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Theorem nnsum4primesodd 39026
Description: If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)
Assertion
Ref Expression
nnsum4primesodd  |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  6
)  /\  N  e. Odd  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
Distinct variable group:    f, N, k, m

Proof of Theorem nnsum4primesodd
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4422 . . . . . 6  |-  ( m  =  N  ->  (
5  <  m  <->  5  <  N ) )
2 eleq1 2528 . . . . . 6  |-  ( m  =  N  ->  (
m  e. GoldbachOdd  <->  N  e. GoldbachOdd  ) )
31, 2imbi12d 326 . . . . 5  |-  ( m  =  N  ->  (
( 5  <  m  ->  m  e. GoldbachOdd  )  <->  ( 5  <  N  ->  N  e. GoldbachOdd  ) ) )
43rspcv 3158 . . . 4  |-  ( N  e. Odd  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  (
5  <  N  ->  N  e. GoldbachOdd  ) ) )
54adantl 472 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( 5  < 
N  ->  N  e. GoldbachOdd  ) ) )
6 eluz2 11199 . . . . . 6  |-  ( N  e.  ( ZZ>= `  6
)  <->  ( 6  e.  ZZ  /\  N  e.  ZZ  /\  6  <_  N ) )
7 5lt6 10820 . . . . . . . . 9  |-  5  <  6
8 5re 10721 . . . . . . . . . . 11  |-  5  e.  RR
98a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  5  e.  RR )
10 6re 10723 . . . . . . . . . . 11  |-  6  e.  RR
1110a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  6  e.  RR )
12 zre 10975 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  RR )
13 ltletr 9756 . . . . . . . . . 10  |-  ( ( 5  e.  RR  /\  6  e.  RR  /\  N  e.  RR )  ->  (
( 5  <  6  /\  6  <_  N )  ->  5  <  N
) )
149, 11, 12, 13syl3anc 1276 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 5  <  6  /\  6  <_  N )  ->  5  <  N
) )
157, 14mpani 687 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
6  <_  N  ->  5  <  N ) )
1615imp 435 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  6  <_  N )  -> 
5  <  N )
17163adant1 1032 . . . . . 6  |-  ( ( 6  e.  ZZ  /\  N  e.  ZZ  /\  6  <_  N )  ->  5  <  N )
186, 17sylbi 200 . . . . 5  |-  ( N  e.  ( ZZ>= `  6
)  ->  5  <  N )
1918adantr 471 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  5  <  N )
20 pm2.27 40 . . . 4  |-  ( 5  <  N  ->  (
( 5  <  N  ->  N  e. GoldbachOdd  )  ->  N  e. GoldbachOdd  ) )
2119, 20syl 17 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  ( (
5  <  N  ->  N  e. GoldbachOdd  )  ->  N  e. GoldbachOdd  ) )
22 isgbo 38988 . . . . 5  |-  ( N  e. GoldbachOdd 
<->  ( N  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  N  =  ( ( p  +  q )  +  r ) ) )
23 1ex 9669 . . . . . . . . . . . . . . 15  |-  1  e.  _V
24 2ex 10714 . . . . . . . . . . . . . . 15  |-  2  e.  _V
25 3ex 10718 . . . . . . . . . . . . . . 15  |-  3  e.  _V
26 vex 3060 . . . . . . . . . . . . . . 15  |-  p  e. 
_V
27 vex 3060 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
28 vex 3060 . . . . . . . . . . . . . . 15  |-  r  e. 
_V
29 1ne2 10856 . . . . . . . . . . . . . . 15  |-  1  =/=  2
30 1re 9673 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
31 1lt3 10812 . . . . . . . . . . . . . . . 16  |-  1  <  3
3230, 31ltneii 9778 . . . . . . . . . . . . . . 15  |-  1  =/=  3
33 2re 10712 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
34 2lt3 10811 . . . . . . . . . . . . . . . 16  |-  2  <  3
3533, 34ltneii 9778 . . . . . . . . . . . . . . 15  |-  2  =/=  3
3623, 24, 25, 26, 27, 28, 29, 32, 35ftp 6104 . . . . . . . . . . . . . 14  |-  { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } : {
1 ,  2 ,  3 } --> { p ,  q ,  r }
3736a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : { 1 ,  2 ,  3 } --> { p ,  q ,  r } )
38 1p2e3 10768 . . . . . . . . . . . . . . . . 17  |-  ( 1  +  2 )  =  3
3938eqcomi 2471 . . . . . . . . . . . . . . . 16  |-  3  =  ( 1  +  2 )
4039oveq2i 6331 . . . . . . . . . . . . . . 15  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
41 1z 11001 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
42 fztp 11887 . . . . . . . . . . . . . . . 16  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
4341, 42ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( 1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
44 eqid 2462 . . . . . . . . . . . . . . . 16  |-  1  =  1
45 id 22 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  1  =  1 )
46 1p1e2 10756 . . . . . . . . . . . . . . . . . 18  |-  ( 1  +  1 )  =  2
4746a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  (
1  +  1 )  =  2 )
4838a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  (
1  +  2 )  =  3 )
4945, 47, 48tpeq123d 4079 . . . . . . . . . . . . . . . 16  |-  ( 1  =  1  ->  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }  =  { 1 ,  2 ,  3 } )
5044, 49ax-mp 5 . . . . . . . . . . . . . . 15  |-  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }  =  { 1 ,  2 ,  3 }
5140, 43, 503eqtri 2488 . . . . . . . . . . . . . 14  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
5251feq2i 5747 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> { p ,  q ,  r }  <->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : { 1 ,  2 ,  3 } --> { p ,  q ,  r } )
5337, 52sylibr 217 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> { p ,  q ,  r } )
54 df-3an 993 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime ) )
5526, 27, 28tpss 4150 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5654, 55bitr3i 259 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5756biimpi 199 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { p ,  q ,  r } 
C_  Prime )
5853, 57fssd 5765 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> Prime )
59 prmex 14683 . . . . . . . . . . . . 13  |-  Prime  e.  _V
60 ovex 6348 . . . . . . . . . . . . 13  |-  ( 1 ... 3 )  e. 
_V
6159, 60pm3.2i 461 . . . . . . . . . . . 12  |-  ( Prime  e.  _V  /\  ( 1 ... 3 )  e. 
_V )
62 elmapg 7516 . . . . . . . . . . . 12  |-  ( ( Prime  e.  _V  /\  ( 1 ... 3
)  e.  _V )  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  e.  ( Prime  ^m  (
1 ... 3 ) )  <->  { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> Prime ) )
6361, 62mp1i 13 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. }  e.  ( Prime  ^m  ( 1 ... 3 ) )  <->  { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> Prime ) )
6458, 63mpbird 240 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  e.  ( Prime  ^m  (
1 ... 3 ) ) )
65 fveq1 5891 . . . . . . . . . . . . 13  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  ->  ( f `  k )  =  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
6665sumeq2sdv 13825 . . . . . . . . . . . 12  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  ->  sum_ k  e.  ( 1 ... 3 ) ( f `  k
)  =  sum_ k  e.  ( 1 ... 3
) ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
6766eqeq2d 2472 . . . . . . . . . . 11  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  ->  ( ( ( p  +  q )  +  r )  = 
sum_ k  e.  ( 1 ... 3 ) ( f `  k
)  <->  ( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3 ) ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) ) )
6867adantl 472 . . . . . . . . . 10  |-  ( ( ( ( p  e. 
Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  /\  f  =  { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } )  -> 
( ( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3 ) ( f `  k )  <-> 
( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3
) ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) ) )
6951a1i 11 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( 1 ... 3 )  =  {
1 ,  2 ,  3 } )
7069sumeq1d 13822 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  sum_ k  e.  ( 1 ... 3 ) ( { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } `
 k )  = 
sum_ k  e.  {
1 ,  2 ,  3 }  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
71 fveq2 5892 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
) )
7223, 26fvtp1 6140 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  2  /\  1  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p )
7329, 32, 72mp2an 683 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p
7471, 73syl6eq 2512 . . . . . . . . . . . 12  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  p )
75 fveq2 5892 . . . . . . . . . . . . 13  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
) )
7624, 27fvtp2 6141 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  2  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q )
7729, 35, 76mp2an 683 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q
7875, 77syl6eq 2512 . . . . . . . . . . . 12  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  q )
79 fveq2 5892 . . . . . . . . . . . . 13  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
) )
8025, 28fvtp3 6142 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  3  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r )
8132, 35, 80mp2an 683 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r
8279, 81syl6eq 2512 . . . . . . . . . . . 12  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  r )
83 prmz 14681 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
8483zcnd 11075 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  CC )
85 prmz 14681 . . . . . . . . . . . . . . 15  |-  ( q  e.  Prime  ->  q  e.  ZZ )
8685zcnd 11075 . . . . . . . . . . . . . 14  |-  ( q  e.  Prime  ->  q  e.  CC )
87 prmz 14681 . . . . . . . . . . . . . . 15  |-  ( r  e.  Prime  ->  r  e.  ZZ )
8887zcnd 11075 . . . . . . . . . . . . . 14  |-  ( r  e.  Prime  ->  r  e.  CC )
8984, 86, 883anim123i 1199 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
90893expa 1215 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
91 2z 11003 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
92 3z 11004 . . . . . . . . . . . . . 14  |-  3  e.  ZZ
9341, 91, 923pm3.2i 1192 . . . . . . . . . . . . 13  |-  ( 1  e.  ZZ  /\  2  e.  ZZ  /\  3  e.  ZZ )
9493a1i 11 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( 1  e.  ZZ  /\  2  e.  ZZ  /\  3  e.  ZZ ) )
9529a1i 11 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  1  =/=  2
)
9632a1i 11 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  1  =/=  3
)
9735a1i 11 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  2  =/=  3
)
9874, 78, 82, 90, 94, 95, 96, 97sumtp 13865 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  sum_ k  e.  {
1 ,  2 ,  3 }  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( ( p  +  q )  +  r ) )
9970, 98eqtr2d 2497 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3 ) ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
10064, 68, 99rspcedvd 3167 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) ( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) )
101 eqeq1 2466 . . . . . . . . . 10  |-  ( N  =  ( ( p  +  q )  +  r )  ->  ( N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k )  <->  ( (
p  +  q )  +  r )  = 
sum_ k  e.  ( 1 ... 3 ) ( f `  k
) ) )
102101rexbidv 2913 . . . . . . . . 9  |-  ( N  =  ( ( p  +  q )  +  r )  ->  ( E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3 ) ( f `  k
)  <->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) ( ( p  +  q )  +  r )  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
103100, 102syl5ibrcom 230 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( N  =  ( ( p  +  q )  +  r )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
104103rexlimdva 2891 . . . . . . 7  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  ( E. r  e.  Prime  N  =  ( ( p  +  q )  +  r )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
105104rexlimivv 2896 . . . . . 6  |-  ( E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  N  =  ( ( p  +  q )  +  r )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) )
106105adantl 472 . . . . 5  |-  ( ( N  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  N  =  ( ( p  +  q )  +  r ) )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3 ) ( f `  k
) )
10722, 106sylbi 200 . . . 4  |-  ( N  e. GoldbachOdd  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) )
108107a1i 11 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  ( N  e. GoldbachOdd 
->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
1095, 21, 1083syld 57 . 2  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
110109com12 32 1  |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  6
)  /\  N  e. Odd  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   E.wrex 2750   _Vcvv 3057    C_ wss 3416   {ctp 3984   <.cop 3986   class class class wbr 4418   -->wf 5601   ` cfv 5605  (class class class)co 6320    ^m cmap 7503   CCcc 9568   RRcr 9569   1c1 9571    + caddc 9573    < clt 9706    <_ cle 9707   2c2 10692   3c3 10693   5c5 10695   6c6 10696   ZZcz 10971   ZZ>=cuz 11193   ...cfz 11819   sum_csu 13807   Primecprime 14677   Odd codd 38889   GoldbachOdd cgbo 38982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-oi 8056  df-card 8404  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-4 10703  df-5 10704  df-6 10705  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-fz 11820  df-fzo 11953  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-clim 13607  df-sum 13808  df-prm 14678  df-gbo 38985
This theorem is referenced by:  nnsum4primeseven  39030  wtgoldbnnsum4prm  39032  bgoldbnnsum3prm  39034
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