tgoldbachlt - Mathbox for Alexander van der Vekens

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Theorem tgoldbachlt 49117

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 49116. (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 12653	. . . 4 ⊢ 8 ∈ ℕ₀
2		8nn 12462	. . . 4 ⊢ 8 ∈ ℕ
3	1, 2	decnncl 12862	. . 3 ⊢ ;88 ∈ ℕ
4		10nn 12858	. . . 4 ⊢ ;10 ∈ ℕ
5		2nn0 12647	. . . . 5 ⊢ 2 ∈ ℕ₀
6		9nn0 12654	. . . . 5 ⊢ 9 ∈ ℕ₀
7	5, 6	deccl 12853	. . . 4 ⊢ ;29 ∈ ℕ₀
8		nnexpcl 14241	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;29 ∈ ℕ₀) → (;10↑;29) ∈ ℕ)
9	4, 7, 8	mp2an 705	. . 3 ⊢ (;10↑;29) ∈ ℕ
10	3, 9	nnmulcli 12384	. 2 ⊢ (;88 · (;10↑;29)) ∈ ℕ
11		id 23	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → (;88 · (;10↑;29)) ∈ ℕ)
12		breq2 5110	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((8 · (;10↑;30)) < 𝑚 ↔ (8 · (;10↑;30)) < (;88 · (;10↑;29))))
13		breq2 5110	. . . . . . . 8 ⊢ (𝑚 = (;88 · (;10↑;29)) → (𝑛 < 𝑚 ↔ 𝑛 < (;88 · (;10↑;29))))
14	13	anbi2d 642	. . . . . . 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 3189	. . . . 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 644	. . . 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 487	. . 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 781	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ Odd )
20		simprl 783	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 7 < 𝑛)
21		simprr 785	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 < (;88 · (;10↑;29)))
22		tgblthelfgott 49116	. . . . . . 7 ⊢ ((𝑛 ∈ Odd ∧ 7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1399	. . . . . 6 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ GoldbachOdd )
24	23	ex 418	. . . . 5 ⊢ (((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) → ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
25	24	ralrimiva 3158	. . . 4 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
26	2, 9	nnmulcli 12384	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℕ
27	26	nngt0i 12401	. . . . . 6 ⊢ 0 < (8 · (;10↑;29))
28	26	nnrei 12368	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℝ
29		3nn0 12648	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
30		0nn0 12645	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
31	29, 30	deccl 12853	. . . . . . . . . 10 ⊢ ;30 ∈ ℕ₀
32		nnexpcl 14241	. . . . . . . . . 10 ⊢ ((;10 ∈ ℕ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℕ)
33	4, 31, 32	mp2an 705	. . . . . . . . 9 ⊢ (;10↑;30) ∈ ℕ
34	2, 33	nnmulcli 12384	. . . . . . . 8 ⊢ (8 · (;10↑;30)) ∈ ℕ
35	34	nnrei 12368	. . . . . . 7 ⊢ (8 · (;10↑;30)) ∈ ℝ
36	28, 35	ltaddposi 11889	. . . . . 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 12841	. . . . . . 7 ⊢ ;88 = ((;10 · 8) + 8)
39	38	oveq1i 7450	. . . . . 6 ⊢ (;88 · (;10↑;29)) = (((;10 · 8) + 8) · (;10↑;29))
40	4, 2	nnmulcli 12384	. . . . . . . 8 ⊢ (;10 · 8) ∈ ℕ
41	40	nncni 12369	. . . . . . 7 ⊢ (;10 · 8) ∈ ℂ
42		8cn 12464	. . . . . . 7 ⊢ 8 ∈ ℂ
43	9	nncni 12369	. . . . . . 7 ⊢ (;10↑;29) ∈ ℂ
44	41, 42, 43	adddiri 11346	. . . . . 6 ⊢ (((;10 · 8) + 8) · (;10↑;29)) = (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29)))
45	41, 43	mulcomi 11341	. . . . . . . . 9 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑;29) · (;10 · 8))
46	4	nncni 12369	. . . . . . . . . 10 ⊢ ;10 ∈ ℂ
47	43, 46, 42	mulassi 11344	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑;29) · (;10 · 8))
48		nncn 12367	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;10 ∈ ℂ)
49	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;29 ∈ ℕ₀)
50	48, 49	expp1d 14314	. . . . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ ((;10↑;29) · ;10) = (;10↑(;29 + 1))
53	52	oveq1i 7450	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑(;29 + 1)) · 8)
54	45, 47, 53	3eqtr2i 2792	. . . . . . . 8 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑(;29 + 1)) · 8)
55	54	oveq1i 7450	. . . . . . 7 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29)))
56		2p1e3 12508	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
57		eqid 2763	. . . . . . . . . . 11 ⊢ ;29 = ;29
58	5, 56, 57	decsucc 12884	. . . . . . . . . 10 ⊢ (;29 + 1) = ;30
59	58	oveq2i 7451	. . . . . . . . 9 ⊢ (;10↑(;29 + 1)) = (;10↑;30)
60	59	oveq1i 7450	. . . . . . . 8 ⊢ ((;10↑(;29 + 1)) · 8) = ((;10↑;30) · 8)
61	60	oveq1i 7450	. . . . . . 7 ⊢ (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29))) = (((;10↑;30) · 8) + (8 · (;10↑;29)))
62	33	nncni 12369	. . . . . . . 8 ⊢ (;10↑;30) ∈ ℂ
63		mulcom 11310	. . . . . . . . 9 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → ((;10↑;30) · 8) = (8 · (;10↑;30)))
64	63	oveq1d 7455	. . . . . . . 8 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29))))
65	62, 42, 64	mp2an 705	. . . . . . 7 ⊢ (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
66	55, 61, 65	3eqtri 2790	. . . . . 6 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
67	39, 44, 66	3eqtri 2790	. . . . 5 ⊢ (;88 · (;10↑;29)) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
68	37, 67	breqtrri 5135	. . . 4 ⊢ (8 · (;10↑;30)) < (;88 · (;10↑;29))
69	25, 68	jctil 529	. . 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 3582	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 class class class wbr 5106 (class class class)co 7440 ℂcc 11222 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 < clt 11367 ℕcn 12359 2c2 12421 3c3 12422 7c7 12426 8c8 12427 9c9 12428 ℕ₀cn0 12630 cdc 12838 ↑cexp 14228 Odd codd 48926 GoldbachOdd cgbo 49048

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302 ax-bgbltosilva 49111 ax-hgprmladder 49115

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-sup 9458 df-inf 9459 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-rp 13145 df-ico 13506 df-fz 13664 df-fzo 13813 df-seq 14169 df-exp 14229 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-dvds 16449 df-prm 16873 df-iccp 48699 df-even 48927 df-odd 48928 df-gbe 49049 df-gbo 49051

This theorem is used by: tgoldbach 49118

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