Theorem bgoldbachlt 45265

Description: The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 45262. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)

Assertion

Ref	Expression
bgoldbachlt	⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Distinct variable group:   𝑚,𝑛

Proof of Theorem bgoldbachlt

Step	Hyp	Ref	Expression
1		4nn 12056	. . 3 ⊢ 4 ∈ ℕ
2		10nn 12453	. . . 4 ⊢ ;10 ∈ ℕ
3		1nn0 12249	. . . . 5 ⊢ 1 ∈ ℕ0
4		8nn0 12256	. . . . 5 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12452	. . . 4 ⊢ ;18 ∈ ℕ0
6		nnexpcl 13795	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (;10↑;18) ∈ ℕ)
7	2, 5, 6	mp2an 689	. . 3 ⊢ (;10↑;18) ∈ ℕ
8	1, 7	nnmulcli 11998	. 2 ⊢ (4 · (;10↑;18)) ∈ ℕ
9		id 22	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℕ)
10		breq2 5078	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 · (;10↑;18)) ≤ 𝑚 ↔ (4 · (;10↑;18)) ≤ (4 · (;10↑;18))))
11		breq2 5078	. . . . . . . 8 ⊢ (𝑚 = (4 · (;10↑;18)) → (𝑛 < 𝑚 ↔ 𝑛 < (4 · (;10↑;18))))
12	11	anbi2d 629	. . . . . . 7 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))))
13	12	imbi1d 342	. . . . . 6 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
14	13	ralbidv 3112	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → (∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
15	10, 14	anbi12d 631	. . . 4 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))))
16	15	adantl 482	. . 3 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑚 = (4 · (;10↑;18))) → (((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) ↔ ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))))
17		nnre 11980	. . . . 5 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℝ)
18	17	leidd 11541	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ≤ (4 · (;10↑;18)))
19		simplr 766	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ Even )
20		simprl 768	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 4 < 𝑛)
21		evenz 45082	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
22	21	zred 12426	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23		ltle 11063	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (;10↑;18)) ∈ ℝ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
24	22, 17, 23	syl2anr 597	. . . . . . . . 9 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
25	24	a1d 25	. . . . . . . 8 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → (4 < 𝑛 → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18)))))
26	25	imp32 419	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ≤ (4 · (;10↑;18)))
27		ax-bgbltosilva 45262	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )
28	19, 20, 26, 27	syl3anc 1370	. . . . . 6 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ GoldbachEven )
29	28	ex 413	. . . . 5 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
30	29	ralrimiva 3103	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	18, 30	jca 512	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ((4 · (;10↑;18)) ≤ (4 · (;10↑;18)) ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
32	9, 16, 31	rspcedvd 3563	. 2 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )))
33	8, 32	ax-mp 5	1 ⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   class class class wbr 5074  (class class class)co 7275  ℝcr 10870  0cc0 10871  1c1 10872   · cmul 10876   < clt 11009   ≤ cle 11010  ℕcn 11973  4c4 12030  8c8 12034  ℕ₀cn0 12233  ;cdc 12437  ↑cexp 13782   Even ceven 45076   GoldbachEven cgbe 45197

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-bgbltosilva 45262

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-seq 13722  df-exp 13783  df-even 45078

This theorem is referenced by: (None)