tgoldbachlt - Mathbox for Alexander van der Vekens

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Theorem tgoldbachlt 48504

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 48503. (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 12527	. . . 4 ⊢ 8 ∈ ℕ₀
2		8nn 12336	. . . 4 ⊢ 8 ∈ ℕ
3	1, 2	decnncl 12735	. . 3 ⊢ ;88 ∈ ℕ
4		10nn 12731	. . . 4 ⊢ ;10 ∈ ℕ
5		2nn0 12521	. . . . 5 ⊢ 2 ∈ ℕ₀
6		9nn0 12528	. . . . 5 ⊢ 9 ∈ ℕ₀
7	5, 6	deccl 12726	. . . 4 ⊢ ;29 ∈ ℕ₀
8		nnexpcl 14110	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;29 ∈ ℕ₀) → (;10↑;29) ∈ ℕ)
9	4, 7, 8	mp2an 704	. . 3 ⊢ (;10↑;29) ∈ ℕ
10	3, 9	nnmulcli 12258	. 2 ⊢ (;88 · (;10↑;29)) ∈ ℕ
11		id 23	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → (;88 · (;10↑;29)) ∈ ℕ)
12		breq2 5117	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((8 · (;10↑;30)) < 𝑚 ↔ (8 · (;10↑;30)) < (;88 · (;10↑;29))))
13		breq2 5117	. . . . . . . 8 ⊢ (𝑚 = (;88 · (;10↑;29)) → (𝑛 < 𝑚 ↔ 𝑛 < (;88 · (;10↑;29))))
14	13	anbi2d 641	. . . . . . 7 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((7 < 𝑛 ∧ 𝑛 < 𝑚) ↔ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))))
15	14	imbi1d 344	. . . . . 6 ⊢ (𝑚 = (;88 · (;10↑;29)) → (((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd ) ↔ ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )))
16	15	ralbidv 3194	. . . . 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 643	. . . 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 486	. . 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 780	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ Odd )
20		simprl 782	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 7 < 𝑛)
21		simprr 784	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 < (;88 · (;10↑;29)))
22		tgblthelfgott 48503	. . . . . . 7 ⊢ ((𝑛 ∈ Odd ∧ 7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1396	. . . . . 6 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ GoldbachOdd )
24	23	ex 417	. . . . 5 ⊢ (((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) → ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
25	24	ralrimiva 3163	. . . 4 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
26	2, 9	nnmulcli 12258	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℕ
27	26	nngt0i 12275	. . . . . 6 ⊢ 0 < (8 · (;10↑;29))
28	26	nnrei 12242	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℝ
29		3nn0 12522	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
30		0nn0 12519	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
31	29, 30	deccl 12726	. . . . . . . . . 10 ⊢ ;30 ∈ ℕ₀
32		nnexpcl 14110	. . . . . . . . . 10 ⊢ ((;10 ∈ ℕ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℕ)
33	4, 31, 32	mp2an 704	. . . . . . . . 9 ⊢ (;10↑;30) ∈ ℕ
34	2, 33	nnmulcli 12258	. . . . . . . 8 ⊢ (8 · (;10↑;30)) ∈ ℕ
35	34	nnrei 12242	. . . . . . 7 ⊢ (8 · (;10↑;30)) ∈ ℝ
36	28, 35	ltaddposi 11763	. . . . . 6 ⊢ (0 < (8 · (;10↑;29)) ↔ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29))))
37	27, 36	mpbi 233	. . . . 5 ⊢ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29)))
38		dfdec10 12714	. . . . . . 7 ⊢ ;88 = ((;10 · 8) + 8)
39	38	oveq1i 7421	. . . . . 6 ⊢ (;88 · (;10↑;29)) = (((;10 · 8) + 8) · (;10↑;29))
40	4, 2	nnmulcli 12258	. . . . . . . 8 ⊢ (;10 · 8) ∈ ℕ
41	40	nncni 12243	. . . . . . 7 ⊢ (;10 · 8) ∈ ℂ
42		8cn 12338	. . . . . . 7 ⊢ 8 ∈ ℂ
43	9	nncni 12243	. . . . . . 7 ⊢ (;10↑;29) ∈ ℂ
44	41, 42, 43	adddiri 11222	. . . . . 6 ⊢ (((;10 · 8) + 8) · (;10↑;29)) = (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29)))
45	41, 43	mulcomi 11217	. . . . . . . . 9 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑;29) · (;10 · 8))
46	4	nncni 12243	. . . . . . . . . 10 ⊢ ;10 ∈ ℂ
47	43, 46, 42	mulassi 11220	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑;29) · (;10 · 8))
48		nncn 12241	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;10 ∈ ℂ)
49	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;29 ∈ ℕ₀)
50	48, 49	expp1d 14183	. . . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ ((;10↑;29) · ;10) = (;10↑(;29 + 1))
53	52	oveq1i 7421	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑(;29 + 1)) · 8)
54	45, 47, 53	3eqtr2i 2798	. . . . . . . 8 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑(;29 + 1)) · 8)
55	54	oveq1i 7421	. . . . . . 7 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29)))
56		2p1e3 12382	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
57		eqid 2769	. . . . . . . . . . 11 ⊢ ;29 = ;29
58	5, 56, 57	decsucc 12757	. . . . . . . . . 10 ⊢ (;29 + 1) = ;30
59	58	oveq2i 7422	. . . . . . . . 9 ⊢ (;10↑(;29 + 1)) = (;10↑;30)
60	59	oveq1i 7421	. . . . . . . 8 ⊢ ((;10↑(;29 + 1)) · 8) = ((;10↑;30) · 8)
61	60	oveq1i 7421	. . . . . . 7 ⊢ (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29))) = (((;10↑;30) · 8) + (8 · (;10↑;29)))
62	33	nncni 12243	. . . . . . . 8 ⊢ (;10↑;30) ∈ ℂ
63		mulcom 11186	. . . . . . . . 9 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → ((;10↑;30) · 8) = (8 · (;10↑;30)))
64	63	oveq1d 7426	. . . . . . . 8 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29))))
65	62, 42, 64	mp2an 704	. . . . . . 7 ⊢ (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
66	55, 61, 65	3eqtri 2796	. . . . . 6 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
67	39, 44, 66	3eqtri 2796	. . . . 5 ⊢ (;88 · (;10↑;29)) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
68	37, 67	breqtrri 5142	. . . 4 ⊢ (8 · (;10↑;30)) < (;88 · (;10↑;29))
69	25, 68	jctil 528	. . 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 3592	. 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 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 class class class wbr 5113 (class class class)co 7411 ℂcc 11098 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 ℕcn 12233 2c2 12295 3c3 12296 7c7 12300 8c8 12301 9c9 12302 ℕ₀cn0 12504 cdc 12711 ↑cexp 14097 Odd codd 48313 GoldbachOdd cgbo 48435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-bgbltosilva 48498 ax-hgprmladder 48502

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-inf 9403 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-rp 13017 df-ico 13378 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-prm 16730 df-iccp 48086 df-even 48314 df-odd 48315 df-gbe 48436 df-gbo 48438

This theorem is referenced by: tgoldbach 48505

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