Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbgoldbm Structured version   Visualization version   GIF version

Theorem sbgoldbm 44289
Description: If the strong binary Goldbach conjecture is valid, the modern version of the original formulation of the Goldbach conjecture also holds: Every integer greater than 5 can be expressed as the sum of three primes. (Contributed by AV, 24-Dec-2021.)
Assertion
Ref Expression
sbgoldbm (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
Distinct variable group:   𝑛,𝑝,𝑞,𝑟

Proof of Theorem sbgoldbm
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 breq2 5037 . . . 4 (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2 eleq1w 2875 . . . 4 (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
31, 2imbi12d 348 . . 3 (𝑛 = 𝑚 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚𝑚 ∈ GoldbachEven )))
43cbvralvw 3399 . 2 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ))
5 eluz2 12241 . . . . 5 (𝑛 ∈ (ℤ‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6 zeoALTV 44175 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7 sgoldbeven3prm 44288 . . . . . . . . . 10 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
87expdcom 418 . . . . . . . . 9 (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9 sbgoldbwt 44282 . . . . . . . . . . 11 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
10 rspa 3174 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
11 df-6 11696 . . . . . . . . . . . . . . . . . . . . 21 6 = (5 + 1)
1211breq1i 5040 . . . . . . . . . . . . . . . . . . . 20 (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13 5nn 11715 . . . . . . . . . . . . . . . . . . . . . . 23 5 ∈ ℕ
1413nnzi 11998 . . . . . . . . . . . . . . . . . . . . . 22 5 ∈ ℤ
15 oddz 44136 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16 zltp1le 12024 . . . . . . . . . . . . . . . . . . . . . 22 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1714, 15, 16sylancr 590 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1817biimprd 251 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
1912, 18syl5bi 245 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
2019imp 410 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21 isgbow 44257 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2221simprbi 500 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ GoldbachOddW → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
2322a1i 11 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → (𝑛 ∈ GoldbachOddW → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2420, 23embantd 59 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → ((5 < 𝑛𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2524ex 416 . . . . . . . . . . . . . . . 16 (𝑛 ∈ Odd → (6 ≤ 𝑛 → ((5 < 𝑛𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
2625com23 86 . . . . . . . . . . . . . . 15 (𝑛 ∈ Odd → ((5 < 𝑛𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
2726adantl 485 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → ((5 < 𝑛𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
2810, 27mpd 15 . . . . . . . . . . . . 13 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2928ex 416 . . . . . . . . . . . 12 (∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) → (𝑛 ∈ Odd → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
3029com23 86 . . . . . . . . . . 11 (∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → (𝑛 ∈ Odd → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
319, 30syl 17 . . . . . . . . . 10 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → (6 ≤ 𝑛 → (𝑛 ∈ Odd → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
3231com13 88 . . . . . . . . 9 (𝑛 ∈ Odd → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
338, 32jaoi 854 . . . . . . . 8 ((𝑛 ∈ Even ∨ 𝑛 ∈ Odd ) → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
346, 33syl 17 . . . . . . 7 (𝑛 ∈ ℤ → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
3534imp 410 . . . . . 6 ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36353adant1 1127 . . . . 5 ((6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
375, 36sylbi 220 . . . 4 (𝑛 ∈ (ℤ‘6) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
3837impcom 411 . . 3 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
3938ralrimiva 3152 . 2 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
404, 39sylbi 220 1 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110   class class class wbr 5033  cfv 6328  (class class class)co 7139  1c1 10531   + caddc 10533   < clt 10668  cle 10669  4c4 11686  5c5 11687  6c6 11688  cz 11973  cuz 12235  cprime 16008   Even ceven 44129   Odd codd 44130   GoldbachEven cgbe 44250   GoldbachOddW cgbow 44251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-dvds 15603  df-prm 16009  df-even 44131  df-odd 44132  df-gbe 44253  df-gbow 44254
This theorem is referenced by:  sbgoldbmb  44291  sbgoldbo  44292
  Copyright terms: Public domain W3C validator