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Theorem sbgoldbm 43385
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 4960 . . . 4 (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2 eleq1w 2863 . . . 4 (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
31, 2imbi12d 346 . . 3 (𝑛 = 𝑚 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚𝑚 ∈ GoldbachEven )))
43cbvralv 3400 . 2 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ))
5 eluz2 12088 . . . . 5 (𝑛 ∈ (ℤ‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6 zeoALTV 43271 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7 sgoldbeven3prm 43384 . . . . . . . . . 10 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
87expdcom 415 . . . . . . . . 9 (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9 sbgoldbwt 43378 . . . . . . . . . . 11 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
10 rspa 3171 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
11 df-6 11541 . . . . . . . . . . . . . . . . . . . . 21 6 = (5 + 1)
1211breq1i 4963 . . . . . . . . . . . . . . . . . . . 20 (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13 5nn 11560 . . . . . . . . . . . . . . . . . . . . . . 23 5 ∈ ℕ
1413nnzi 11844 . . . . . . . . . . . . . . . . . . . . . 22 5 ∈ ℤ
15 oddz 43232 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16 zltp1le 11870 . . . . . . . . . . . . . . . . . . . . . 22 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1714, 15, 16sylancr 587 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1817biimprd 249 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
1912, 18syl5bi 243 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
2019imp 407 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21 isgbow 43353 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2221simprbi 497 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . 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 852 . . . . . . . 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 407 . . . . . 6 ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36353adant1 1121 . . . . 5 ((6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
375, 36sylbi 218 . . . 4 (𝑛 ∈ (ℤ‘6) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
3837impcom 408 . . 3 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
3938ralrimiva 3147 . 2 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
404, 39sylbi 218 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 207  wa 396  wo 842  w3a 1078   = wceq 1520  wcel 2079  wral 3103  wrex 3104   class class class wbr 4956  cfv 6217  (class class class)co 7007  1c1 10373   + caddc 10375   < clt 10510  cle 10511  4c4 11531  5c5 11532  6c6 11533  cz 11818  cuz 12082  cprime 15832   Even ceven 43225   Odd codd 43226   GoldbachEven cgbe 43346   GoldbachOddW cgbow 43347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-2o 7945  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-sup 8742  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-n0 11735  df-z 11819  df-uz 12083  df-rp 12229  df-fz 12732  df-seq 13208  df-exp 13268  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-dvds 15429  df-prm 15833  df-even 43227  df-odd 43228  df-gbe 43349  df-gbow 43350
This theorem is referenced by:  sbgoldbmb  43387  sbgoldbo  43388
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