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Theorem nnsum4primesodd 38761
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 4427 . . . . . 6  |-  ( m  =  N  ->  (
5  <  m  <->  5  <  N ) )
2 eleq1 2495 . . . . . 6  |-  ( m  =  N  ->  (
m  e. GoldbachOdd  <->  N  e. GoldbachOdd  ) )
31, 2imbi12d 321 . . . . 5  |-  ( m  =  N  ->  (
( 5  <  m  ->  m  e. GoldbachOdd  )  <->  ( 5  <  N  ->  N  e. GoldbachOdd  ) ) )
43rspcv 3178 . . . 4  |-  ( N  e. Odd  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  (
5  <  N  ->  N  e. GoldbachOdd  ) ) )
54adantl 467 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
6 )  /\  N  e. Odd  )  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( 5  < 
N  ->  N  e. GoldbachOdd  ) ) )
6 eluz2 11172 . . . . . 6  |-  ( N  e.  ( ZZ>= `  6
)  <->  ( 6  e.  ZZ  /\  N  e.  ZZ  /\  6  <_  N ) )
7 5lt6 10793 . . . . . . . . 9  |-  5  <  6
8 5re 10695 . . . . . . . . . . 11  |-  5  e.  RR
98a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  5  e.  RR )
10 6re 10697 . . . . . . . . . . 11  |-  6  e.  RR
1110a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  6  e.  RR )
12 zre 10948 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  RR )
13 ltletr 9732 . . . . . . . . . 10  |-  ( ( 5  e.  RR  /\  6  e.  RR  /\  N  e.  RR )  ->  (
( 5  <  6  /\  6  <_  N )  ->  5  <  N
) )
149, 11, 12, 13syl3anc 1264 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 5  <  6  /\  6  <_  N )  ->  5  <  N
) )
157, 14mpani 680 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
6  <_  N  ->  5  <  N ) )
1615imp 430 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  6  <_  N )  -> 
5  <  N )
17163adant1 1023 . . . . . 6  |-  ( ( 6  e.  ZZ  /\  N  e.  ZZ  /\  6  <_  N )  ->  5  <  N )
186, 17sylbi 198 . . . . 5  |-  ( N  e.  ( ZZ>= `  6
)  ->  5  <  N )
1918adantr 466 . . . 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 38723 . . . . 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 9645 . . . . . . . . . . . . . . 15  |-  1  e.  _V
24 2ex 10688 . . . . . . . . . . . . . . 15  |-  2  e.  _V
25 3ex 10692 . . . . . . . . . . . . . . 15  |-  3  e.  _V
26 vex 3083 . . . . . . . . . . . . . . 15  |-  p  e. 
_V
27 vex 3083 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
28 vex 3083 . . . . . . . . . . . . . . 15  |-  r  e. 
_V
29 1ne2 10829 . . . . . . . . . . . . . . 15  |-  1  =/=  2
30 1re 9649 . . . . . . . . . . . . . . . 16  |-  1  e.  RR
31 1lt3 10785 . . . . . . . . . . . . . . . 16  |-  1  <  3
3230, 31ltneii 9754 . . . . . . . . . . . . . . 15  |-  1  =/=  3
33 2re 10686 . . . . . . . . . . . . . . . 16  |-  2  e.  RR
34 2lt3 10784 . . . . . . . . . . . . . . . 16  |-  2  <  3
3533, 34ltneii 9754 . . . . . . . . . . . . . . 15  |-  2  =/=  3
3623, 24, 25, 26, 27, 28, 29, 32, 35ftp 6090 . . . . . . . . . . . . . 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 10741 . . . . . . . . . . . . . . . . 17  |-  ( 1  +  2 )  =  3
3938eqcomi 2435 . . . . . . . . . . . . . . . 16  |-  3  =  ( 1  +  2 )
4039oveq2i 6316 . . . . . . . . . . . . . . 15  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
41 1z 10974 . . . . . . . . . . . . . . . 16  |-  1  e.  ZZ
42 fztp 11859 . . . . . . . . . . . . . . . 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 2422 . . . . . . . . . . . . . . . 16  |-  1  =  1
45 id 22 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  1  =  1 )
46 1p1e2 10730 . . . . . . . . . . . . . . . . . 18  |-  ( 1  +  1 )  =  2
4746a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  (
1  +  1 )  =  2 )
4838a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  (
1  +  2 )  =  3 )
4945, 47, 48tpeq123d 4094 . . . . . . . . . . . . . . . 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 2455 . . . . . . . . . . . . . 14  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
5251feq2i 5739 . . . . . . . . . . . . 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 215 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> { p ,  q ,  r } )
54 df-3an 984 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime ) )
5526, 27, 28tpss 4165 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5654, 55bitr3i 254 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5756biimpi 197 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { p ,  q ,  r } 
C_  Prime )
5853, 57fssd 5755 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> Prime )
59 prmex 14627 . . . . . . . . . . . . 13  |-  Prime  e.  _V
60 ovex 6333 . . . . . . . . . . . . 13  |-  ( 1 ... 3 )  e. 
_V
6159, 60pm3.2i 456 . . . . . . . . . . . 12  |-  ( Prime  e.  _V  /\  ( 1 ... 3 )  e. 
_V )
62 elmapg 7496 . . . . . . . . . . . 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 235 . . . . . . . . . 10  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  e.  ( Prime  ^m  (
1 ... 3 ) ) )
65 fveq1 5880 . . . . . . . . . . . . 13  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  ->  ( f `  k )  =  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
6665sumeq2sdv 13769 . . . . . . . . . . . 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 2436 . . . . . . . . . . 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 467 . . . . . . . . . 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 13766 . . . . . . . . . . 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 5881 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
) )
7223, 26fvtp1 6126 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  2  /\  1  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p )
7329, 32, 72mp2an 676 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p
7471, 73syl6eq 2479 . . . . . . . . . . . 12  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  p )
75 fveq2 5881 . . . . . . . . . . . . 13  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
) )
7624, 27fvtp2 6127 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  2  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q )
7729, 35, 76mp2an 676 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q
7875, 77syl6eq 2479 . . . . . . . . . . . 12  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  q )
79 fveq2 5881 . . . . . . . . . . . . 13  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
) )
8025, 28fvtp3 6128 . . . . . . . . . . . . . 14  |-  ( ( 1  =/=  3  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r )
8132, 35, 80mp2an 676 . . . . . . . . . . . . 13  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r
8279, 81syl6eq 2479 . . . . . . . . . . . 12  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  r )
83 prmz 14625 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  ZZ )
8483zcnd 11048 . . . . . . . . . . . . . 14  |-  ( p  e.  Prime  ->  p  e.  CC )
85 prmz 14625 . . . . . . . . . . . . . . 15  |-  ( q  e.  Prime  ->  q  e.  ZZ )
8685zcnd 11048 . . . . . . . . . . . . . 14  |-  ( q  e.  Prime  ->  q  e.  CC )
87 prmz 14625 . . . . . . . . . . . . . . 15  |-  ( r  e.  Prime  ->  r  e.  ZZ )
8887zcnd 11048 . . . . . . . . . . . . . 14  |-  ( r  e.  Prime  ->  r  e.  CC )
8984, 86, 883anim123i 1190 . . . . . . . . . . . . 13  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
90893expa 1205 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
91 2z 10976 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
92 3z 10977 . . . . . . . . . . . . . 14  |-  3  e.  ZZ
9341, 91, 923pm3.2i 1183 . . . . . . . . . . . . 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 13809 . . . . . . . . . . 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 2464 . . . . . . . . . 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 3187 . . . . . . . . 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 2426 . . . . . . . . . 10  |-  ( N  =  ( ( p  +  q )  +  r )  ->  ( N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k )  <->  ( (
p  +  q )  +  r )  = 
sum_ k  e.  ( 1 ... 3 ) ( f `  k
) ) )
102101rexbidv 2936 . . . . . . . . 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 225 . . . . . . . 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 2914 . . . . . . 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 2919 . . . . . 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 467 . . . . 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 198 . . . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080    C_ wss 3436   {ctp 4002   <.cop 4004   class class class wbr 4423   -->wf 5597   ` cfv 5601  (class class class)co 6305    ^m cmap 7483   CCcc 9544   RRcr 9545   1c1 9547    + caddc 9549    < clt 9682    <_ cle 9683   2c2 10666   3c3 10667   5c5 10669   6c6 10670   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791   sum_csu 13751   Primecprime 14621   Odd codd 38624   GoldbachOdd cgbo 38717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-inf2 8155  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-sup 7965  df-oi 8034  df-card 8381  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-seq 12220  df-exp 12279  df-hash 12522  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13551  df-sum 13752  df-prm 14622  df-gbo 38720
This theorem is referenced by:  nnsum4primeseven  38765  wtgoldbnnsum4prm  38767  bgoldbnnsum3prm  38769
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