tgoldbachlt - Mathbox for Alexander van der Vekens

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Theorem tgoldbachlt 45156

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 45155. (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 12186	. . . 4 ⊢ 8 ∈ ℕ₀
2		8nn 11998	. . . 4 ⊢ 8 ∈ ℕ
3	1, 2	decnncl 12386	. . 3 ⊢ ;88 ∈ ℕ
4		10nn 12382	. . . 4 ⊢ ;10 ∈ ℕ
5		2nn0 12180	. . . . 5 ⊢ 2 ∈ ℕ₀
6		9nn0 12187	. . . . 5 ⊢ 9 ∈ ℕ₀
7	5, 6	deccl 12381	. . . 4 ⊢ ;29 ∈ ℕ₀
8		nnexpcl 13723	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;29 ∈ ℕ₀) → (;10↑;29) ∈ ℕ)
9	4, 7, 8	mp2an 688	. . 3 ⊢ (;10↑;29) ∈ ℕ
10	3, 9	nnmulcli 11928	. 2 ⊢ (;88 · (;10↑;29)) ∈ ℕ
11		id 22	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → (;88 · (;10↑;29)) ∈ ℕ)
12		breq2 5074	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((8 · (;10↑;30)) < 𝑚 ↔ (8 · (;10↑;30)) < (;88 · (;10↑;29))))
13		breq2 5074	. . . . . . . 8 ⊢ (𝑚 = (;88 · (;10↑;29)) → (𝑛 < 𝑚 ↔ 𝑛 < (;88 · (;10↑;29))))
14	13	anbi2d 628	. . . . . . 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 3120	. . . . 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 630	. . . 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 765	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 ∈ Odd )
20		simprl 767	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 7 < 𝑛)
21		simprr 769	. . . . . . 7 ⊢ ((((;88 · (;10↑;29)) ∈ ℕ ∧ 𝑛 ∈ Odd ) ∧ (7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29)))) → 𝑛 < (;88 · (;10↑;29)))
22		tgblthelfgott 45155	. . . . . . 7 ⊢ ((𝑛 ∈ Odd ∧ 7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1369	. . . . . 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 3107	. . . 4 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
26	2, 9	nnmulcli 11928	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℕ
27	26	nngt0i 11942	. . . . . 6 ⊢ 0 < (8 · (;10↑;29))
28	26	nnrei 11912	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℝ
29		3nn0 12181	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
30		0nn0 12178	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
31	29, 30	deccl 12381	. . . . . . . . . 10 ⊢ ;30 ∈ ℕ₀
32		nnexpcl 13723	. . . . . . . . . 10 ⊢ ((;10 ∈ ℕ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℕ)
33	4, 31, 32	mp2an 688	. . . . . . . . 9 ⊢ (;10↑;30) ∈ ℕ
34	2, 33	nnmulcli 11928	. . . . . . . 8 ⊢ (8 · (;10↑;30)) ∈ ℕ
35	34	nnrei 11912	. . . . . . 7 ⊢ (8 · (;10↑;30)) ∈ ℝ
36	28, 35	ltaddposi 11454	. . . . . 6 ⊢ (0 < (8 · (;10↑;29)) ↔ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29))))
37	27, 36	mpbi 229	. . . . 5 ⊢ (8 · (;10↑;30)) < ((8 · (;10↑;30)) + (8 · (;10↑;29)))
38		dfdec10 12369	. . . . . . 7 ⊢ ;88 = ((;10 · 8) + 8)
39	38	oveq1i 7265	. . . . . 6 ⊢ (;88 · (;10↑;29)) = (((;10 · 8) + 8) · (;10↑;29))
40	4, 2	nnmulcli 11928	. . . . . . . 8 ⊢ (;10 · 8) ∈ ℕ
41	40	nncni 11913	. . . . . . 7 ⊢ (;10 · 8) ∈ ℂ
42		8cn 12000	. . . . . . 7 ⊢ 8 ∈ ℂ
43	9	nncni 11913	. . . . . . 7 ⊢ (;10↑;29) ∈ ℂ
44	41, 42, 43	adddiri 10919	. . . . . 6 ⊢ (((;10 · 8) + 8) · (;10↑;29)) = (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29)))
45	41, 43	mulcomi 10914	. . . . . . . . 9 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑;29) · (;10 · 8))
46	4	nncni 11913	. . . . . . . . . 10 ⊢ ;10 ∈ ℂ
47	43, 46, 42	mulassi 10917	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑;29) · (;10 · 8))
48		nncn 11911	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;10 ∈ ℂ)
49	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;29 ∈ ℕ₀)
50	48, 49	expp1d 13793	. . . . . . . . . . . 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 2747	. . . . . . . . . 10 ⊢ ((;10↑;29) · ;10) = (;10↑(;29 + 1))
53	52	oveq1i 7265	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑(;29 + 1)) · 8)
54	45, 47, 53	3eqtr2i 2772	. . . . . . . 8 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑(;29 + 1)) · 8)
55	54	oveq1i 7265	. . . . . . 7 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29)))
56		2p1e3 12045	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
57		eqid 2738	. . . . . . . . . . 11 ⊢ ;29 = ;29
58	5, 56, 57	decsucc 12407	. . . . . . . . . 10 ⊢ (;29 + 1) = ;30
59	58	oveq2i 7266	. . . . . . . . 9 ⊢ (;10↑(;29 + 1)) = (;10↑;30)
60	59	oveq1i 7265	. . . . . . . 8 ⊢ ((;10↑(;29 + 1)) · 8) = ((;10↑;30) · 8)
61	60	oveq1i 7265	. . . . . . 7 ⊢ (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29))) = (((;10↑;30) · 8) + (8 · (;10↑;29)))
62	33	nncni 11913	. . . . . . . 8 ⊢ (;10↑;30) ∈ ℂ
63		mulcom 10888	. . . . . . . . 9 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → ((;10↑;30) · 8) = (8 · (;10↑;30)))
64	63	oveq1d 7270	. . . . . . . 8 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29))))
65	62, 42, 64	mp2an 688	. . . . . . 7 ⊢ (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
66	55, 61, 65	3eqtri 2770	. . . . . 6 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
67	39, 44, 66	3eqtri 2770	. . . . 5 ⊢ (;88 · (;10↑;29)) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
68	37, 67	breqtrri 5097	. . . 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 3555	. 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 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 class class class wbr 5070 (class class class)co 7255 ℂcc 10800 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ℕcn 11903 2c2 11958 3c3 11959 7c7 11963 8c8 11964 9c9 11965 ℕ₀cn0 12163 cdc 12366 ↑cexp 13710 Odd codd 44965 GoldbachOdd cgbo 45087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-bgbltosilva 45150 ax-hgprmladder 45154

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-rp 12660 df-ico 13014 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-dvds 15892 df-prm 16305 df-iccp 44754 df-even 44966 df-odd 44967 df-gbe 45088 df-gbo 45090

This theorem is referenced by: tgoldbach 45157

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