Theorem sbgoldbm 49027

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 5110	. . . 4 ⊢ (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2		eleq1w 2847	. . . 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 3244	. 2 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ))
5		eluz2 12984	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6		zeoALTV 48913	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7		sgoldbeven3prm 49026	. . . . . . . . . 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 49020	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
10		rspa 3255	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
11		df-6 12422	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 6 = (5 + 1)
12	11	breq1i 5113	. . . . . . . . . . . . . . . . . . . 20 ⊢ (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13		5nn 12442	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 5 ∈ ℕ
14	13	nnzi 12733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℤ
15		oddz 48874	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16		zltp1le 12759	. . . . . . . . . . . . . . . . . . . . . 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 48995	. . . . . . . . . . . . . . . . . . . 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 3158	. 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 1569   ∈ wcel 2146  ∀wral 3080  ∃wrex 3090   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  1c1 11214   + caddc 11216   < clt 11356   ≤ cle 11357  4c4 12412  5c5 12413  6c6 12414  ℤcz 12706  ℤ_≥cuz 12978  ℙcprime 16861   Even ceven 48867   Odd codd 48868   GoldbachEven cgbe 48988   GoldbachOddW cgbow 48989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-pre-sup 11291

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-2o 8491  df-er 8731  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-sup 9447  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-div 11987  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-7 12423  df-n0 12620  df-z 12707  df-uz 12979  df-rp 13134  df-fz 13653  df-seq 14158  df-exp 14218  df-cj 15281  df-re 15282  df-im 15283  df-sqrt 15417  df-abs 15418  df-dvds 16438  df-prm 16862  df-even 48869  df-odd 48870  df-gbe 48991  df-gbow 48992

This theorem is used by:  sbgoldbmb  49029  sbgoldbo  49030