Theorem sbgoldbm 41672

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 4657	. . . 4 ⊢ (𝑛 = 𝑚 → (4 < 𝑛 ↔ 4 < 𝑚))
2		eleq1w 2684	. . . 4 ⊢ (𝑛 = 𝑚 → (𝑛 ∈ GoldbachEven ↔ 𝑚 ∈ GoldbachEven ))
3	1, 2	imbi12d 334	. . 3 ⊢ (𝑛 = 𝑚 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ (4 < 𝑚 → 𝑚 ∈ GoldbachEven )))
4	3	cbvralv 3171	. 2 ⊢ (∀𝑛 ∈ Even (4 < 𝑛 → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ))
5		eluz2 11693	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘6) ↔ (6 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 6 ≤ 𝑛))
6		zeoALTV 41581	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
7		sgoldbeven3prm 41671	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ((𝑛 ∈ Even ∧ 6 ≤ 𝑛) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
8	7	expdcom 455	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
9		sbgoldbwt 41665	. . . . . . . . . . 11 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
10		rspa 2930	. . . . . . . . . . . . . 14 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ))
11		df-6 11083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 6 = (5 + 1)
12	11	breq1i 4660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (6 ≤ 𝑛 ↔ (5 + 1) ≤ 𝑛)
13		5nn 11188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 5 ∈ ℕ
14	13	nnzi 11401	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 5 ∈ ℤ
15		oddz 41544	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ Odd → 𝑛 ∈ ℤ)
16		zltp1le 11427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((5 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
17	14, 15, 16	sylancr 695	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ Odd → (5 < 𝑛 ↔ (5 + 1) ≤ 𝑛))
18	17	biimprd 238	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ Odd → ((5 + 1) ≤ 𝑛 → 5 < 𝑛))
19	12, 18	syl5bi 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ Odd → (6 ≤ 𝑛 → 5 < 𝑛))
20	19	imp 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → 5 < 𝑛)
21		isgbow 41640	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ GoldbachOddW ↔ (𝑛 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
22	21	simprbi 480	. . . . . . . . . . . . . . . . . . 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 59	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ Odd ∧ 6 ≤ 𝑛) → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
25	24	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ Odd → (6 ≤ 𝑛 → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
26	25	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ Odd → ((5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
27	26	adantl 482	. . . . . . . . . . . . . 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 15	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) ∧ 𝑛 ∈ Odd ) → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
29	28	ex 450	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ Odd (5 < 𝑛 → 𝑛 ∈ GoldbachOddW ) → (𝑛 ∈ Odd → (6 ≤ 𝑛 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
30	29	com23 86	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (6 ≤ 𝑛 → (𝑛 ∈ Odd → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
32	31	com13 88	. . . . . . . . 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 394	. . . . . . . 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 17	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (6 ≤ 𝑛 → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))))
35	34	imp 445	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ 6 ≤ 𝑛) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
36	35	3adant1 1079	. . . . 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 207	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘6) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟)))
38	37	impcom 446	. . 3 ⊢ ((∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ≥‘6)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
39	38	ralrimiva 2966	. 2 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘6)∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑛 = ((𝑝 + 𝑞) + 𝑟))
40	4, 39	sylbi 207	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 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075  4c4 11072  5c5 11073  6c6 11074  ℤcz 11377  ℤ_≥cuz 11687  ℙcprime 15385   Even ceven 41537   Odd codd 41538   GoldbachEven cgbe 41633   GoldbachOddW cgbow 41634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-even 41539  df-odd 41540  df-gbe 41636  df-gbow 41637

This theorem is referenced by:  sbgoldbmb  41674  sbgoldbo  41675