Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nnsum3primesgbe Structured version   Visualization version   Unicode version

Theorem nnsum3primesgbe 38897
Description: Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)
Assertion
Ref Expression
nnsum3primesgbe  |-  ( N  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) )
Distinct variable group:    N, d, f, k

Proof of Theorem nnsum3primesgbe
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isgbe 38862 . 2  |-  ( N  e. GoldbachEven 
<->  ( N  e. Even  /\  E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) ) )
2 2nn 10774 . . . . . . . 8  |-  2  e.  NN
32a1i 11 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  -> 
2  e.  NN )
4 oveq2 6303 . . . . . . . . . . 11  |-  ( d  =  2  ->  (
1 ... d )  =  ( 1 ... 2
) )
5 df-2 10675 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
65oveq2i 6306 . . . . . . . . . . . 12  |-  ( 1 ... 2 )  =  ( 1 ... (
1  +  1 ) )
7 1z 10974 . . . . . . . . . . . . 13  |-  1  e.  ZZ
8 fzpr 11858 . . . . . . . . . . . . 13  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) } )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  ( 1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) }
10 1p1e2 10730 . . . . . . . . . . . . 13  |-  ( 1  +  1 )  =  2
1110preq2i 4058 . . . . . . . . . . . 12  |-  { 1 ,  ( 1  +  1 ) }  =  { 1 ,  2 }
126, 9, 113eqtri 2479 . . . . . . . . . . 11  |-  ( 1 ... 2 )  =  { 1 ,  2 }
134, 12syl6eq 2503 . . . . . . . . . 10  |-  ( d  =  2  ->  (
1 ... d )  =  { 1 ,  2 } )
1413oveq2d 6311 . . . . . . . . 9  |-  ( d  =  2  ->  ( Prime  ^m  ( 1 ... d ) )  =  ( Prime  ^m  { 1 ,  2 } ) )
15 breq1 4408 . . . . . . . . . 10  |-  ( d  =  2  ->  (
d  <_  3  <->  2  <_  3 ) )
1613sumeq1d 13779 . . . . . . . . . . 11  |-  ( d  =  2  ->  sum_ k  e.  ( 1 ... d
) ( f `  k )  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
)
1716eqeq2d 2463 . . . . . . . . . 10  |-  ( d  =  2  ->  ( N  =  sum_ k  e.  ( 1 ... d
) ( f `  k )  <->  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
) )
1815, 17anbi12d 718 . . . . . . . . 9  |-  ( d  =  2  ->  (
( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k ) )  <->  ( 2  <_ 
3  /\  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
) ) )
1914, 18rexeqbidv 3004 . . . . . . . 8  |-  ( d  =  2  ->  ( E. f  e.  ( Prime  ^m  ( 1 ... d ) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d
) ( f `  k ) )  <->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
) ) )
2019adantl 468 . . . . . . 7  |-  ( ( ( ( p  e. 
Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  /\  d  =  2 )  ->  ( E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) )  <->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
) ) )
21 1ne2 10829 . . . . . . . . . . . . 13  |-  1  =/=  2
22 1ex 9643 . . . . . . . . . . . . . 14  |-  1  e.  _V
23 2ex 10688 . . . . . . . . . . . . . 14  |-  2  e.  _V
24 vex 3050 . . . . . . . . . . . . . 14  |-  p  e. 
_V
25 vex 3050 . . . . . . . . . . . . . 14  |-  q  e. 
_V
2622, 23, 24, 25fpr 6077 . . . . . . . . . . . . 13  |-  ( 1  =/=  2  ->  { <. 1 ,  p >. , 
<. 2 ,  q
>. } : { 1 ,  2 } --> { p ,  q } )
2721, 26mp1i 13 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  { <. 1 ,  p >. , 
<. 2 ,  q
>. } : { 1 ,  2 } --> { p ,  q } )
28 prssi 4131 . . . . . . . . . . . 12  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  { p ,  q }  C_  Prime )
2927, 28fssd 5743 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  { <. 1 ,  p >. , 
<. 2 ,  q
>. } : { 1 ,  2 } --> Prime )
30 prmex 14640 . . . . . . . . . . . . 13  |-  Prime  e.  _V
31 prex 4645 . . . . . . . . . . . . 13  |-  { 1 ,  2 }  e.  _V
3230, 31pm3.2i 457 . . . . . . . . . . . 12  |-  ( Prime  e.  _V  /\  { 1 ,  2 }  e.  _V )
33 elmapg 7490 . . . . . . . . . . . 12  |-  ( ( Prime  e.  _V  /\  { 1 ,  2 }  e.  _V )  -> 
( { <. 1 ,  p >. ,  <. 2 ,  q >. }  e.  ( Prime  ^m  { 1 ,  2 } )  <->  { <. 1 ,  p >. ,  <. 2 ,  q
>. } : { 1 ,  2 } --> Prime )
)
3432, 33mp1i 13 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. }  e.  ( Prime  ^m  { 1 ,  2 } )  <->  { <. 1 ,  p >. ,  <. 2 ,  q >. } : { 1 ,  2 } --> Prime ) )
3529, 34mpbird 236 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  { <. 1 ,  p >. , 
<. 2 ,  q
>. }  e.  ( Prime  ^m  { 1 ,  2 } ) )
36 fveq1 5869 . . . . . . . . . . . . . . 15  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. }  ->  ( f `  k )  =  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  k ) )
3736adantr 467 . . . . . . . . . . . . . 14  |-  ( ( f  =  { <. 1 ,  p >. , 
<. 2 ,  q
>. }  /\  k  e. 
{ 1 ,  2 } )  ->  (
f `  k )  =  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  k ) )
3837sumeq2dv 13781 . . . . . . . . . . . . 13  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. }  ->  sum_ k  e.  { 1 ,  2 }  (
f `  k )  =  sum_ k  e.  {
1 ,  2 }  ( { <. 1 ,  p >. ,  <. 2 ,  q >. } `  k ) )
3938eqeq1d 2455 . . . . . . . . . . . 12  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. }  ->  (
sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q )  <->  sum_ k  e. 
{ 1 ,  2 }  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  k )  =  ( p  +  q ) ) )
4039anbi2d 711 . . . . . . . . . . 11  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. }  ->  ( ( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) )  <-> 
( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( { <. 1 ,  p >. ,  <. 2 ,  q >. } `  k )  =  ( p  +  q ) ) ) )
4140adantl 468 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  f  =  { <. 1 ,  p >. , 
<. 2 ,  q
>. } )  ->  (
( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) )  <-> 
( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( { <. 1 ,  p >. ,  <. 2 ,  q >. } `  k )  =  ( p  +  q ) ) ) )
42 prmz 14638 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e.  ZZ )
43 prmz 14638 . . . . . . . . . . . 12  |-  ( q  e.  Prime  ->  q  e.  ZZ )
44 fveq2 5870 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  k )  =  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  1 ) )
4522, 24fvpr1 6112 . . . . . . . . . . . . . . 15  |-  ( 1  =/=  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  1 )  =  p )
4621, 45ax-mp 5 . . . . . . . . . . . . . 14  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. } `  1 )  =  p
4744, 46syl6eq 2503 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  k )  =  p )
48 fveq2 5870 . . . . . . . . . . . . . 14  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  k )  =  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  2 ) )
4923, 25fvpr2 6113 . . . . . . . . . . . . . . 15  |-  ( 1  =/=  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  2 )  =  q )
5021, 49ax-mp 5 . . . . . . . . . . . . . 14  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. } `  2 )  =  q
5148, 50syl6eq 2503 . . . . . . . . . . . . 13  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  k )  =  q )
52 zcn 10949 . . . . . . . . . . . . . 14  |-  ( p  e.  ZZ  ->  p  e.  CC )
53 zcn 10949 . . . . . . . . . . . . . 14  |-  ( q  e.  ZZ  ->  q  e.  CC )
5452, 53anim12i 570 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  ZZ )  ->  ( p  e.  CC  /\  q  e.  CC ) )
557, 2pm3.2i 457 . . . . . . . . . . . . . 14  |-  ( 1  e.  ZZ  /\  2  e.  NN )
5655a1i 11 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  ZZ )  ->  ( 1  e.  ZZ  /\  2  e.  NN ) )
5721a1i 11 . . . . . . . . . . . . 13  |-  ( ( p  e.  ZZ  /\  q  e.  ZZ )  ->  1  =/=  2 )
5847, 51, 54, 56, 57sumpr 13821 . . . . . . . . . . . 12  |-  ( ( p  e.  ZZ  /\  q  e.  ZZ )  -> 
sum_ k  e.  {
1 ,  2 }  ( { <. 1 ,  p >. ,  <. 2 ,  q >. } `  k )  =  ( p  +  q ) )
5942, 43, 58syl2an 480 . . . . . . . . . . 11  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  sum_ k  e.  { 1 ,  2 }  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. } `  k )  =  ( p  +  q ) )
60 2re 10686 . . . . . . . . . . . 12  |-  2  e.  RR
61 3re 10690 . . . . . . . . . . . 12  |-  3  e.  RR
62 2lt3 10784 . . . . . . . . . . . 12  |-  2  <  3
6360, 61, 62ltleii 9762 . . . . . . . . . . 11  |-  2  <_  3
6459, 63jctil 540 . . . . . . . . . 10  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  (
2  <_  3  /\  sum_ k  e.  { 1 ,  2 }  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. } `  k )  =  ( p  +  q ) ) )
6535, 41, 64rspcedvd 3157 . . . . . . . . 9  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) ) )
6665adantr 467 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  ->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  sum_ k  e.  { 1 ,  2 }  (
f `  k )  =  ( p  +  q ) ) )
67 eqeq1 2457 . . . . . . . . . . . . 13  |-  ( N  =  ( p  +  q )  ->  ( N  =  sum_ k  e. 
{ 1 ,  2 }  ( f `  k )  <->  ( p  +  q )  = 
sum_ k  e.  {
1 ,  2 }  ( f `  k
) ) )
68 eqcom 2460 . . . . . . . . . . . . 13  |-  ( ( p  +  q )  =  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  <->  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) )
6967, 68syl6bb 265 . . . . . . . . . . . 12  |-  ( N  =  ( p  +  q )  ->  ( N  =  sum_ k  e. 
{ 1 ,  2 }  ( f `  k )  <->  sum_ k  e. 
{ 1 ,  2 }  ( f `  k )  =  ( p  +  q ) ) )
7069anbi2d 711 . . . . . . . . . . 11  |-  ( N  =  ( p  +  q )  ->  (
( 2  <_  3  /\  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
)  <->  ( 2  <_ 
3  /\  sum_ k  e. 
{ 1 ,  2 }  ( f `  k )  =  ( p  +  q ) ) ) )
7170rexbidv 2903 . . . . . . . . . 10  |-  ( N  =  ( p  +  q )  ->  ( E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e. 
{ 1 ,  2 }  ( f `  k ) )  <->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) ) ) )
72713ad2ant3 1032 . . . . . . . . 9  |-  ( ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) )  ->  ( E. f  e.  ( Prime  ^m 
{ 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e.  {
1 ,  2 }  ( f `  k
) )  <->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) ) ) )
7372adantl 468 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  -> 
( E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e.  { 1 ,  2 }  (
f `  k )
)  <->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  sum_ k  e.  {
1 ,  2 }  ( f `  k
)  =  ( p  +  q ) ) ) )
7466, 73mpbird 236 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  ->  E. f  e.  ( Prime  ^m  { 1 ,  2 } ) ( 2  <_  3  /\  N  =  sum_ k  e. 
{ 1 ,  2 }  ( f `  k ) ) )
753, 20, 74rspcedvd 3157 . . . . . 6  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) )
7675a1d 26 . . . . 5  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  -> 
( N  e. Even  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) ) )
7776ex 436 . . . 4  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  (
( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) )  ->  ( N  e. Even  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  (
1 ... d ) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k ) ) ) ) )
7877rexlimivv 2886 . . 3  |-  ( E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) )  -> 
( N  e. Even  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) ) )
7978impcom 432 . 2  |-  ( ( N  e. Even  /\  E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  N  =  ( p  +  q ) ) )  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) )
801, 79sylbi 199 1  |-  ( N  e. GoldbachEven  ->  E. d  e.  NN  E. f  e.  ( Prime  ^m  ( 1 ... d
) ) ( d  <_  3  /\  N  =  sum_ k  e.  ( 1 ... d ) ( f `  k
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   E.wrex 2740   _Vcvv 3047   {cpr 3972   <.cop 3976   class class class wbr 4405   -->wf 5581   ` cfv 5585  (class class class)co 6295    ^m cmap 7477   CCcc 9542   1c1 9545    + caddc 9547    <_ cle 9681   NNcn 10616   2c2 10666   3c3 10667   ZZcz 10944   ...cfz 11791   sum_csu 13764   Primecprime 14634   Even ceven 38763   Odd codd 38764   GoldbachEven cgbe 38856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  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-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-prm 14635  df-gbe 38859
This theorem is referenced by:  nnsum4primesgbe  38898  nnsum3primesle9  38899  bgoldbnnsum3prm  38909
  Copyright terms: Public domain W3C validator