Theorem sbgoldbm 48472

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 5117	. . . 4 ⊢ (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2		eleq1w 2852	. . . 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 3249	. 2 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ))
5		eluz2 12868	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6		zeoALTV 48358	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7		sgoldbeven3prm 48471	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
8	7	expdcom 419	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9		sbgoldbwt 48465	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
10		rspa 3260	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
11		df-6 12307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 6 = (5 + 1)
12	11	breq1i 5120	. . . . . . . . . . . . . . . . . . . 20 ⊢ (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13		5nn 12327	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 5 ∈ ℕ
14	13	nnzi 12618	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℤ
15		oddz 48319	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16		zltp1le 12644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
17	14, 15, 16	sylancr 598	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21		isgbow 48440	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
22	21	simprbi 502	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 870	. . . . . . . 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 411	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36	35	3adant1 1146	. . . . 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 412	. . 3 ⊢ ((∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ≥‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
39	38	ralrimiva 3163	. 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))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  1c1 11101   + caddc 11103   < clt 11243   ≤ cle 11244  4c4 12297  5c5 12298  6c6 12299  ℤcz 12591  ℤ_≥cuz 12862  ℙcprime 16729   Even ceven 48312   Odd codd 48313   GoldbachEven cgbe 48433   GoldbachOddW cgbow 48434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9402  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-n0 12505  df-z 12592  df-uz 12863  df-rp 13017  df-fz 13536  df-seq 14038  df-exp 14098  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-dvds 16311  df-prm 16730  df-even 48314  df-odd 48315  df-gbe 48436  df-gbow 48437

This theorem is referenced by:  sbgoldbmb  48474  sbgoldbo  48475