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Theorem nnsum4primesoddALTV 38983
Description: If the (strong) ternary Goldbach conjecture is valid, then every odd integer greater than 7 is the sum of 3 primes. (Contributed by AV, 26-Jul-2020.)
Assertion
Ref Expression
nnsum4primesoddALTV  |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= `  8 )  /\  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 nnsum4primesoddALTV
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4378 . . . . . 6  |-  ( m  =  N  ->  (
7  <  m  <->  7  <  N ) )
2 eleq1 2518 . . . . . 6  |-  ( m  =  N  ->  (
m  e. GoldbachOddALTV 
<->  N  e. GoldbachOddALTV  ) )
31, 2imbi12d 326 . . . . 5  |-  ( m  =  N  ->  (
( 7  <  m  ->  m  e. GoldbachOddALTV  )  <->  ( 7  <  N  ->  N  e. GoldbachOddALTV 
) ) )
43rspcv 3114 . . . 4  |-  ( N  e. Odd  ->  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  (
7  <  N  ->  N  e. GoldbachOddALTV  ) ) )
54adantl 472 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
8 )  /\  N  e. Odd  )  ->  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( 7  <  N  ->  N  e. GoldbachOddALTV  ) ) )
6 eluz2 11155 . . . . . 6  |-  ( N  e.  ( ZZ>= `  8
)  <->  ( 8  e.  ZZ  /\  N  e.  ZZ  /\  8  <_  N ) )
7 7lt8 10787 . . . . . . . . 9  |-  7  <  8
8 7re 10681 . . . . . . . . . . 11  |-  7  e.  RR
98a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  7  e.  RR )
10 8re 10683 . . . . . . . . . . 11  |-  8  e.  RR
1110a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  8  e.  RR )
12 zre 10931 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  RR )
13 ltletr 9712 . . . . . . . . . 10  |-  ( ( 7  e.  RR  /\  8  e.  RR  /\  N  e.  RR )  ->  (
( 7  <  8  /\  8  <_  N )  ->  7  <  N
) )
149, 11, 12, 13syl3anc 1271 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
( 7  <  8  /\  8  <_  N )  ->  7  <  N
) )
157, 14mpani 687 . . . . . . . 8  |-  ( N  e.  ZZ  ->  (
8  <_  N  ->  7  <  N ) )
1615imp 435 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  8  <_  N )  -> 
7  <  N )
17163adant1 1027 . . . . . 6  |-  ( ( 8  e.  ZZ  /\  N  e.  ZZ  /\  8  <_  N )  ->  7  <  N )
186, 17sylbi 200 . . . . 5  |-  ( N  e.  ( ZZ>= `  8
)  ->  7  <  N )
1918adantr 471 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
8 )  /\  N  e. Odd  )  ->  7  <  N )
20 pm2.27 40 . . . 4  |-  ( 7  <  N  ->  (
( 7  <  N  ->  N  e. GoldbachOddALTV  )  ->  N  e. GoldbachOddALTV 
) )
2119, 20syl 17 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
8 )  /\  N  e. Odd  )  ->  ( (
7  <  N  ->  N  e. GoldbachOddALTV  )  ->  N  e. GoldbachOddALTV  )
)
22 isgboa 38945 . . . . 5  |-  ( N  e. GoldbachOddALTV  <-> 
( N  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  N  =  ( ( p  +  q )  +  r ) ) ) )
23 1ex 9625 . . . . . . . . . . . . . . . 16  |-  1  e.  _V
24 2ex 10670 . . . . . . . . . . . . . . . 16  |-  2  e.  _V
25 3ex 10674 . . . . . . . . . . . . . . . 16  |-  3  e.  _V
26 vex 3016 . . . . . . . . . . . . . . . 16  |-  p  e. 
_V
27 vex 3016 . . . . . . . . . . . . . . . 16  |-  q  e. 
_V
28 vex 3016 . . . . . . . . . . . . . . . 16  |-  r  e. 
_V
29 1ne2 10812 . . . . . . . . . . . . . . . 16  |-  1  =/=  2
30 1re 9629 . . . . . . . . . . . . . . . . 17  |-  1  e.  RR
31 1lt3 10768 . . . . . . . . . . . . . . . . 17  |-  1  <  3
3230, 31ltneii 9734 . . . . . . . . . . . . . . . 16  |-  1  =/=  3
33 2re 10668 . . . . . . . . . . . . . . . . 17  |-  2  e.  RR
34 2lt3 10767 . . . . . . . . . . . . . . . . 17  |-  2  <  3
3533, 34ltneii 9734 . . . . . . . . . . . . . . . 16  |-  2  =/=  3
3623, 24, 25, 26, 27, 28, 29, 32, 35ftp 6060 . . . . . . . . . . . . . . 15  |-  { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } : {
1 ,  2 ,  3 } --> { p ,  q ,  r }
3736a1i 11 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : { 1 ,  2 ,  3 } --> { p ,  q ,  r } )
38 1p2e3 10724 . . . . . . . . . . . . . . . . . 18  |-  ( 1  +  2 )  =  3
3938eqcomi 2461 . . . . . . . . . . . . . . . . 17  |-  3  =  ( 1  +  2 )
4039oveq2i 6287 . . . . . . . . . . . . . . . 16  |-  ( 1 ... 3 )  =  ( 1 ... (
1  +  2 ) )
41 1z 10957 . . . . . . . . . . . . . . . . 17  |-  1  e.  ZZ
42 fztp 11843 . . . . . . . . . . . . . . . . 17  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) } )
4341, 42ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( 1 ... ( 1  +  2 ) )  =  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }
44 eqid 2452 . . . . . . . . . . . . . . . . 17  |-  1  =  1
45 id 22 . . . . . . . . . . . . . . . . . 18  |-  ( 1  =  1  ->  1  =  1 )
46 1p1e2 10712 . . . . . . . . . . . . . . . . . . 19  |-  ( 1  +  1 )  =  2
4746a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( 1  =  1  ->  (
1  +  1 )  =  2 )
4838a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( 1  =  1  ->  (
1  +  2 )  =  3 )
4945, 47, 48tpeq123d 4035 . . . . . . . . . . . . . . . . 17  |-  ( 1  =  1  ->  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }  =  { 1 ,  2 ,  3 } )
5044, 49ax-mp 5 . . . . . . . . . . . . . . . 16  |-  { 1 ,  ( 1  +  1 ) ,  ( 1  +  2 ) }  =  { 1 ,  2 ,  3 }
5140, 43, 503eqtri 2478 . . . . . . . . . . . . . . 15  |-  ( 1 ... 3 )  =  { 1 ,  2 ,  3 }
5251feq2i 5703 . . . . . . . . . . . . . 14  |-  ( {
<. 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 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> { p ,  q ,  r } )
54 df-3an 988 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime ) )
5526, 27, 28tpss 4106 . . . . . . . . . . . . . . 15  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5654, 55bitr3i 259 . . . . . . . . . . . . . 14  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  <->  { p ,  q ,  r }  C_  Prime )
5756biimpi 199 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { p ,  q ,  r } 
C_  Prime )
5853, 57fssd 5721 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. } : ( 1 ... 3 ) --> Prime )
59 prmex 14639 . . . . . . . . . . . . . 14  |-  Prime  e.  _V
60 ovex 6304 . . . . . . . . . . . . . 14  |-  ( 1 ... 3 )  e. 
_V
6159, 60pm3.2i 461 . . . . . . . . . . . . 13  |-  ( Prime  e.  _V  /\  ( 1 ... 3 )  e. 
_V )
62 elmapg 7472 . . . . . . . . . . . . 13  |-  ( ( 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 . . . . . . . . . . . 12  |-  ( ( ( 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 . . . . . . . . . . 11  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  e.  ( Prime  ^m  (
1 ... 3 ) ) )
65 fveq1 5847 . . . . . . . . . . . . . 14  |-  ( f  =  { <. 1 ,  p >. ,  <. 2 ,  q >. ,  <. 3 ,  r >. }  ->  ( f `  k )  =  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
) )
6665sumeq2sdv 13781 . . . . . . . . . . . . 13  |-  ( 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 2462 . . . . . . . . . . . 12  |-  ( 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 . . . . . . . . . . 11  |-  ( ( ( ( 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 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( 1 ... 3 )  =  {
1 ,  2 ,  3 } )
7069sumeq1d 13778 . . . . . . . . . . . 12  |-  ( ( ( 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 5848 . . . . . . . . . . . . . 14  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
) )
7223, 26fvtp1 6096 . . . . . . . . . . . . . . 15  |-  ( ( 1  =/=  2  /\  1  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p )
7329, 32, 72mp2an 683 . . . . . . . . . . . . . 14  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  1
)  =  p
7471, 73syl6eq 2502 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  p )
75 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
) )
7624, 27fvtp2 6097 . . . . . . . . . . . . . . 15  |-  ( ( 1  =/=  2  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q )
7729, 35, 76mp2an 683 . . . . . . . . . . . . . 14  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  2
)  =  q
7875, 77syl6eq 2502 . . . . . . . . . . . . 13  |-  ( k  =  2  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  q )
79 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
) )
8025, 28fvtp3 6098 . . . . . . . . . . . . . . 15  |-  ( ( 1  =/=  3  /\  2  =/=  3 )  ->  ( { <. 1 ,  p >. , 
<. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r )
8132, 35, 80mp2an 683 . . . . . . . . . . . . . 14  |-  ( {
<. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  3
)  =  r
8279, 81syl6eq 2502 . . . . . . . . . . . . 13  |-  ( k  =  3  ->  ( { <. 1 ,  p >. ,  <. 2 ,  q
>. ,  <. 3 ,  r >. } `  k
)  =  r )
83 prmz 14637 . . . . . . . . . . . . . . . 16  |-  ( p  e.  Prime  ->  p  e.  ZZ )
8483zcnd 11031 . . . . . . . . . . . . . . 15  |-  ( p  e.  Prime  ->  p  e.  CC )
85 prmz 14637 . . . . . . . . . . . . . . . 16  |-  ( q  e.  Prime  ->  q  e.  ZZ )
8685zcnd 11031 . . . . . . . . . . . . . . 15  |-  ( q  e.  Prime  ->  q  e.  CC )
87 prmz 14637 . . . . . . . . . . . . . . . 16  |-  ( r  e.  Prime  ->  r  e.  ZZ )
8887zcnd 11031 . . . . . . . . . . . . . . 15  |-  ( r  e.  Prime  ->  r  e.  CC )
8984, 86, 883anim123i 1194 . . . . . . . . . . . . . 14  |-  ( ( p  e.  Prime  /\  q  e.  Prime  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
90893expa 1210 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( p  e.  CC  /\  q  e.  CC  /\  r  e.  CC ) )
91 2z 10959 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
92 3z 10960 . . . . . . . . . . . . . . 15  |-  3  e.  ZZ
9341, 91, 923pm3.2i 1187 . . . . . . . . . . . . . 14  |-  ( 1  e.  ZZ  /\  2  e.  ZZ  /\  3  e.  ZZ )
9493a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( 1  e.  ZZ  /\  2  e.  ZZ  /\  3  e.  ZZ ) )
9529a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  1  =/=  2
)
9632a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  1  =/=  3
)
9735a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  2  =/=  3
)
9874, 78, 82, 90, 94, 95, 96, 97sumtp 13821 . . . . . . . . . . . 12  |-  ( ( ( 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 2487 . . . . . . . . . . 11  |-  ( ( ( 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 3123 . . . . . . . . . 10  |-  ( ( ( 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 2456 . . . . . . . . . . 11  |-  ( N  =  ( ( p  +  q )  +  r )  ->  ( N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k )  <->  ( (
p  +  q )  +  r )  = 
sum_ k  e.  ( 1 ... 3 ) ( f `  k
) ) )
102101rexbidv 2873 . . . . . . . . . 10  |-  ( 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 . . . . . . . . 9  |-  ( ( ( 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 ) ) )
104103adantld 473 . . . . . . . 8  |-  ( ( ( p  e.  Prime  /\  q  e.  Prime )  /\  r  e.  Prime )  ->  ( ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd  )  /\  N  =  (
( p  +  q )  +  r ) )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
105104rexlimdva 2852 . . . . . . 7  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  ( E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  N  =  ( ( p  +  q )  +  r ) )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
106105rexlimivv 2857 . . . . . 6  |-  ( E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  N  =  ( ( p  +  q )  +  r ) )  ->  E. f  e.  ( Prime  ^m  (
1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) )
107106adantl 472 . . . . 5  |-  ( ( N  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  N  =  ( ( p  +  q )  +  r ) ) )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3 ) ( f `  k
) )
10822, 107sylbi 200 . . . 4  |-  ( N  e. GoldbachOddALTV  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) )
109108a1i 11 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
8 )  /\  N  e. Odd  )  ->  ( N  e. GoldbachOddALTV  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
1105, 21, 1093syld 57 . 2  |-  ( ( N  e.  ( ZZ>= ` 
8 )  /\  N  e. Odd  )  ->  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  E. f  e.  ( Prime  ^m  ( 1 ... 3 ) ) N  =  sum_ k  e.  ( 1 ... 3
) ( f `  k ) ) )
111110com12 32 1  |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= `  8 )  /\  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 986    = wceq 1448    e. wcel 1891    =/= wne 2622   A.wral 2737   E.wrex 2738   _Vcvv 3013    C_ wss 3372   {ctp 3940   <.cop 3942   class class class wbr 4374   -->wf 5557   ` cfv 5561  (class class class)co 6276    ^m cmap 7459   CCcc 9524   RRcr 9525   1c1 9527    + caddc 9529    < clt 9662    <_ cle 9663   2c2 10648   3c3 10649   7c7 10653   8c8 10654   ZZcz 10927   ZZ>=cuz 11149   ...cfz 11775   sum_csu 13763   Primecprime 14633   Odd codd 38845   GoldbachOddALTV cgboa 38939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-inf2 8133  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603  ax-pre-sup 9604
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-fal 1454  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-se 4772  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-isom 5570  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-sup 7943  df-oi 8012  df-card 8360  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-n0 10860  df-z 10928  df-uz 11150  df-rp 11293  df-fz 11776  df-fzo 11909  df-seq 12208  df-exp 12267  df-hash 12510  df-cj 13173  df-re 13174  df-im 13175  df-sqrt 13309  df-abs 13310  df-clim 13563  df-sum 13764  df-prm 14634  df-gboa 38942
This theorem is referenced by:  nnsum4primesevenALTV  38987
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