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Theorem sbgoldbm 47153
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 5156 . . . 4 (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2 eleq1w 2812 . . . 4 (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
31, 2imbi12d 343 . . 3 (𝑛 = 𝑚 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚𝑚 ∈ GoldbachEven )))
43cbvralvw 3232 . 2 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ))
5 eluz2 12866 . . . . 5 (𝑛 ∈ (ℤ‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6 zeoALTV 47039 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7 sgoldbeven3prm 47152 . . . . . . . . . 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 47146 . . . . . . . . . . 11 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
10 rspa 3243 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
11 df-6 12317 . . . . . . . . . . . . . . . . . . . . 21 6 = (5 + 1)
1211breq1i 5159 . . . . . . . . . . . . . . . . . . . 20 (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13 5nn 12336 . . . . . . . . . . . . . . . . . . . . . . 23 5 ∈ ℕ
1413nnzi 12624 . . . . . . . . . . . . . . . . . . . . . 22 5 ∈ ℤ
15 oddz 47000 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16 zltp1le 12650 . . . . . . . . . . . . . . . . . . . . . 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 47121 . . . . . . . . . . . . . . . . . . . 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 3143 . 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 1533  wcel 2098  wral 3058  wrex 3067   class class class wbr 5152  cfv 6553  (class class class)co 7426  1c1 11147   + caddc 11149   < clt 11286  cle 11287  4c4 12307  5c5 12308  6c6 12309  cz 12596  cuz 12860  cprime 16649   Even ceven 46993   Odd codd 46994   GoldbachEven cgbe 47114   GoldbachOddW cgbow 47115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-seq 14007  df-exp 14067  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-dvds 16239  df-prm 16650  df-even 46995  df-odd 46996  df-gbe 47117  df-gbow 47118
This theorem is referenced by:  sbgoldbmb  47155  sbgoldbo  47156
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