tgoldbachlt - Mathbox for Alexander van der Vekens

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Theorem tgoldbachlt 47690

Description: The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big 𝑚 greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 47689. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)

**Assertion**
Ref	Expression
tgoldbachlt	⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ))

Distinct variable group: 𝑚,𝑛

**Proof of Theorem tgoldbachlt**
Step	Hyp	Ref	Expression
1		8nn0 12576	. . . 4 ⊢ 8 ∈ ℕ₀
2		8nn 12388	. . . 4 ⊢ 8 ∈ ℕ
3	1, 2	decnncl 12778	. . 3 ⊢ ;88 ∈ ℕ
4		10nn 12774	. . . 4 ⊢ ;10 ∈ ℕ
5		2nn0 12570	. . . . 5 ⊢ 2 ∈ ℕ₀
6		9nn0 12577	. . . . 5 ⊢ 9 ∈ ℕ₀
7	5, 6	deccl 12773	. . . 4 ⊢ ;29 ∈ ℕ₀
8		nnexpcl 14125	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;29 ∈ ℕ₀) → (;10↑;29) ∈ ℕ)
9	4, 7, 8	mp2an 691	. . 3 ⊢ (;10↑;29) ∈ ℕ
10	3, 9	nnmulcli 12318	. 2 ⊢ (;88 · (;10↑;29)) ∈ ℕ
11		id 22	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → (;88 · (;10↑;29)) ∈ ℕ)
12		breq2 5170	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((8 · (;10↑;30)) < 𝑚 ↔ (8 · (;10↑;30)) < (;88 · (;10↑;29))))
13		breq2 5170	. . . . . . . 8 ⊢ (𝑚 = (;88 · (;10↑;29)) → (𝑛 < 𝑚 ↔ 𝑛 < (;88 · (;10↑;29))))
14	13	anbi2d 629	. . . . . . 7 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((7 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))))
15	14	imbi1d 341	. . . . . 6 ⊢ (𝑚 = (;88 · (;10↑;29)) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) ↔ ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )))
16	15	ralbidv 3184	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → (∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) ↔ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )))
17	12, 16	anbi12d 631	. . . 4 ⊢ (𝑚 = (;88 · (;10↑;29)) → (((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )) ↔ ((8 · (;10↑;30)) < (;88 · (;10↑;29)) ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))))
18	17	adantl 481	. . 3 ⊢ (((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑚 = (;88 · (;10↑;29))) → (((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )) ↔ ((8 · (;10↑;30)) < (;88 · (;10↑;29)) ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))))
19		simplr 768	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ Odd )
20		simprl 770	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 7 < 𝑛)
21		simprr 772	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 < (;88 · (;10↑;29)))
22		tgblthelfgott 47689	. . . . . . 7 ⊢ ((𝑛 ∈ Odd ∧ 7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1371	. . . . . 6 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ GoldbachOdd )
24	23	ex 412	. . . . 5 ⊢ (((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) → ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
25	24	ralrimiva 3152	. . . 4 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
26	2, 9	nnmulcli 12318	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℕ
27	26	nngt0i 12332	. . . . . 6 ⊢ 0 < (8 · (;10↑;29))
28	26	nnrei 12302	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℝ
29		3nn0 12571	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
30		0nn0 12568	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
31	29, 30	deccl 12773	. . . . . . . . . 10 ⊢ ;30 ∈ ℕ₀
32		nnexpcl 14125	. . . . . . . . . 10 ⊢ ((;10 ∈ ℕ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℕ)
33	4, 31, 32	mp2an 691	. . . . . . . . 9 ⊢ (;10↑;30) ∈ ℕ
34	2, 33	nnmulcli 12318	. . . . . . . 8 ⊢ (8 · (;10↑;30)) ∈ ℕ
35	34	nnrei 12302	. . . . . . 7 ⊢ (8 · (;10↑;30)) ∈ ℝ
36	28, 35	ltaddposi 11839	. . . . . 6 ⊢ (0 < (8 · (;10↑;29)) ↔ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29))))
37	27, 36	mpbi 230	. . . . 5 ⊢ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29)))
38		dfdec10 12761	. . . . . . 7 ⊢ ;88 = ((;10 · 8) + 8)
39	38	oveq1i 7458	. . . . . 6 ⊢ (;88 · (;10↑;29)) = (((;10 · 8) + 8) · (;10↑;29))
40	4, 2	nnmulcli 12318	. . . . . . . 8 ⊢ (;10 · 8) ∈ ℕ
41	40	nncni 12303	. . . . . . 7 ⊢ (;10 · 8) ∈ ℂ
42		8cn 12390	. . . . . . 7 ⊢ 8 ∈ ℂ
43	9	nncni 12303	. . . . . . 7 ⊢ (;10↑;29) ∈ ℂ
44	41, 42, 43	adddiri 11303	. . . . . 6 ⊢ (((;10 · 8) + 8) · (;10↑;29)) = (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29)))
45	41, 43	mulcomi 11298	. . . . . . . . 9 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑;29) · (;10 · 8))
46	4	nncni 12303	. . . . . . . . . 10 ⊢ ;10 ∈ ℂ
47	43, 46, 42	mulassi 11301	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑;29) · (;10 · 8))
48		nncn 12301	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;10 ∈ ℂ)
49	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;29 ∈ ℕ₀)
50	48, 49	expp1d 14197	. . . . . . . . . . . 12 ⊢ (;10 ∈ ℕ → (;10↑(;29 + 1)) = ((;10↑;29) · ;10))
51	4, 50	ax-mp 5	. . . . . . . . . . 11 ⊢ (;10↑(;29 + 1)) = ((;10↑;29) · ;10)
52	51	eqcomi 2749	. . . . . . . . . 10 ⊢ ((;10↑;29) · ;10) = (;10↑(;29 + 1))
53	52	oveq1i 7458	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑(;29 + 1)) · 8)
54	45, 47, 53	3eqtr2i 2774	. . . . . . . 8 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑(;29 + 1)) · 8)
55	54	oveq1i 7458	. . . . . . 7 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29)))
56		2p1e3 12435	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
57		eqid 2740	. . . . . . . . . . 11 ⊢ ;29 = ;29
58	5, 56, 57	decsucc 12799	. . . . . . . . . 10 ⊢ (;29 + 1) = ;30
59	58	oveq2i 7459	. . . . . . . . 9 ⊢ (;10↑(;29 + 1)) = (;10↑;30)
60	59	oveq1i 7458	. . . . . . . 8 ⊢ ((;10↑(;29 + 1)) · 8) = ((;10↑;30) · 8)
61	60	oveq1i 7458	. . . . . . 7 ⊢ (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29))) = (((;10↑;30) · 8) + (8 · (;10↑;29)))
62	33	nncni 12303	. . . . . . . 8 ⊢ (;10↑;30) ∈ ℂ
63		mulcom 11270	. . . . . . . . 9 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → ((;10↑;30) · 8) = (8 · (;10↑;30)))
64	63	oveq1d 7463	. . . . . . . 8 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29))))
65	62, 42, 64	mp2an 691	. . . . . . 7 ⊢ (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
66	55, 61, 65	3eqtri 2772	. . . . . 6 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
67	39, 44, 66	3eqtri 2772	. . . . 5 ⊢ (;88 · (;10↑;29)) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
68	37, 67	breqtrri 5193	. . . 4 ⊢ (8 · (;10↑;30)) < (;88 · (;10↑;29))
69	25, 68	jctil 519	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ((8 · (;10↑;30)) < (;88 · (;10↑;29)) ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )))
70	11, 18, 69	rspcedvd 3637	. 2 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )))
71	10, 70	ax-mp 5	1 ⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 class class class wbr 5166 (class class class)co 7448 ℂcc 11182 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ℕcn 12293 2c2 12348 3c3 12349 7c7 12353 8c8 12354 9c9 12355 ℕ₀cn0 12553 cdc 12758 ↑cexp 14112 Odd codd 47499 GoldbachOdd cgbo 47621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-bgbltosilva 47684 ax-hgprmladder 47688

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-ico 13413 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-dvds 16303 df-prm 16719 df-iccp 47288 df-even 47500 df-odd 47501 df-gbe 47622 df-gbo 47624

This theorem is referenced by: tgoldbach 47691

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