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Theorem evengpop3 41686
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. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> E. o e. GoldbachOddW N = ( o + 3 ) ) )
Distinct variable groups:   m, N   o, N
Proof of Theorem evengpop3
StepHypRef Expression
1 3odd 41617 . . . . . . 7 |- 3 e. Odd
21a1i 11 . . . . . 6 |- ( N e. ( ZZ>= ` 9 ) -> 3 e. Odd )
32anim1i 592 . . . . 5 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( 3 e. Odd /\ N e. Even ) )
43ancomd 467 . . . 4 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( N e. Even /\ 3 e. Odd ) )
5 emoo 41613 . . . 4 |- ( ( N e. Even /\ 3 e. Odd ) -> ( N - 3 ) e. Odd )
64, 5syl 17 . . 3 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( N - 3 ) e. Odd )
7 breq2 4657 . . . . 5 |- ( m = ( N - 3 ) -> ( 5 < m <-> 5 < ( N - 3 ) ) )
8 eleq1 2689 . . . . 5 |- ( m = ( N - 3 ) -> ( m e. GoldbachOddW <-> ( N - 3 ) e. GoldbachOddW ) )
97, 8imbi12d 334 . . . 4 |- ( m = ( N - 3 ) -> ( ( 5 < m -> m e. GoldbachOddW ) <-> ( 5 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddW ) ) )
109adantl 482 . . 3 |- ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ m = ( N - 3 ) ) -> ( ( 5 < m -> m e. GoldbachOddW ) <-> ( 5 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddW ) ) )
116, 10rspcdv 3312 . 2 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( 5 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddW ) ) )
12 eluz2 11693 . . . . 5 |- ( N e. ( ZZ>= ` 9 ) <-> ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) )
13 5p3e8 11166 . . . . . . . 8 |- ( 5 + 3 ) = 8
14 8p1e9 11158 . . . . . . . . 9 |- ( 8 + 1 ) = 9
15 9cn 11108 . . . . . . . . . 10 |- 9 e. CC
16 ax-1cn 9994 . . . . . . . . . 10 |- 1 e. CC
17 8cn 11106 . . . . . . . . . 10 |- 8 e. CC
1815, 16, 17subadd2i 10369 . . . . . . . . 9 |- ( ( 9 - 1 ) = 8 <-> ( 8 + 1 ) = 9 )
1914, 18mpbir 221 . . . . . . . 8 |- ( 9 - 1 ) = 8
2013, 19eqtr4i 2647 . . . . . . 7 |- ( 5 + 3 ) = ( 9 - 1 )
21 zlem1lt 11429 . . . . . . . 8 |- ( ( 9 e. ZZ /\ N e. ZZ ) -> ( 9 <_ N <-> ( 9 - 1 ) < N ) )
2221biimp3a 1432 . . . . . . 7 |- ( ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) -> ( 9 - 1 ) < N )
2320, 22syl5eqbr 4688 . . . . . 6 |- ( ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) -> ( 5 + 3 ) < N )
24 5re 11099 . . . . . . . . . 10 |- 5 e. RR
2524a1i 11 . . . . . . . . 9 |- ( N e. ZZ -> 5 e. RR )
26 3re 11094 . . . . . . . . . 10 |- 3 e. RR
2726a1i 11 . . . . . . . . 9 |- ( N e. ZZ -> 3 e. RR )
28 zre 11381 . . . . . . . . 9 |- ( N e. ZZ -> N e. RR )
2925, 27, 283jca 1242 . . . . . . . 8 |- ( N e. ZZ -> ( 5 e. RR /\ 3 e. RR /\ N e. RR ) )
30293ad2ant2 1083 . . . . . . 7 |- ( ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) -> ( 5 e. RR /\ 3 e. RR /\ N e. RR ) )
31 ltaddsub 10502 . . . . . . 7 |- ( ( 5 e. RR /\ 3 e. RR /\ N e. RR ) -> ( ( 5 + 3 ) < N <-> 5 < ( N - 3 ) ) )
3230, 31syl 17 . . . . . 6 |- ( ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) -> ( ( 5 + 3 ) < N <-> 5 < ( N - 3 ) ) )
3323, 32mpbid 222 . . . . 5 |- ( ( 9 e. ZZ /\ N e. ZZ /\ 9 <_ N ) -> 5 < ( N - 3 ) )
3412, 33sylbi 207 . . . 4 |- ( N e. ( ZZ>= ` 9 ) -> 5 < ( N - 3 ) )
3534adantr 481 . . 3 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> 5 < ( N - 3 ) )
36 simpr 477 . . . . 5 |- ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddW ) -> ( N - 3 ) e. GoldbachOddW )
37 oveq1 6657 . . . . . . 7 |- ( o = ( N - 3 ) -> ( o + 3 ) = ( ( N - 3 ) + 3 ) )
3837eqeq2d 2632 . . . . . 6 |- ( o = ( N - 3 ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
3938adantl 482 . . . . 5 |- ( ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddW ) /\ o = ( N - 3 ) ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
40 eluzelcn 11699 . . . . . . . . 9 |- ( N e. ( ZZ>= ` 9 ) -> N e. CC )
41 3cn 11095 . . . . . . . . . 10 |- 3 e. CC
4241a1i 11 . . . . . . . . 9 |- ( N e. ( ZZ>= ` 9 ) -> 3 e. CC )
4340, 42jca 554 . . . . . . . 8 |- ( N e. ( ZZ>= ` 9 ) -> ( N e. CC /\ 3 e. CC ) )
4443adantr 481 . . . . . . 7 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( N e. CC /\ 3 e. CC ) )
4544adantr 481 . . . . . 6 |- ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddW ) -> ( N e. CC /\ 3 e. CC ) )
46 npcan 10290 . . . . . . 7 |- ( ( N e. CC /\ 3 e. CC ) -> ( ( N - 3 ) + 3 ) = N )
4746eqcomd 2628 . . . . . 6 |- ( ( N e. CC /\ 3 e. CC ) -> N = ( ( N - 3 ) + 3 ) )
4845, 47syl 17 . . . . 5 |- ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddW ) -> N = ( ( N - 3 ) + 3 ) )
4936, 39, 48rspcedvd 3317 . . . 4 |- ( ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddW ) -> E. o e. GoldbachOddW N = ( o + 3 ) )
5049ex 450 . . 3 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( ( N - 3 ) e. GoldbachOddW -> E. o e. GoldbachOddW N = ( o + 3 ) ) )
5135, 50embantd 59 . 2 |- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( ( 5 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddW ) -> E. o e. GoldbachOddW N = ( o + 3 ) ) )
5211, 51syldc 48 1 |- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> E. o e. GoldbachOddW N = ( o + 3 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   3c3 11071   5c5 11073   8c8 11076   9c9 11077   ZZcz 11377   ZZ>=cuz 11687   Even ceven 41537   Odd codd 41538   GoldbachOddW cgbow 41634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-even 41539  df-odd 41540
This theorem is referenced by:  nnsum4primeseven  41688
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