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Theorem sbgoldbm 46066
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 5113 . . . 4 (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2 eleq1w 2817 . . . 4 (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
31, 2imbi12d 345 . . 3 (𝑛 = 𝑚 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚𝑚 ∈ GoldbachEven )))
43cbvralvw 3224 . 2 (∀𝑛 ∈ Even (4 < 𝑛𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ))
5 eluz2 12777 . . . . 5 (𝑛 ∈ (ℤ‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6 zeoALTV 45952 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7 sgoldbeven3prm 46065 . . . . . . . . . 10 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
87expdcom 416 . . . . . . . . 9 (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9 sbgoldbwt 46059 . . . . . . . . . . 11 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ))
10 rspa 3230 . . . . . . . . . . . . . 14 ((∀𝑛 ∈ Odd (5 < 𝑛𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛𝑛 ∈ GoldbachOddW ))
11 df-6 12228 . . . . . . . . . . . . . . . . . . . . 21 6 = (5 + 1)
1211breq1i 5116 . . . . . . . . . . . . . . . . . . . 20 (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13 5nn 12247 . . . . . . . . . . . . . . . . . . . . . . 23 5 ∈ ℕ
1413nnzi 12535 . . . . . . . . . . . . . . . . . . . . . 22 5 ∈ ℤ
15 oddz 45913 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16 zltp1le 12561 . . . . . . . . . . . . . . . . . . . . . 22 ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1714, 15, 16sylancr 588 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
1817biimprd 248 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
1912, 18biimtrid 241 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
2019imp 408 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21 isgbow 46034 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
2221simprbi 498 . . . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 856 . . . . . . . 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 408 . . . . . 6 ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36353adant1 1131 . . . . 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 409 . . 3 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
3938ralrimiva 3140 . 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070   class class class wbr 5109  cfv 6500  (class class class)co 7361  1c1 11060   + caddc 11062   < clt 11197  cle 11198  4c4 12218  5c5 12219  6c6 12220  cz 12507  cuz 12771  cprime 16555   Even ceven 45906   Odd codd 45907   GoldbachEven cgbe 46027   GoldbachOddW cgbow 46028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-fz 13434  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-dvds 16145  df-prm 16556  df-even 45908  df-odd 45909  df-gbe 46030  df-gbow 46031
This theorem is referenced by:  sbgoldbmb  46068  sbgoldbo  46069
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