tgoldbachlt - Mathbox for Alexander van der Vekens

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Theorem tgoldbachlt 47821

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 47820. (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 12472	. . . 4 ⊢ 8 ∈ ℕ₀
2		8nn 12288	. . . 4 ⊢ 8 ∈ ℕ
3	1, 2	decnncl 12676	. . 3 ⊢ ;88 ∈ ℕ
4		10nn 12672	. . . 4 ⊢ ;10 ∈ ℕ
5		2nn0 12466	. . . . 5 ⊢ 2 ∈ ℕ₀
6		9nn0 12473	. . . . 5 ⊢ 9 ∈ ℕ₀
7	5, 6	deccl 12671	. . . 4 ⊢ ;29 ∈ ℕ₀
8		nnexpcl 14046	. . . 4 ⊢ ((;10 ∈ ℕ ∧ ;29 ∈ ℕ₀) → (;10↑;29) ∈ ℕ)
9	4, 7, 8	mp2an 692	. . 3 ⊢ (;10↑;29) ∈ ℕ
10	3, 9	nnmulcli 12218	. 2 ⊢ (;88 · (;10↑;29)) ∈ ℕ
11		id 22	. . 3 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → (;88 · (;10↑;29)) ∈ ℕ)
12		breq2 5114	. . . . 5 ⊢ (𝑚 = (;88 · (;10↑;29)) → ((8 · (;10↑;30)) < 𝑚 ↔ (8 · (;10↑;30)) < (;88 · (;10↑;29))))
13		breq2 5114	. . . . . . . 8 ⊢ (𝑚 = (;88 · (;10↑;29)) → (𝑛 < 𝑚 ↔ 𝑛 < (;88 · (;10↑;29))))
14	13	anbi2d 630	. . . . . . 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 3157	. . . . 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 632	. . . 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 47820	. . . . . . 7 ⊢ ((𝑛 ∈ Odd ∧ 7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1373	. . . . . 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 3126	. . . 4 ⊢ ((;88 · (;10↑;29)) ∈ ℕ → ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (;88 · (;10↑;29))) → 𝑛 ∈ GoldbachOdd ))
26	2, 9	nnmulcli 12218	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℕ
27	26	nngt0i 12232	. . . . . 6 ⊢ 0 < (8 · (;10↑;29))
28	26	nnrei 12202	. . . . . . 7 ⊢ (8 · (;10↑;29)) ∈ ℝ
29		3nn0 12467	. . . . . . . . . . 11 ⊢ 3 ∈ ℕ₀
30		0nn0 12464	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ₀
31	29, 30	deccl 12671	. . . . . . . . . 10 ⊢ ;30 ∈ ℕ₀
32		nnexpcl 14046	. . . . . . . . . 10 ⊢ ((;10 ∈ ℕ ∧ ;30 ∈ ℕ₀) → (;10↑;30) ∈ ℕ)
33	4, 31, 32	mp2an 692	. . . . . . . . 9 ⊢ (;10↑;30) ∈ ℕ
34	2, 33	nnmulcli 12218	. . . . . . . 8 ⊢ (8 · (;10↑;30)) ∈ ℕ
35	34	nnrei 12202	. . . . . . 7 ⊢ (8 · (;10↑;30)) ∈ ℝ
36	28, 35	ltaddposi 11734	. . . . . 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 12659	. . . . . . 7 ⊢ ;88 = ((;10 · 8) + 8)
39	38	oveq1i 7400	. . . . . 6 ⊢ (;88 · (;10↑;29)) = (((;10 · 8) + 8) · (;10↑;29))
40	4, 2	nnmulcli 12218	. . . . . . . 8 ⊢ (;10 · 8) ∈ ℕ
41	40	nncni 12203	. . . . . . 7 ⊢ (;10 · 8) ∈ ℂ
42		8cn 12290	. . . . . . 7 ⊢ 8 ∈ ℂ
43	9	nncni 12203	. . . . . . 7 ⊢ (;10↑;29) ∈ ℂ
44	41, 42, 43	adddiri 11194	. . . . . 6 ⊢ (((;10 · 8) + 8) · (;10↑;29)) = (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29)))
45	41, 43	mulcomi 11189	. . . . . . . . 9 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑;29) · (;10 · 8))
46	4	nncni 12203	. . . . . . . . . 10 ⊢ ;10 ∈ ℂ
47	43, 46, 42	mulassi 11192	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑;29) · (;10 · 8))
48		nncn 12201	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;10 ∈ ℂ)
49	7	a1i 11	. . . . . . . . . . . . 13 ⊢ (;10 ∈ ℕ → ;29 ∈ ℕ₀)
50	48, 49	expp1d 14119	. . . . . . . . . . . 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 2739	. . . . . . . . . 10 ⊢ ((;10↑;29) · ;10) = (;10↑(;29 + 1))
53	52	oveq1i 7400	. . . . . . . . 9 ⊢ (((;10↑;29) · ;10) · 8) = ((;10↑(;29 + 1)) · 8)
54	45, 47, 53	3eqtr2i 2759	. . . . . . . 8 ⊢ ((;10 · 8) · (;10↑;29)) = ((;10↑(;29 + 1)) · 8)
55	54	oveq1i 7400	. . . . . . 7 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29)))
56		2p1e3 12330	. . . . . . . . . . 11 ⊢ (2 + 1) = 3
57		eqid 2730	. . . . . . . . . . 11 ⊢ ;29 = ;29
58	5, 56, 57	decsucc 12697	. . . . . . . . . 10 ⊢ (;29 + 1) = ;30
59	58	oveq2i 7401	. . . . . . . . 9 ⊢ (;10↑(;29 + 1)) = (;10↑;30)
60	59	oveq1i 7400	. . . . . . . 8 ⊢ ((;10↑(;29 + 1)) · 8) = ((;10↑;30) · 8)
61	60	oveq1i 7400	. . . . . . 7 ⊢ (((;10↑(;29 + 1)) · 8) + (8 · (;10↑;29))) = (((;10↑;30) · 8) + (8 · (;10↑;29)))
62	33	nncni 12203	. . . . . . . 8 ⊢ (;10↑;30) ∈ ℂ
63		mulcom 11161	. . . . . . . . 9 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → ((;10↑;30) · 8) = (8 · (;10↑;30)))
64	63	oveq1d 7405	. . . . . . . 8 ⊢ (((;10↑;30) ∈ ℂ ∧ 8 ∈ ℂ) → (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29))))
65	62, 42, 64	mp2an 692	. . . . . . 7 ⊢ (((;10↑;30) · 8) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
66	55, 61, 65	3eqtri 2757	. . . . . 6 ⊢ (((;10 · 8) · (;10↑;29)) + (8 · (;10↑;29))) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
67	39, 44, 66	3eqtri 2757	. . . . 5 ⊢ (;88 · (;10↑;29)) = ((8 · (;10↑;30)) + (8 · (;10↑;29)))
68	37, 67	breqtrri 5137	. . . 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 3593	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 class class class wbr 5110 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 ℕcn 12193 2c2 12248 3c3 12249 7c7 12253 8c8 12254 9c9 12255 ℕ₀cn0 12449 cdc 12656 ↑cexp 14033 Odd codd 47630 GoldbachOdd cgbo 47752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-bgbltosilva 47815 ax-hgprmladder 47819

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-ico 13319 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-prm 16649 df-iccp 47419 df-even 47631 df-odd 47632 df-gbe 47753 df-gbo 47755

This theorem is referenced by: tgoldbach 47822

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