Theorem tgoldbachlt 41704

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 41703. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)

Assertion

Ref	Expression
tgoldbachlt	|- E. m e. NN ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m /\ A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) )

Distinct variable group:   m, n

Proof of Theorem tgoldbachlt

Step	Hyp	Ref	Expression
1		8nn0 11315	. . . 4 |- 8 e. NN0
2		8nn 11191	. . . 4 |- 8 e. NN
3	1, 2	decnncl 11518	. . 3 |- ; 8 8 e. NN
4		10nn 11514	. . . 4 |- ; 1 0 e. NN
5		2nn0 11309	. . . . 5 |- 2 e. NN0
6		9nn0 11316	. . . . 5 |- 9 e. NN0
7	5, 6	deccl 11512	. . . 4 |- ; 2 9 e. NN0
8		nnexpcl 12873	. . . 4 |- ( (; 1 0 e. NN /\ ; 2 9 e. NN0 ) -> (; 1 0 ^; 2 9 ) e. NN )
9	4, 7, 8	mp2an 708	. . 3 |- (; 1 0 ^; 2 9 ) e. NN
10	3, 9	nnmulcli 11044	. 2 |- (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN
11		id 22	. . 3 |- ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN -> (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN )
12		breq2 4657	. . . . 5 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m <-> ( 8 x. (; 1 0 ^; 3 0 ) ) < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) )
13		breq2 4657	. . . . . . . 8 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( n < m <-> n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) )
14	13	anbi2d 740	. . . . . . 7 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( ( 7 < n /\ n < m ) <-> ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) ) )
15	14	imbi1d 331	. . . . . 6 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) <-> ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) ) )
16	15	ralbidv 2986	. . . . 5 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) <-> A. n e. Odd ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) ) )
17	12, 16	anbi12d 747	. . . 4 |- ( m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) -> ( ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m /\ A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) ) <-> ( ( 8 x. (; 1 0 ^; 3 0 ) ) < (; 8 8 x. (; 1 0 ^; 2 9 ) ) /\ A. n e. Odd ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) ) ) )
18	17	adantl 482	. . 3 |- ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ m = (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> ( ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m /\ A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) ) <-> ( ( 8 x. (; 1 0 ^; 3 0 ) ) < (; 8 8 x. (; 1 0 ^; 2 9 ) ) /\ A. n e. Odd ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) ) ) )
19		simplr 792	. . . . . . 7 |- ( ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ n e. Odd ) /\ ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) ) -> n e. Odd )
20		simprl 794	. . . . . . 7 |- ( ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ n e. Odd ) /\ ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) ) -> 7 < n )
21		simprr 796	. . . . . . 7 |- ( ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ n e. Odd ) /\ ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) ) -> n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) )
22		tgblthelfgott 41703	. . . . . . 7 |- ( ( n e. Odd /\ 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd )
23	19, 20, 21, 22	syl3anc 1326	. . . . . 6 |- ( ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ n e. Odd ) /\ ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) ) -> n e. GoldbachOdd )
24	23	ex 450	. . . . 5 |- ( ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN /\ n e. Odd ) -> ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) )
25	24	ralrimiva 2966	. . . 4 |- ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN -> A. n e. Odd ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) )
26	2, 9	nnmulcli 11044	. . . . . . 7 |- ( 8 x. (; 1 0 ^; 2 9 ) ) e. NN
27	26	nngt0i 11054	. . . . . 6 |- 0 < ( 8 x. (; 1 0 ^; 2 9 ) )
28	26	nnrei 11029	. . . . . . 7 |- ( 8 x. (; 1 0 ^; 2 9 ) ) e. RR
29		3nn0 11310	. . . . . . . . . . 11 |- 3 e. NN0
30		0nn0 11307	. . . . . . . . . . 11 |- 0 e. NN0
31	29, 30	deccl 11512	. . . . . . . . . 10 |- ; 3 0 e. NN0
32		nnexpcl 12873	. . . . . . . . . 10 |- ( (; 1 0 e. NN /\ ; 3 0 e. NN0 ) -> (; 1 0 ^; 3 0 ) e. NN )
33	4, 31, 32	mp2an 708	. . . . . . . . 9 |- (; 1 0 ^; 3 0 ) e. NN
34	2, 33	nnmulcli 11044	. . . . . . . 8 |- ( 8 x. (; 1 0 ^; 3 0 ) ) e. NN
35	34	nnrei 11029	. . . . . . 7 |- ( 8 x. (; 1 0 ^; 3 0 ) ) e. RR
36	28, 35	ltaddposi 10577	. . . . . 6 |- ( 0 < ( 8 x. (; 1 0 ^; 2 9 ) ) <-> ( 8 x. (; 1 0 ^; 3 0 ) ) < ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) )
37	27, 36	mpbi 220	. . . . 5 |- ( 8 x. (; 1 0 ^; 3 0 ) ) < ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
38		dfdec10 11497	. . . . . . 7 |- ; 8 8 = ( (; 1 0 x. 8 ) + 8 )
39	38	oveq1i 6660	. . . . . 6 |- (; 8 8 x. (; 1 0 ^; 2 9 ) ) = ( ( (; 1 0 x. 8 ) + 8 ) x. (; 1 0 ^; 2 9 ) )
40	4, 2	nnmulcli 11044	. . . . . . . 8 |- (; 1 0 x. 8 ) e. NN
41	40	nncni 11030	. . . . . . 7 |- (; 1 0 x. 8 ) e. CC
42		8cn 11106	. . . . . . 7 |- 8 e. CC
43	9	nncni 11030	. . . . . . 7 |- (; 1 0 ^; 2 9 ) e. CC
44	41, 42, 43	adddiri 10051	. . . . . 6 |- ( ( (; 1 0 x. 8 ) + 8 ) x. (; 1 0 ^; 2 9 ) ) = ( ( (; 1 0 x. 8 ) x. (; 1 0 ^; 2 9 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
45	41, 43	mulcomi 10046	. . . . . . . . 9 |- ( (; 1 0 x. 8 ) x. (; 1 0 ^; 2 9 ) ) = ( (; 1 0 ^; 2 9 ) x. (; 1 0 x. 8 ) )
46	4	nncni 11030	. . . . . . . . . 10 |- ; 1 0 e. CC
47	43, 46, 42	mulassi 10049	. . . . . . . . 9 |- ( ( (; 1 0 ^; 2 9 ) x. ; 1 0 ) x. 8 ) = ( (; 1 0 ^; 2 9 ) x. (; 1 0 x. 8 ) )
48		nncn 11028	. . . . . . . . . . . . 13 |- (; 1 0 e. NN -> ; 1 0 e. CC )
49	7	a1i 11	. . . . . . . . . . . . 13 |- (; 1 0 e. NN -> ; 2 9 e. NN0 )
50	48, 49	expp1d 13009	. . . . . . . . . . . 12 |- (; 1 0 e. NN -> (; 1 0 ^ (; 2 9 + 1 ) ) = ( (; 1 0 ^; 2 9 ) x. ; 1 0 ) )
51	4, 50	ax-mp 5	. . . . . . . . . . 11 |- (; 1 0 ^ (; 2 9 + 1 ) ) = ( (; 1 0 ^; 2 9 ) x. ; 1 0 )
52	51	eqcomi 2631	. . . . . . . . . 10 |- ( (; 1 0 ^; 2 9 ) x. ; 1 0 ) = (; 1 0 ^ (; 2 9 + 1 ) )
53	52	oveq1i 6660	. . . . . . . . 9 |- ( ( (; 1 0 ^; 2 9 ) x. ; 1 0 ) x. 8 ) = ( (; 1 0 ^ (; 2 9 + 1 ) ) x. 8 )
54	45, 47, 53	3eqtr2i 2650	. . . . . . . 8 |- ( (; 1 0 x. 8 ) x. (; 1 0 ^; 2 9 ) ) = ( (; 1 0 ^ (; 2 9 + 1 ) ) x. 8 )
55	54	oveq1i 6660	. . . . . . 7 |- ( ( (; 1 0 x. 8 ) x. (; 1 0 ^; 2 9 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) = ( ( (; 1 0 ^ (; 2 9 + 1 ) ) x. 8 ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
56		2p1e3 11151	. . . . . . . . . . 11 |- ( 2 + 1 ) = 3
57		eqid 2622	. . . . . . . . . . 11 |- ; 2 9 = ; 2 9
58	5, 56, 57	decsucc 11550	. . . . . . . . . 10 |- (; 2 9 + 1 ) = ; 3 0
59	58	oveq2i 6661	. . . . . . . . 9 |- (; 1 0 ^ (; 2 9 + 1 ) ) = (; 1 0 ^; 3 0 )
60	59	oveq1i 6660	. . . . . . . 8 |- ( (; 1 0 ^ (; 2 9 + 1 ) ) x. 8 ) = ( (; 1 0 ^; 3 0 ) x. 8 )
61	60	oveq1i 6660	. . . . . . 7 |- ( ( (; 1 0 ^ (; 2 9 + 1 ) ) x. 8 ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) = ( ( (; 1 0 ^; 3 0 ) x. 8 ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
62	33	nncni 11030	. . . . . . . 8 |- (; 1 0 ^; 3 0 ) e. CC
63		mulcom 10022	. . . . . . . . 9 |- ( ( (; 1 0 ^; 3 0 ) e. CC /\ 8 e. CC ) -> ( (; 1 0 ^; 3 0 ) x. 8 ) = ( 8 x. (; 1 0 ^; 3 0 ) ) )
64	63	oveq1d 6665	. . . . . . . 8 |- ( ( (; 1 0 ^; 3 0 ) e. CC /\ 8 e. CC ) -> ( ( (; 1 0 ^; 3 0 ) x. 8 ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) = ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) )
65	62, 42, 64	mp2an 708	. . . . . . 7 |- ( ( (; 1 0 ^; 3 0 ) x. 8 ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) = ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
66	55, 61, 65	3eqtri 2648	. . . . . 6 |- ( ( (; 1 0 x. 8 ) x. (; 1 0 ^; 2 9 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) ) = ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
67	39, 44, 66	3eqtri 2648	. . . . 5 |- (; 8 8 x. (; 1 0 ^; 2 9 ) ) = ( ( 8 x. (; 1 0 ^; 3 0 ) ) + ( 8 x. (; 1 0 ^; 2 9 ) ) )
68	37, 67	breqtrri 4680	. . . 4 |- ( 8 x. (; 1 0 ^; 3 0 ) ) < (; 8 8 x. (; 1 0 ^; 2 9 ) )
69	25, 68	jctil 560	. . 3 |- ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN -> ( ( 8 x. (; 1 0 ^; 3 0 ) ) < (; 8 8 x. (; 1 0 ^; 2 9 ) ) /\ A. n e. Odd ( ( 7 < n /\ n < (; 8 8 x. (; 1 0 ^; 2 9 ) ) ) -> n e. GoldbachOdd ) ) )
70	11, 18, 69	rspcedvd 3317	. 2 |- ( (; 8 8 x. (; 1 0 ^; 2 9 ) ) e. NN -> E. m e. NN ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m /\ A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) ) )
71	10, 70	ax-mp 5	1 |- E. m e. NN ( ( 8 x. (; 1 0 ^; 3 0 ) ) < m /\ A. n e. Odd ( ( 7 < n /\ n < m ) -> n e. GoldbachOdd ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   NNcn 11020   2c2 11070   3c3 11071   7c7 11075   8c8 11076   9c9 11077   NN0cn0 11292  ;cdc 11493   ^cexp 12860   Odd codd 41538   GoldbachOdd cgbo 41635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-bgbltosilva 41698  ax-hgprmladder 41702

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-prm 15386  df-iccp 41350  df-even 41539  df-odd 41540  df-gbe 41636  df-gbo 41638

This theorem is referenced by:  tgoldbach  41705