Theorem sbgoldbm 48627

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 < 𝑛 → 𝑛 ∈ 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.

Step	Hyp	Ref	Expression
1		breq2 5115	. . . 4 ⊢ (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2		eleq1w 2848	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
3	1, 2	imbi12d 347	. . 3 ⊢ (𝑛 = 𝑚 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚 → 𝑚 ∈ GoldbachEven )))
4	3	cbvralvw 3245	. 2 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ))
5		eluz2 12889	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6		zeoALTV 48513	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7		sgoldbeven3prm 48626	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
8	7	expdcom 420	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9		sbgoldbwt 48620	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
10		rspa 3256	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
11		df-6 12327	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 6 = (5 + 1)
12	11	breq1i 5118	. . . . . . . . . . . . . . . . . . . 20 ⊢ (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13		5nn 12347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 5 ∈ ℕ
14	13	nnzi 12638	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℤ
15		oddz 48474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16		zltp1le 12664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
17	14, 15, 16	sylancr 599	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
18	17	biimprd 251	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
19	12, 18	biimtrid 245	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
20	19	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21		isgbow 48595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
22	21	simprbi 503	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ GoldbachOddW → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
23	22	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → (𝑛 ∈ GoldbachOddW → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
24	20, 23	embantd 60	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
25	24	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ Odd → (6 ≤ 𝑛 → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
26	25	com23 87	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ Odd → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
27	26	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
28	10, 27	mpd 16	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
29	28	ex 418	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (𝑛 ∈ Odd → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
30	29	com23 87	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → (𝑛 ∈ Odd → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
31	9, 30	syl 18	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (6 ≤ 𝑛 → (𝑛 ∈ Odd → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
32	31	com13 89	. . . . . . . . 9 ⊢ (𝑛 ∈ Odd → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
33	8, 32	jaoi 871	. . . . . . . 8 ⊢ ((𝑛 ∈ Even ∨ 𝑛 ∈ Odd ) → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
34	6, 33	syl 18	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
35	34	imp 412	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36	35	3adant1 1148	. . . . 5 ⊢ ((6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
37	5, 36	sylbi 220	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘6) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
38	37	impcom 413	. . 3 ⊢ ((∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ≥‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
39	38	ralrimiva 3159	. 2 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
40	4, 39	sylbi 220	1 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   class class class wbr 5111  ‘cfv 6541  (class class class)co 7420  1c1 11121   + caddc 11123   < clt 11263   ≤ cle 11264  4c4 12317  5c5 12318  6c6 12319  ℤcz 12611  ℤ_≥cuz 12883  ℙcprime 16756   Even ceven 48467   Odd codd 48468   GoldbachEven cgbe 48588   GoldbachOddW cgbow 48589

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7743  ax-cnex 11176  ax-resscn 11177  ax-1cn 11178  ax-icn 11179  ax-addcl 11180  ax-addrcl 11181  ax-mulcl 11182  ax-mulrcl 11183  ax-mulcom 11184  ax-addass 11185  ax-mulass 11186  ax-distr 11187  ax-i2m1 11188  ax-1ne0 11189  ax-1rid 11190  ax-rnegex 11191  ax-rrecex 11192  ax-cnre 11193  ax-pre-lttri 11194  ax-pre-lttrn 11195  ax-pre-ltadd 11196  ax-pre-mulgt0 11197  ax-pre-sup 11198

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7870  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8460  df-2o 8461  df-er 8701  df-en 8951  df-dom 8952  df-sdom 8953  df-fin 8954  df-sup 9410  df-pnf 11265  df-mnf 11266  df-xr 11267  df-ltxr 11268  df-le 11269  df-sub 11463  df-neg 11464  df-div 11892  df-nn 12254  df-2 12323  df-3 12324  df-4 12325  df-5 12326  df-6 12327  df-7 12328  df-n0 12525  df-z 12612  df-uz 12884  df-rp 13038  df-fz 13557  df-seq 14061  df-exp 14121  df-cj 15179  df-re 15180  df-im 15181  df-sqrt 15315  df-abs 15316  df-dvds 16338  df-prm 16757  df-even 48469  df-odd 48470  df-gbe 48591  df-gbow 48592

This theorem is used by:  sbgoldbmb  48629  sbgoldbo  48630