Theorem bgoldbachlt 43985

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 43982. (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 11723	. . 3 ⊢ 4 ∈ ℕ
2		10nn 12117	. . . 4 ⊢ ;10 ∈ ℕ
3		1nn0 11916	. . . . 5 ⊢ 1 ∈ ℕ0
4		8nn0 11923	. . . . 5 ⊢ 8 ∈ ℕ0
5	3, 4	deccl 12116	. . . 4 ⊢ ;18 ∈ ℕ0
6		nnexpcl 13445	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;18 ∈ ℕ0) → (;10↑;18) ∈ ℕ)
7	2, 5, 6	mp2an 690	. . 3 ⊢ (;10↑;18) ∈ ℕ
8	1, 7	nnmulcli 11665	. 2 ⊢ (4 · (;10↑;18)) ∈ ℕ
9		id 22	. . 3 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℕ)
10		breq2 5072	. . . . 5 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 · (;10↑;18)) ≤ 𝑚 ↔ (4 · (;10↑;18)) ≤ (4 · (;10↑;18))))
11		breq2 5072	. . . . . . . 8 ⊢ (𝑚 = (4 · (;10↑;18)) → (𝑛 < 𝑚 ↔ 𝑛 < (4 · (;10↑;18))))
12	11	anbi2d 630	. . . . . . 7 ⊢ (𝑚 = (4 · (;10↑;18)) → ((4 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))))
13	12	imbi1d 344	. . . . . 6 ⊢ (𝑚 = (4 · (;10↑;18)) → (((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven ) ↔ ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )))
14	13	ralbidv 3199	. . . . 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 632	. . . 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 484	. . 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 11647	. . . . 5 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ∈ ℝ)
18	17	leidd 11208	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → (4 · (;10↑;18)) ≤ (4 · (;10↑;18)))
19		simplr 767	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ Even )
20		simprl 769	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 4 < 𝑛)
21		evenz 43802	. . . . . . . . . . 11 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℤ)
22	21	zred 12090	. . . . . . . . . 10 ⊢ (𝑛 ∈ Even → 𝑛 ∈ ℝ)
23		ltle 10731	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℝ ∧ (4 · (;10↑;18)) ∈ ℝ) → (𝑛 < (4 · (;10↑;18)) → 𝑛 ≤ (4 · (;10↑;18))))
24	22, 17, 23	syl2anr 598	. . . . . . . . 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 421	. . . . . . 7 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ≤ (4 · (;10↑;18)))
27		ax-bgbltosilva 43982	. . . . . . 7 ⊢ ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≤ (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven )
28	19, 20, 26, 27	syl3anc 1367	. . . . . 6 ⊢ ((((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) ∧ (4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18)))) → 𝑛 ∈ GoldbachEven )
29	28	ex 415	. . . . 5 ⊢ (((4 · (;10↑;18)) ∈ ℕ ∧ 𝑛 ∈ Even ) → ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
30	29	ralrimiva 3184	. . . 4 ⊢ ((4 · (;10↑;18)) ∈ ℕ → ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 · (;10↑;18))) → 𝑛 ∈ GoldbachEven ))
31	18, 30	jca 514	. . 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 3628	. 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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   class class class wbr 5068  (class class class)co 7158  ℝcr 10538  0cc0 10539  1c1 10540   · cmul 10544   < clt 10677   ≤ cle 10678  ℕcn 11640  4c4 11697  8c8 11701  ℕ₀cn0 11900  ;cdc 12101  ↑cexp 13432   Even ceven 43796   GoldbachEven cgbe 43917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-bgbltosilva 43982

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-seq 13373  df-exp 13433  df-even 43798

This theorem is referenced by: (None)