Theorem bgoldbachlt 41701

Description: The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big m). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 41698. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.)

Assertion

Ref	Expression
bgoldbachlt	|- E. m e. NN ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m /\ A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) )

Distinct variable group:   m, n

Proof of Theorem bgoldbachlt

Step	Hyp	Ref	Expression
1		4nn 11187	. . 3 |- 4 e. NN
2		10nn 11514	. . . 4 |- ; 1 0 e. NN
3		1nn0 11308	. . . . 5 |- 1 e. NN0
4		8nn0 11315	. . . . 5 |- 8 e. NN0
5	3, 4	deccl 11512	. . . 4 |- ; 1 8 e. NN0
6		nnexpcl 12873	. . . 4 |- ( (; 1 0 e. NN /\ ; 1 8 e. NN0 ) -> (; 1 0 ^; 1 8 ) e. NN )
7	2, 5, 6	mp2an 708	. . 3 |- (; 1 0 ^; 1 8 ) e. NN
8	1, 7	nnmulcli 11044	. 2 |- ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN
9		id 22	. . 3 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN )
10		breq2 4657	. . . . 5 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m <-> ( 4 x. (; 1 0 ^; 1 8 ) ) <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) )
11		breq2 4657	. . . . . . . 8 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( n < m <-> n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) )
12	11	anbi2d 740	. . . . . . 7 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( ( 4 < n /\ n < m ) <-> ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) )
13	12	imbi1d 331	. . . . . 6 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) <-> ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) ) )
14	13	ralbidv 2986	. . . . 5 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) <-> A. n e. Even ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) ) )
15	10, 14	anbi12d 747	. . . 4 |- ( m = ( 4 x. (; 1 0 ^; 1 8 ) ) -> ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m /\ A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) ) <-> ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ ( 4 x. (; 1 0 ^; 1 8 ) ) /\ A. n e. Even ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) ) ) )
16	15	adantl 482	. . 3 |- ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ m = ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m /\ A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) ) <-> ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ ( 4 x. (; 1 0 ^; 1 8 ) ) /\ A. n e. Even ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) ) ) )
17		nnre 11027	. . . . 5 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> ( 4 x. (; 1 0 ^; 1 8 ) ) e. RR )
18	17	leidd 10594	. . . 4 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> ( 4 x. (; 1 0 ^; 1 8 ) ) <_ ( 4 x. (; 1 0 ^; 1 8 ) ) )
19		simplr 792	. . . . . . 7 |- ( ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) /\ ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) -> n e. Even )
20		simprl 794	. . . . . . 7 |- ( ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) /\ ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) -> 4 < n )
21		evenz 41543	. . . . . . . . . . 11 |- ( n e. Even -> n e. ZZ )
22	21	zred 11482	. . . . . . . . . 10 |- ( n e. Even -> n e. RR )
23		ltle 10126	. . . . . . . . . 10 |- ( ( n e. RR /\ ( 4 x. (; 1 0 ^; 1 8 ) ) e. RR ) -> ( n < ( 4 x. (; 1 0 ^; 1 8 ) ) -> n <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) )
24	22, 17, 23	syl2anr 495	. . . . . . . . 9 |- ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) -> ( n < ( 4 x. (; 1 0 ^; 1 8 ) ) -> n <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) )
25	24	a1d 25	. . . . . . . 8 |- ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) -> ( 4 < n -> ( n < ( 4 x. (; 1 0 ^; 1 8 ) ) -> n <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) )
26	25	imp32 449	. . . . . . 7 |- ( ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) /\ ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) -> n <_ ( 4 x. (; 1 0 ^; 1 8 ) ) )
27		ax-bgbltosilva 41698	. . . . . . 7 |- ( ( n e. Even /\ 4 < n /\ n <_ ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven )
28	19, 20, 26, 27	syl3anc 1326	. . . . . 6 |- ( ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) /\ ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) ) -> n e. GoldbachEven )
29	28	ex 450	. . . . 5 |- ( ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN /\ n e. Even ) -> ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) )
30	29	ralrimiva 2966	. . . 4 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> A. n e. Even ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) )
31	18, 30	jca 554	. . 3 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ ( 4 x. (; 1 0 ^; 1 8 ) ) /\ A. n e. Even ( ( 4 < n /\ n < ( 4 x. (; 1 0 ^; 1 8 ) ) ) -> n e. GoldbachEven ) ) )
32	9, 16, 31	rspcedvd 3317	. 2 |- ( ( 4 x. (; 1 0 ^; 1 8 ) ) e. NN -> E. m e. NN ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m /\ A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) ) )
33	8, 32	ax-mp 5	1 |- E. m e. NN ( ( 4 x. (; 1 0 ^; 1 8 ) ) <_ m /\ A. n e. Even ( ( 4 < n /\ n < m ) -> n e. GoldbachEven ) )

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   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   NNcn 11020   4c4 11072   8c8 11076   NN0cn0 11292  ;cdc 11493   ^cexp 12860   Even ceven 41537   GoldbachEven cgbe 41633

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-bgbltosilva 41698

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  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-seq 12802  df-exp 12861  df-even 41539

This theorem is referenced by: (None)