2an 653 . . . . . . . . . . . . . . . 16  |-  ( ( ( abs `  0
) ^ 2 )  +  ( ( abs `  0 ) ^
2 ) )  e.  S
6562, 64eqeltrri 2367 . . . . . . . . . . . . . . 15  |-  0  e.  S
66 snssi 3775 . . . . . . . . . . . . . . 15  |-  ( 0  e.  S  ->  { 0 }  C_  S )
6765, 66ax-mp 8 . . . . . . . . . . . . . 14  |-  { 0 }  C_  S
6859, 67eqsstri 3221 . . . . . . . . . . . . 13  |-  ( 0 ... 0 )  C_  S
6968a1i 10 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... 0 )  C_  S )
70 0p1e1 9855 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
7170oveq1i 5884 . . . . . . . . . . . . 13  |-  ( ( 0  +  1 ) ... ( j  - 
1 ) )  =  ( 1 ... (
j  -  1 ) )
72 simpr 447 . . . . . . . . . . . . . 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 3183 . . . . . . . . . . . . . 14  |-  ( ( 1 ... ( j  -  1 ) ) 
C_  S  <->  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)
7472, 73sylibr 203 . . . . . . . . . . . . 13  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 1 ... ( j  - 
1 ) )  C_  S )
7571, 74syl5eqss 3235 . . . . . . . . . . . 12  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( (
0  +  1 ) ... ( j  - 
1 ) )  C_  S )
7669, 75unssd 3364 . . . . . . . . . . 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 3225 . . . . . . . . . 10  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  ( 0 ... ( 2  x.  ( ( j  - 
1 )  /  2
) ) )  C_  S )
78 oveq1 5881 . . . . . . . . . . . 12  |-  ( k  =  i  ->  (
k  x.  j )  =  ( i  x.  j ) )
7978eleq1d 2362 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( k  x.  j
)  e.  S  <->  ( i  x.  j )  e.  S
) )
8079cbvrabv 2800 . . . . . . . . . 10  |-  { k  e.  NN  |  ( k  x.  j )  e.  S }  =  { i  e.  NN  |  ( i  x.  j )  e.  S }
81 eqid 2296 . . . . . . . . . 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 13025 . . . . . . . . 9  |-  ( ( j  e.  ( Prime  \  { 2 } )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8330, 82sylanbr 459 . . . . . . . 8  |-  ( ( ( j  e.  Prime  /\  j  =/=  2 )  /\  A. m  e.  ( 1 ... (
j  -  1 ) ) m  e.  S
)  ->  j  e.  S )
8483an32s 779 . . . . . . 7  |-  ( ( ( j  e.  Prime  /\ 
A. m  e.  ( 1 ... ( j  -  1 ) ) m  e.  S )  /\  j  =/=  2
)  ->  j  e.  S )
8510, 10oveq12i 5886 . . . . . . . . . 10  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  ( 1  +  1 )
86 df-2 9820 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
8785, 86eqtr4i 2319 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  =  2
88254sqlem4a 13014 . . . . . . . . . 10  |-  ( ( 1  e.  ZZ [
_i ]  /\  1  e.  ZZ [ _i ]
)  ->  ( (
( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S )
8921, 21, 88mp2an 653 . . . . . . . . 9  |-  ( ( ( abs `  1
) ^ 2 )  +  ( ( abs `  1 ) ^
2 ) )  e.  S
9087, 89eqeltrri 2367 . . . . . . . 8  |-  2  e.  S
9190a1i 10 . . . . . . 7  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
2  e.  S )
9229, 84, 91pm2.61ne 2534 . . . . . 6  |-  ( ( j  e.  Prime  /\  A. m  e.  ( 1 ... ( j  - 
1 ) ) m  e.  S )  -> 
j  e.  S )
9325mul4sq 13017 . . . . . . 7  |-  ( ( m  e.  S  /\  i  e.  S )  ->  ( m  x.  i
)  e.  S )
9493a1i 10 . . . . . 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 12785 . . . . 5  |-  ( k  e.  NN  ->  k  e.  S )
96 id 19 . . . . . 6  |-  ( k  =  0  ->  k  =  0 )
9796, 65syl6eqel 2384 . . . . 5  |-  ( k  =  0  ->  k  e.  S )
9895, 97jaoi 368 . . . 4  |-  ( ( k  e.  NN  \/  k  =  0 )  ->  k  e.  S
)
991, 98sylbi 187 . . 3  |-  ( k  e.  NN0  ->  k  e.  S )
10099ssriv 3197 . 2  |-  NN0  C_  S
101254sqlem1 13011 . 2  |-  S  C_  NN0
102100, 101eqssi 3208 1  |-  NN0  =  S
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120   abscabs 11735   Primecprime 12774   ZZ [ _i ]cgz 12992
This theorem is referenced by:  4sq  13027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-gz 12993
  Copyright terms: Public domain W3C validator