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Theorem evengpop3 38649
Description: If the (weak) ternary Goldbach conjecture is valid, then every even integer greater than 8 is the sum of an odd Goldbach number and 3. (Contributed by AV, 24-Jul-2020.)
Assertion
Ref Expression
evengpop3  |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9
)  /\  N  e. Even  )  ->  E. o  e. GoldbachOdd  N  =  ( o  +  3 ) ) )
Distinct variable groups:    m, N    o, N

Proof of Theorem evengpop3
StepHypRef Expression
1 3odd 38591 . . . . . . . 8  |-  3  e. Odd
21a1i 11 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  9
)  ->  3  e. Odd  )
32anim1i 571 . . . . . 6  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( 3  e. Odd  /\  N  e. Even  ) )
43ancomd 453 . . . . 5  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( N  e. Even  /\  3  e. Odd  )
)
5 emoo 38587 . . . . 5  |-  ( ( N  e. Even  /\  3  e. Odd  )  ->  ( N  -  3 )  e. Odd 
)
64, 5syl 17 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( N  -  3 )  e. Odd 
)
7 breq2 4425 . . . . . 6  |-  ( m  =  ( N  - 
3 )  ->  (
5  <  m  <->  5  <  ( N  -  3 ) ) )
8 eleq1 2495 . . . . . 6  |-  ( m  =  ( N  - 
3 )  ->  (
m  e. GoldbachOdd  <->  ( N  - 
3 )  e. GoldbachOdd  ) )
97, 8imbi12d 322 . . . . 5  |-  ( m  =  ( N  - 
3 )  ->  (
( 5  <  m  ->  m  e. GoldbachOdd  )  <->  ( 5  <  ( N  - 
3 )  ->  ( N  -  3 )  e. GoldbachOdd  ) ) )
109adantl 468 . . . 4  |-  ( ( ( N  e.  (
ZZ>= `  9 )  /\  N  e. Even  )  /\  m  =  ( N  - 
3 ) )  -> 
( ( 5  < 
m  ->  m  e. GoldbachOdd  )  <-> 
( 5  <  ( N  -  3 )  ->  ( N  - 
3 )  e. GoldbachOdd  ) ) )
116, 10rspcdv 3186 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( 5  < 
( N  -  3 )  ->  ( N  -  3 )  e. GoldbachOdd  ) ) )
12 eluz2 11167 . . . . . 6  |-  ( N  e.  ( ZZ>= `  9
)  <->  ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N ) )
13 5p3e8 10750 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
14 8p1e9 10742 . . . . . . . . . 10  |-  ( 8  +  1 )  =  9
15 9cn 10699 . . . . . . . . . . 11  |-  9  e.  CC
16 ax-1cn 9599 . . . . . . . . . . 11  |-  1  e.  CC
17 8cn 10697 . . . . . . . . . . 11  |-  8  e.  CC
1815, 16, 17subadd2i 9965 . . . . . . . . . 10  |-  ( ( 9  -  1 )  =  8  <->  ( 8  +  1 )  =  9 )
1914, 18mpbir 213 . . . . . . . . 9  |-  ( 9  -  1 )  =  8
2013, 19eqtr4i 2455 . . . . . . . 8  |-  ( 5  +  3 )  =  ( 9  -  1 )
21 zlem1lt 10990 . . . . . . . . 9  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ )  ->  ( 9  <_  N  <->  ( 9  -  1 )  <  N ) )
2221biimp3a 1365 . . . . . . . 8  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N )  ->  (
9  -  1 )  <  N )
2320, 22syl5eqbr 4455 . . . . . . 7  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N )  ->  (
5  +  3 )  <  N )
24 5re 10690 . . . . . . . . . . 11  |-  5  e.  RR
2524a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  5  e.  RR )
26 3re 10685 . . . . . . . . . . 11  |-  3  e.  RR
2726a1i 11 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  3  e.  RR )
28 zre 10943 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  N  e.  RR )
2925, 27, 283jca 1186 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
5  e.  RR  /\  3  e.  RR  /\  N  e.  RR ) )
30293ad2ant2 1028 . . . . . . . 8  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N )  ->  (
5  e.  RR  /\  3  e.  RR  /\  N  e.  RR ) )
31 ltaddsub 10090 . . . . . . . 8  |-  ( ( 5  e.  RR  /\  3  e.  RR  /\  N  e.  RR )  ->  (
( 5  +  3 )  <  N  <->  5  <  ( N  -  3 ) ) )
3230, 31syl 17 . . . . . . 7  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N )  ->  (
( 5  +  3 )  <  N  <->  5  <  ( N  -  3 ) ) )
3323, 32mpbid 214 . . . . . 6  |-  ( ( 9  e.  ZZ  /\  N  e.  ZZ  /\  9  <_  N )  ->  5  <  ( N  -  3 ) )
3412, 33sylbi 199 . . . . 5  |-  ( N  e.  ( ZZ>= `  9
)  ->  5  <  ( N  -  3 ) )
3534adantr 467 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  5  <  ( N  -  3 ) )
36 simpr 463 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  9 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOdd  )  ->  ( N  -  3 )  e. GoldbachOdd  )
37 oveq1 6310 . . . . . . . 8  |-  ( o  =  ( N  - 
3 )  ->  (
o  +  3 )  =  ( ( N  -  3 )  +  3 ) )
3837eqeq2d 2437 . . . . . . 7  |-  ( o  =  ( N  - 
3 )  ->  ( N  =  ( o  +  3 )  <->  N  =  ( ( N  - 
3 )  +  3 ) ) )
3938adantl 468 . . . . . 6  |-  ( ( ( ( N  e.  ( ZZ>= `  9 )  /\  N  e. Even  )  /\  ( N  -  3
)  e. GoldbachOdd  )  /\  o  =  ( N  - 
3 ) )  -> 
( N  =  ( o  +  3 )  <-> 
N  =  ( ( N  -  3 )  +  3 ) ) )
40 eluzelcn 11172 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  9
)  ->  N  e.  CC )
41 3cn 10686 . . . . . . . . . . 11  |-  3  e.  CC
4241a1i 11 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  9
)  ->  3  e.  CC )
4340, 42jca 535 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  9
)  ->  ( N  e.  CC  /\  3  e.  CC ) )
4443adantr 467 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( N  e.  CC  /\  3  e.  CC ) )
4544adantr 467 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= `  9 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOdd  )  ->  ( N  e.  CC  /\  3  e.  CC ) )
46 npcan 9886 . . . . . . . 8  |-  ( ( N  e.  CC  /\  3  e.  CC )  ->  ( ( N  - 
3 )  +  3 )  =  N )
4746eqcomd 2431 . . . . . . 7  |-  ( ( N  e.  CC  /\  3  e.  CC )  ->  N  =  ( ( N  -  3 )  +  3 ) )
4845, 47syl 17 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= `  9 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOdd  )  ->  N  =  ( ( N  - 
3 )  +  3 ) )
4936, 39, 48rspcedvd 3188 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= `  9 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOdd  )  ->  E. o  e. GoldbachOdd 
N  =  ( o  +  3 ) )
5049ex 436 . . . 4  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( ( N  -  3 )  e. GoldbachOdd  ->  E. o  e. GoldbachOdd  N  =  ( o  +  3 ) ) )
5135, 50embantd 57 . . 3  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( (
5  <  ( N  -  3 )  -> 
( N  -  3 )  e. GoldbachOdd  )  ->  E. o  e. GoldbachOdd 
N  =  ( o  +  3 ) ) )
5211, 51syld 46 . 2  |-  ( ( N  e.  ( ZZ>= ` 
9 )  /\  N  e. Even  )  ->  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  E. o  e. GoldbachOdd  N  =  ( o  +  3 ) ) )
5352com12 33 1  |-  ( A. m  e. Odd  ( 5  <  m  ->  m  e. GoldbachOdd  )  ->  ( ( N  e.  ( ZZ>= `  9
)  /\  N  e. Even  )  ->  E. o  e. GoldbachOdd  N  =  ( o  +  3 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   E.wrex 2777   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   CCcc 9539   RRcr 9540   1c1 9542    + caddc 9544    < clt 9677    <_ cle 9678    - cmin 9862   3c3 10662   5c5 10664   8c8 10667   9c9 10668   ZZcz 10939   ZZ>=cuz 11161   Even ceven 38509   Odd codd 38510   GoldbachOdd cgbo 38603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-n0 10872  df-z 10940  df-uz 11162  df-even 38511  df-odd 38512
This theorem is referenced by:  nnsum4primeseven  38651
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