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 47390
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 5148 . . . 4 (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2 eleq1w 2809 . . . 4 (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
31, 2imbi12d 343 . . 3 (𝑛 = 𝑚 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚𝑚 ∈ GoldbachEven )))
43cbvralvw 3225 . 2 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ))
5 eluz2 12872 . . . . 5 (𝑛 ∈ (ℤ‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6 zeoALTV 47276 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7 sgoldbeven3prm 47389 . . . . . . . . . 10 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
87expdcom 413 . . . . . . . . 9 (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9 sbgoldbwt 47383 . . . . . . . . . . 11 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
10 rspa 3236 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
11 df-6 12323 . . . . . . . . . . . . . . . . . . . . 21 6 = (5 + 1)
1211breq1i 5151 . . . . . . . . . . . . . . . . . . . 20 (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13 5nn 12342 . . . . . . . . . . . . . . . . . . . . . . 23 5 ∈ ℕ
1413nnzi 12630 . . . . . . . . . . . . . . . . . . . . . 22 5 ∈ ℤ
15 oddz 47237 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16 zltp1le 12656 . . . . . . . . . . . . . . . . . . . . . 22 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1714, 15, 16sylancr 585 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1817biimprd 247 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
1912, 18biimtrid 241 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
2019imp 405 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21 isgbow 47358 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2221simprbi 495 . . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 855 . . . . . . . 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 405 . . . . . 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 216 . . . 4 (𝑛 ∈ (ℤ‘6) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
3837impcom 406 . . 3 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
3938ralrimiva 3136 . 2 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
404, 39sylbi 216 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 205  wa 394  wo 845  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060   class class class wbr 5144  cfv 6544  (class class class)co 7414  1c1 11148   + caddc 11150   < clt 11287  cle 11288  4c4 12313  5c5 12314  6c6 12315  cz 12602  cuz 12866  cprime 16665   Even ceven 47230   Odd codd 47231   GoldbachEven cgbe 47351   GoldbachOddW cgbow 47352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224  ax-pre-sup 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9476  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-div 11911  df-nn 12257  df-2 12319  df-3 12320  df-4 12321  df-5 12322  df-6 12323  df-7 12324  df-n0 12517  df-z 12603  df-uz 12867  df-rp 13021  df-fz 13531  df-seq 14014  df-exp 14074  df-cj 15097  df-re 15098  df-im 15099  df-sqrt 15233  df-abs 15234  df-dvds 16250  df-prm 16666  df-even 47232  df-odd 47233  df-gbe 47354  df-gbow 47355
This theorem is referenced by:  sbgoldbmb  47392  sbgoldbo  47393
  Copyright terms: Public domain W3C validator