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Theorem bgoldbst 38879
Description: If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)
Assertion
Ref Expression
bgoldbst  |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  A. m  e. Odd  (
7  <  m  ->  m  e. GoldbachOddALTV  ) )
Distinct variable group:    m, n

Proof of Theorem bgoldbst
Dummy variables  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 459 . . . . . . 7  |-  ( ( m  e. Odd  /\  7  <  m )  ->  m  e. Odd  )
2 3odd 38835 . . . . . . 7  |-  3  e. Odd
31, 2jctir 541 . . . . . 6  |-  ( ( m  e. Odd  /\  7  <  m )  ->  (
m  e. Odd  /\  3  e. Odd  ) )
4 omoeALTV 38814 . . . . . 6  |-  ( ( m  e. Odd  /\  3  e. Odd  )  ->  ( m  -  3 )  e. Even 
)
5 breq2 4406 . . . . . . . 8  |-  ( n  =  ( m  - 
3 )  ->  (
4  <  n  <->  4  <  ( m  -  3 ) ) )
6 eleq1 2517 . . . . . . . 8  |-  ( n  =  ( m  - 
3 )  ->  (
n  e. GoldbachEven  <->  ( m  - 
3 )  e. GoldbachEven  ) )
75, 6imbi12d 322 . . . . . . 7  |-  ( n  =  ( m  - 
3 )  ->  (
( 4  <  n  ->  n  e. GoldbachEven  )  <->  ( 4  <  ( m  - 
3 )  ->  (
m  -  3 )  e. GoldbachEven  ) ) )
87rspcv 3146 . . . . . 6  |-  ( ( m  -  3 )  e. Even  ->  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  (
4  <  ( m  -  3 )  -> 
( m  -  3 )  e. GoldbachEven  ) ) )
93, 4, 83syl 18 . . . . 5  |-  ( ( m  e. Odd  /\  7  <  m )  ->  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  ( 4  <  ( m  - 
3 )  ->  (
m  -  3 )  e. GoldbachEven  ) ) )
10 4p3e7 10745 . . . . . . . . 9  |-  ( 4  +  3 )  =  7
1110breq1i 4409 . . . . . . . 8  |-  ( ( 4  +  3 )  <  m  <->  7  <  m )
12 4re 10686 . . . . . . . . . . 11  |-  4  e.  RR
1312a1i 11 . . . . . . . . . 10  |-  ( m  e. Odd  ->  4  e.  RR )
14 3re 10683 . . . . . . . . . . 11  |-  3  e.  RR
1514a1i 11 . . . . . . . . . 10  |-  ( m  e. Odd  ->  3  e.  RR )
16 oddz 38760 . . . . . . . . . . 11  |-  ( m  e. Odd  ->  m  e.  ZZ )
1716zred 11040 . . . . . . . . . 10  |-  ( m  e. Odd  ->  m  e.  RR )
1813, 15, 17ltaddsubd 10213 . . . . . . . . 9  |-  ( m  e. Odd  ->  ( ( 4  +  3 )  < 
m  <->  4  <  (
m  -  3 ) ) )
1918biimpd 211 . . . . . . . 8  |-  ( m  e. Odd  ->  ( ( 4  +  3 )  < 
m  ->  4  <  ( m  -  3 ) ) )
2011, 19syl5bir 222 . . . . . . 7  |-  ( m  e. Odd  ->  ( 7  < 
m  ->  4  <  ( m  -  3 ) ) )
2120imp 431 . . . . . 6  |-  ( ( m  e. Odd  /\  7  <  m )  ->  4  <  ( m  -  3 ) )
22 pm2.27 40 . . . . . 6  |-  ( 4  <  ( m  - 
3 )  ->  (
( 4  <  (
m  -  3 )  ->  ( m  - 
3 )  e. GoldbachEven  )  -> 
( m  -  3 )  e. GoldbachEven  ) )
2321, 22syl 17 . . . . 5  |-  ( ( m  e. Odd  /\  7  <  m )  ->  (
( 4  <  (
m  -  3 )  ->  ( m  - 
3 )  e. GoldbachEven  )  -> 
( m  -  3 )  e. GoldbachEven  ) )
24 isgbe 38852 . . . . . 6  |-  ( ( m  -  3 )  e. GoldbachEven 
<->  ( ( m  - 
3 )  e. Even  /\  E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  (
m  -  3 )  =  ( p  +  q ) ) ) )
25 3prm 14641 . . . . . . . . . . . . . 14  |-  3  e.  Prime
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  ->  3  e.  Prime )
27 eleq1 2517 . . . . . . . . . . . . . . . 16  |-  ( r  =  3  ->  (
r  e. Odd  <->  3  e. Odd  )
)
28273anbi3d 1345 . . . . . . . . . . . . . . 15  |-  ( r  =  3  ->  (
( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  <->  ( p  e. Odd  /\  q  e. Odd  /\  3  e. Odd  ) ) )
29 oveq2 6298 . . . . . . . . . . . . . . . 16  |-  ( r  =  3  ->  (
( p  +  q )  +  r )  =  ( ( p  +  q )  +  3 ) )
3029eqeq2d 2461 . . . . . . . . . . . . . . 15  |-  ( r  =  3  ->  (
m  =  ( ( p  +  q )  +  r )  <->  m  =  ( ( p  +  q )  +  3 ) ) )
3128, 30anbi12d 717 . . . . . . . . . . . . . 14  |-  ( r  =  3  ->  (
( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd  )  /\  m  =  ( ( p  +  q )  +  r ) )  <->  ( (
p  e. Odd  /\  q  e. Odd  /\  3  e. Odd  )  /\  m  =  (
( p  +  q )  +  3 ) ) ) )
3231adantl 468 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  /\  r  =  3 )  -> 
( ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd  )  /\  m  =  ( (
p  +  q )  +  r ) )  <-> 
( ( p  e. Odd  /\  q  e. Odd  /\  3  e. Odd  )  /\  m  =  ( ( p  +  q )  +  3 ) ) ) )
33 simp1 1008 . . . . . . . . . . . . . . . 16  |-  ( ( p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  p  e. Odd  )
34 simp2 1009 . . . . . . . . . . . . . . . 16  |-  ( ( p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  q  e. Odd  )
352a1i 11 . . . . . . . . . . . . . . . 16  |-  ( ( p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  3  e. Odd  )
3633, 34, 353jca 1188 . . . . . . . . . . . . . . 15  |-  ( ( p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  ( p  e. Odd  /\  q  e. Odd  /\  3  e. Odd  ) )
3736adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  ->  (
p  e. Odd  /\  q  e. Odd  /\  3  e. Odd  )
)
3816zcnd 11041 . . . . . . . . . . . . . . . . . . 19  |-  ( m  e. Odd  ->  m  e.  CC )
3938ad3antrrr 736 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  m  e.  CC )
40 3cn 10684 . . . . . . . . . . . . . . . . . . 19  |-  3  e.  CC
4140a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  3  e.  CC )
42 prmz 14626 . . . . . . . . . . . . . . . . . . . . 21  |-  ( p  e.  Prime  ->  p  e.  ZZ )
43 prmz 14626 . . . . . . . . . . . . . . . . . . . . 21  |-  ( q  e.  Prime  ->  q  e.  ZZ )
44 zaddcl 10977 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( p  e.  ZZ  /\  q  e.  ZZ )  ->  ( p  +  q )  e.  ZZ )
4542, 43, 44syl2an 480 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  (
p  +  q )  e.  ZZ )
4645zcnd 11041 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p  e.  Prime  /\  q  e.  Prime )  ->  (
p  +  q )  e.  CC )
4746adantll 720 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( p  +  q )  e.  CC )
4839, 41, 47subadd2d 10005 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( m  -  3 )  =  ( p  +  q )  <->  ( ( p  +  q )  +  3 )  =  m ) )
4948biimpa 487 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
m  -  3 )  =  ( p  +  q ) )  -> 
( ( p  +  q )  +  3 )  =  m )
5049eqcomd 2457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
m  -  3 )  =  ( p  +  q ) )  ->  m  =  ( (
p  +  q )  +  3 ) )
51503ad2antr3 1175 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  ->  m  =  ( ( p  +  q )  +  3 ) )
5237, 51jca 535 . . . . . . . . . . . . 13  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  ->  (
( p  e. Odd  /\  q  e. Odd  /\  3  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  3 ) ) )
5326, 32, 52rspcedvd 3155 . . . . . . . . . . . 12  |-  ( ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e. 
Prime )  /\  q  e.  Prime )  /\  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) ) )  ->  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd  )  /\  m  =  ( (
p  +  q )  +  r ) ) )
5453ex 436 . . . . . . . . . . 11  |-  ( ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  /\  q  e.  Prime )  ->  ( ( p  e. Odd  /\  q  e. Odd  /\  ( m  -  3 )  =  ( p  +  q ) )  ->  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  r ) ) ) )
5554reximdva 2862 . . . . . . . . . 10  |-  ( ( ( m  e. Odd  /\  7  <  m )  /\  p  e.  Prime )  -> 
( E. q  e. 
Prime  ( p  e. Odd  /\  q  e. Odd  /\  ( m  -  3 )  =  ( p  +  q ) )  ->  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  r ) ) ) )
5655reximdva 2862 . . . . . . . . 9  |-  ( ( m  e. Odd  /\  7  <  m )  ->  ( E. p  e.  Prime  E. q  e.  Prime  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  (
( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  r ) ) ) )
5756, 1jctild 546 . . . . . . . 8  |-  ( ( m  e. Odd  /\  7  <  m )  ->  ( E. p  e.  Prime  E. q  e.  Prime  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  ( m  e. Odd  /\  E. p  e. 
Prime  E. q  e.  Prime  E. r  e.  Prime  (
( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  r ) ) ) ) )
58 isgboa 38854 . . . . . . . 8  |-  ( m  e. GoldbachOddALTV  <-> 
( m  e. Odd  /\  E. p  e.  Prime  E. q  e.  Prime  E. r  e.  Prime  ( ( p  e. Odd  /\  q  e. Odd  /\  r  e. Odd 
)  /\  m  =  ( ( p  +  q )  +  r ) ) ) )
5957, 58syl6ibr 231 . . . . . . 7  |-  ( ( m  e. Odd  /\  7  <  m )  ->  ( E. p  e.  Prime  E. q  e.  Prime  (
p  e. Odd  /\  q  e. Odd  /\  ( m  - 
3 )  =  ( p  +  q ) )  ->  m  e. GoldbachOddALTV  )
)
6059adantld 469 . . . . . 6  |-  ( ( m  e. Odd  /\  7  <  m )  ->  (
( ( m  - 
3 )  e. Even  /\  E. p  e.  Prime  E. q  e.  Prime  ( p  e. Odd  /\  q  e. Odd  /\  (
m  -  3 )  =  ( p  +  q ) ) )  ->  m  e. GoldbachOddALTV  ) )
6124, 60syl5bi 221 . . . . 5  |-  ( ( m  e. Odd  /\  7  <  m )  ->  (
( m  -  3 )  e. GoldbachEven  ->  m  e. GoldbachOddALTV  )
)
629, 23, 613syld 57 . . . 4  |-  ( ( m  e. Odd  /\  7  <  m )  ->  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  m  e. GoldbachOddALTV  )
)
6362com12 32 . . 3  |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  ( ( m  e. Odd  /\  7  <  m )  ->  m  e. GoldbachOddALTV  )
)
6463expd 438 . 2  |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  ( m  e. Odd 
->  ( 7  <  m  ->  m  e. GoldbachOddALTV  ) ) )
6564ralrimiv 2800 1  |-  ( A. n  e. Even  ( 4  <  n  ->  n  e. GoldbachEven  )  ->  A. m  e. Odd  (
7  <  m  ->  m  e. GoldbachOddALTV  ) )
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  (class class class)co 6290   CCcc 9537   RRcr 9538    + caddc 9542    < clt 9675    - cmin 9860   3c3 10660   4c4 10661   7c7 10664   ZZcz 10937   Primecprime 14622   Even ceven 38753   Odd codd 38754   GoldbachEven cgbe 38846   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  ax-pre-sup 9617
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-int 4235  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-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  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-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-prm 14623  df-even 38755  df-odd 38756  df-gbe 38849  df-gboa 38851
This theorem is referenced by: (None)
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