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Theorem evengpoap3 38894
Description: If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.)
Assertion
Ref Expression
evengpoap3  |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
Distinct variable groups:    m, N    o, N

Proof of Theorem evengpoap3
StepHypRef Expression
1 3odd 38835 . . . . . . . 8  |-  3  e. Odd
21a1i 11 . . . . . . 7  |-  ( N  e.  ( ZZ>= ` ; 1 2 )  -> 
3  e. Odd  )
32anim1i 572 . . . . . 6  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( 3  e. Odd  /\  N  e. Even  ) )
43ancomd 453 . . . . 5  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( N  e. Even  /\  3  e. Odd  )
)
5 emoo 38831 . . . . 5  |-  ( ( N  e. Even  /\  3  e. Odd  )  ->  ( N  -  3 )  e. Odd 
)
64, 5syl 17 . . . 4  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( N  -  3 )  e. Odd 
)
7 breq2 4406 . . . . . 6  |-  ( m  =  ( N  - 
3 )  ->  (
7  <  m  <->  7  <  ( N  -  3 ) ) )
8 eleq1 2517 . . . . . 6  |-  ( m  =  ( N  - 
3 )  ->  (
m  e. GoldbachOddALTV 
<->  ( N  -  3 )  e. GoldbachOddALTV  ) )
97, 8imbi12d 322 . . . . 5  |-  ( m  =  ( N  - 
3 )  ->  (
( 7  <  m  ->  m  e. GoldbachOddALTV  )  <->  ( 7  <  ( N  - 
3 )  ->  ( N  -  3 )  e. GoldbachOddALTV  ) ) )
109adantl 468 . . . 4  |-  ( ( ( N  e.  (
ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  m  =  ( N  - 
3 ) )  -> 
( ( 7  < 
m  ->  m  e. GoldbachOddALTV  )  <->  ( 7  <  ( N  -  3 )  -> 
( N  -  3 )  e. GoldbachOddALTV  ) ) )
116, 10rspcdv 3153 . . 3  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( 7  <  ( N  -  3 )  ->  ( N  - 
3 )  e. GoldbachOddALTV  ) ) )
12 eluz2 11165 . . . . . 6  |-  ( N  e.  ( ZZ>= ` ; 1 2 )  <->  (; 1 2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N ) )
13 7p3e10 10755 . . . . . . . 8  |-  ( 7  +  3 )  =  10
14 dec10p 11080 . . . . . . . . . . . . 13  |-  ( 10  +  2 )  = ; 1
2
1514eqcomi 2460 . . . . . . . . . . . 12  |- ; 1 2  =  ( 10  +  2 )
1615breq1i 4409 . . . . . . . . . . 11  |-  (; 1 2  <_  N  <->  ( 10  +  2 )  <_  N )
17 2pos 10701 . . . . . . . . . . . . 13  |-  0  <  2
18 2re 10679 . . . . . . . . . . . . . 14  |-  2  e.  RR
19 10re 10698 . . . . . . . . . . . . . 14  |-  10  e.  RR
2018, 19ltaddposi 10163 . . . . . . . . . . . . 13  |-  ( 0  <  2  <->  10  <  ( 10  +  2 ) )
2117, 20mpbi 212 . . . . . . . . . . . 12  |-  10  <  ( 10  +  2 )
2219a1i 11 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  10  e.  RR )
2318a1i 11 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  2  e.  RR )
2422, 23readdcld 9670 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  ( 10  +  2 )  e.  RR )
25 zre 10941 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  RR )
26 ltletr 9725 . . . . . . . . . . . . 13  |-  ( ( 10  e.  RR  /\  ( 10  +  2
)  e.  RR  /\  N  e.  RR )  ->  ( ( 10  <  ( 10  +  2 )  /\  ( 10  + 
2 )  <_  N
)  ->  10  <  N ) )
2722, 24, 25, 26syl3anc 1268 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
( 10  <  ( 10  +  2 )  /\  ( 10  +  2
)  <_  N )  ->  10  <  N ) )
2821, 27mpani 682 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
( 10  +  2 )  <_  N  ->  10 
<  N ) )
2916, 28syl5bi 221 . . . . . . . . . 10  |-  ( N  e.  ZZ  ->  (; 1 2  <_  N  ->  10  <  N ) )
3029imp 431 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\ ; 1 2  <_  N )  ->  10  <  N )
31303adant1 1026 . . . . . . . 8  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  10  <  N )
3213, 31syl5eqbr 4436 . . . . . . 7  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  ( 7  +  3 )  < 
N )
33 7re 10692 . . . . . . . . 9  |-  7  e.  RR
3433a1i 11 . . . . . . . 8  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  7  e.  RR )
35 3re 10683 . . . . . . . . 9  |-  3  e.  RR
3635a1i 11 . . . . . . . 8  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  3  e.  RR )
37253ad2ant2 1030 . . . . . . . 8  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  N  e.  RR )
3834, 36, 37ltaddsubd 10213 . . . . . . 7  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  ( (
7  +  3 )  <  N  <->  7  <  ( N  -  3 ) ) )
3932, 38mpbid 214 . . . . . 6  |-  ( (; 1
2  e.  ZZ  /\  N  e.  ZZ  /\ ; 1 2  <_  N
)  ->  7  <  ( N  -  3 ) )
4012, 39sylbi 199 . . . . 5  |-  ( N  e.  ( ZZ>= ` ; 1 2 )  -> 
7  <  ( N  -  3 ) )
4140adantr 467 . . . 4  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  7  <  ( N  -  3 ) )
42 simpr 463 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOddALTV  )  ->  ( N  -  3 )  e. GoldbachOddALTV  )
43 oveq1 6297 . . . . . . . 8  |-  ( o  =  ( N  - 
3 )  ->  (
o  +  3 )  =  ( ( N  -  3 )  +  3 ) )
4443eqeq2d 2461 . . . . . . 7  |-  ( o  =  ( N  - 
3 )  ->  ( N  =  ( o  +  3 )  <->  N  =  ( ( N  - 
3 )  +  3 ) ) )
4544adantl 468 . . . . . 6  |-  ( ( ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOddALTV  )  /\  o  =  ( N  -  3 ) )  ->  ( N  =  ( o  +  3 )  <->  N  =  ( ( N  - 
3 )  +  3 ) ) )
46 eluzelcn 11170 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= ` ; 1 2 )  ->  N  e.  CC )
47 3cn 10684 . . . . . . . . . 10  |-  3  e.  CC
4846, 47jctir 541 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= ` ; 1 2 )  -> 
( N  e.  CC  /\  3  e.  CC ) )
4948adantr 467 . . . . . . . 8  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( N  e.  CC  /\  3  e.  CC ) )
5049adantr 467 . . . . . . 7  |-  ( ( ( N  e.  (
ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOddALTV  )  ->  ( N  e.  CC  /\  3  e.  CC ) )
51 npcan 9884 . . . . . . . 8  |-  ( ( N  e.  CC  /\  3  e.  CC )  ->  ( ( N  - 
3 )  +  3 )  =  N )
5251eqcomd 2457 . . . . . . 7  |-  ( ( N  e.  CC  /\  3  e.  CC )  ->  N  =  ( ( N  -  3 )  +  3 ) )
5350, 52syl 17 . . . . . 6  |-  ( ( ( N  e.  (
ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOddALTV  )  ->  N  =  ( ( N  - 
3 )  +  3 ) )
5442, 45, 53rspcedvd 3155 . . . . 5  |-  ( ( ( N  e.  (
ZZ>= ` ; 1 2 )  /\  N  e. Even  )  /\  ( N  -  3 )  e. GoldbachOddALTV  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) )
5554ex 436 . . . 4  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( ( N  -  3 )  e. GoldbachOddALTV  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
5641, 55embantd 56 . . 3  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( (
7  <  ( N  -  3 )  -> 
( N  -  3 )  e. GoldbachOddALTV  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
5711, 56syld 45 . 2  |-  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even 
)  ->  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
5857com12 32 1  |-  ( A. m  e. Odd  ( 7  <  m  ->  m  e. GoldbachOddALTV  )  ->  ( ( N  e.  ( ZZ>= ` ; 1 2 )  /\  N  e. Even  )  ->  E. o  e. GoldbachOddALTV  N  =  ( o  +  3 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   E.wrex 2738   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   2c2 10659   3c3 10660   7c7 10664   10c10 10667   ZZcz 10937  ;cdc 11051   ZZ>=cuz 11159   Even ceven 38753   Odd codd 38754   GoldbachOddALTV cgboa 38848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-even 38755  df-odd 38756
This theorem is referenced by:  nnsum4primesevenALTV  38896
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