Theorem mogoldbb 41673

Description: If the modern version of the original formulation of the Goldbach conjecture is valid, the (weak) binary Goldbach conjecture also holds. (Contributed by AV, 26-Dec-2021.)

Assertion

Ref	Expression
mogoldbb	|- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )

Distinct variable group:   n, p, q, r

Proof of Theorem mogoldbb

Dummy variables x m y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfra1 2941	. 2 |- F/ n A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r )
2		eqeq1 2626	. . . . . . . 8 |- ( n = m -> ( n = ( ( p + q ) + r ) <-> m = ( ( p + q ) + r ) ) )
3	2	rexbidv 3052	. . . . . . 7 |- ( n = m -> ( E. r e. Prime n = ( ( p + q ) + r ) <-> E. r e. Prime m = ( ( p + q ) + r ) ) )
4	3	2rexbidv 3057	. . . . . 6 |- ( n = m -> ( E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) <-> E. p e. Prime E. q e. Prime E. r e. Prime m = ( ( p + q ) + r ) ) )
5	4	cbvralv 3171	. . . . 5 |- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) <-> A. m e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime m = ( ( p + q ) + r ) )
6		6nn 11189	. . . . . . . . 9 |- 6 e. NN
7	6	nnzi 11401	. . . . . . . 8 |- 6 e. ZZ
8	7	a1i 11	. . . . . . 7 |- ( ( n e. Even /\ 2 < n ) -> 6 e. ZZ )
9		evenz 41543	. . . . . . . . 9 |- ( n e. Even -> n e. ZZ )
10		2z 11409	. . . . . . . . . 10 |- 2 e. ZZ
11	10	a1i 11	. . . . . . . . 9 |- ( n e. Even -> 2 e. ZZ )
12	9, 11	zaddcld 11486	. . . . . . . 8 |- ( n e. Even -> ( n + 2 ) e. ZZ )
13	12	adantr 481	. . . . . . 7 |- ( ( n e. Even /\ 2 < n ) -> ( n + 2 ) e. ZZ )
14		4cn 11098	. . . . . . . . . 10 |- 4 e. CC
15		2cn 11091	. . . . . . . . . 10 |- 2 e. CC
16		4p2e6 11162	. . . . . . . . . . 11 |- ( 4 + 2 ) = 6
17	16	eqcomi 2631	. . . . . . . . . 10 |- 6 = ( 4 + 2 )
18	14, 15, 17	mvrraddi 10298	. . . . . . . . 9 |- ( 6 - 2 ) = 4
19		2p2e4 11144	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
20		2evenALTV 41603	. . . . . . . . . . 11 |- 2 e. Even
21		evenltle 41626	. . . . . . . . . . 11 |- ( ( n e. Even /\ 2 e. Even /\ 2 < n ) -> ( 2 + 2 ) <_ n )
22	20, 21	mp3an2 1412	. . . . . . . . . 10 |- ( ( n e. Even /\ 2 < n ) -> ( 2 + 2 ) <_ n )
23	19, 22	syl5eqbrr 4689	. . . . . . . . 9 |- ( ( n e. Even /\ 2 < n ) -> 4 <_ n )
24	18, 23	syl5eqbr 4688	. . . . . . . 8 |- ( ( n e. Even /\ 2 < n ) -> ( 6 - 2 ) <_ n )
25		6re 11101	. . . . . . . . . . . 12 |- 6 e. RR
26	25	a1i 11	. . . . . . . . . . 11 |- ( n e. Even -> 6 e. RR )
27		2re 11090	. . . . . . . . . . . 12 |- 2 e. RR
28	27	a1i 11	. . . . . . . . . . 11 |- ( n e. Even -> 2 e. RR )
29	9	zred 11482	. . . . . . . . . . 11 |- ( n e. Even -> n e. RR )
30	26, 28, 29	3jca 1242	. . . . . . . . . 10 |- ( n e. Even -> ( 6 e. RR /\ 2 e. RR /\ n e. RR ) )
31	30	adantr 481	. . . . . . . . 9 |- ( ( n e. Even /\ 2 < n ) -> ( 6 e. RR /\ 2 e. RR /\ n e. RR ) )
32		lesubadd 10500	. . . . . . . . 9 |- ( ( 6 e. RR /\ 2 e. RR /\ n e. RR ) -> ( ( 6 - 2 ) <_ n <-> 6 <_ ( n + 2 ) ) )
33	31, 32	syl 17	. . . . . . . 8 |- ( ( n e. Even /\ 2 < n ) -> ( ( 6 - 2 ) <_ n <-> 6 <_ ( n + 2 ) ) )
34	24, 33	mpbid 222	. . . . . . 7 |- ( ( n e. Even /\ 2 < n ) -> 6 <_ ( n + 2 ) )
35		eluz2 11693	. . . . . . 7 |- ( ( n + 2 ) e. ( ZZ>= ` 6 ) <-> ( 6 e. ZZ /\ ( n + 2 ) e. ZZ /\ 6 <_ ( n + 2 ) ) )
36	8, 13, 34, 35	syl3anbrc 1246	. . . . . 6 |- ( ( n e. Even /\ 2 < n ) -> ( n + 2 ) e. ( ZZ>= ` 6 ) )
37		eqeq1 2626	. . . . . . . . 9 |- ( m = ( n + 2 ) -> ( m = ( ( p + q ) + r ) <-> ( n + 2 ) = ( ( p + q ) + r ) ) )
38	37	rexbidv 3052	. . . . . . . 8 |- ( m = ( n + 2 ) -> ( E. r e. Prime m = ( ( p + q ) + r ) <-> E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) ) )
39	38	2rexbidv 3057	. . . . . . 7 |- ( m = ( n + 2 ) -> ( E. p e. Prime E. q e. Prime E. r e. Prime m = ( ( p + q ) + r ) <-> E. p e. Prime E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) ) )
40	39	rspcv 3305	. . . . . 6 |- ( ( n + 2 ) e. ( ZZ>= ` 6 ) -> ( A. m e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime m = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) ) )
41	36, 40	syl 17	. . . . 5 |- ( ( n e. Even /\ 2 < n ) -> ( A. m e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime m = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) ) )
42	5, 41	syl5bi 232	. . . 4 |- ( ( n e. Even /\ 2 < n ) -> ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) ) )
43		nfv 1843	. . . . 5 |- F/ p ( n e. Even /\ 2 < n )
44		nfre1 3005	. . . . 5 |- F/ p E. p e. Prime E. q e. Prime n = ( p + q )
45		nfv 1843	. . . . . . 7 |- F/ q ( ( n e. Even /\ 2 < n ) /\ p e. Prime )
46		nfcv 2764	. . . . . . . 8 |- F/_ q Prime
47		nfre1 3005	. . . . . . . 8 |- F/ q E. q e. Prime n = ( p + q )
48	46, 47	nfrex 3007	. . . . . . 7 |- F/ q E. p e. Prime E. q e. Prime n = ( p + q )
49		simplrl 800	. . . . . . . . . . . . 13 |- ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) -> p e. Prime )
50		simplrr 801	. . . . . . . . . . . . 13 |- ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) -> q e. Prime )
51		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) -> r e. Prime )
52	49, 50, 51	3jca 1242	. . . . . . . . . . . 12 |- ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) -> ( p e. Prime /\ q e. Prime /\ r e. Prime ) )
53	52	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> ( p e. Prime /\ q e. Prime /\ r e. Prime ) )
54		simp-4l 806	. . . . . . . . . . 11 |- ( ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> n e. Even )
55		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> ( n + 2 ) = ( ( p + q ) + r ) )
56		mogoldbblem 41629	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) /\ n e. Even /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> E. y e. Prime E. x e. Prime n = ( y + x ) )
57		oveq1 6657	. . . . . . . . . . . . . 14 |- ( p = y -> ( p + q ) = ( y + q ) )
58	57	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( p = y -> ( n = ( p + q ) <-> n = ( y + q ) ) )
59		oveq2 6658	. . . . . . . . . . . . . 14 |- ( q = x -> ( y + q ) = ( y + x ) )
60	59	eqeq2d 2632	. . . . . . . . . . . . 13 |- ( q = x -> ( n = ( y + q ) <-> n = ( y + x ) ) )
61	58, 60	cbvrex2v 3180	. . . . . . . . . . . 12 |- ( E. p e. Prime E. q e. Prime n = ( p + q ) <-> E. y e. Prime E. x e. Prime n = ( y + x ) )
62	56, 61	sylibr 224	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) /\ n e. Even /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) )
63	53, 54, 55, 62	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) /\ r e. Prime ) /\ ( n + 2 ) = ( ( p + q ) + r ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) )
64	63	exp31 630	. . . . . . . . 9 |- ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) -> ( r e. Prime -> ( ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
65	64	rexlimdv 3030	. . . . . . . 8 |- ( ( ( n e. Even /\ 2 < n ) /\ ( p e. Prime /\ q e. Prime ) ) -> ( E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
66	65	expr 643	. . . . . . 7 |- ( ( ( n e. Even /\ 2 < n ) /\ p e. Prime ) -> ( q e. Prime -> ( E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
67	45, 48, 66	rexlimd 3026	. . . . . 6 |- ( ( ( n e. Even /\ 2 < n ) /\ p e. Prime ) -> ( E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
68	67	ex 450	. . . . 5 |- ( ( n e. Even /\ 2 < n ) -> ( p e. Prime -> ( E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
69	43, 44, 68	rexlimd 3026	. . . 4 |- ( ( n e. Even /\ 2 < n ) -> ( E. p e. Prime E. q e. Prime E. r e. Prime ( n + 2 ) = ( ( p + q ) + r ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
70	42, 69	syldc 48	. . 3 |- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> ( ( n e. Even /\ 2 < n ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
71	70	expd 452	. 2 |- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> ( n e. Even -> ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
72	1, 71	ralrimi 2957	1 |- ( A. n e. ( ZZ>= ` 6 ) E. p e. Prime E. q e. Prime E. r e. Prime n = ( ( p + q ) + r ) -> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   4c4 11072   6c6 11074   ZZcz 11377   ZZ>=cuz 11687   Primecprime 15385   Even ceven 41537

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

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-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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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-even 41539  df-odd 41540

This theorem is referenced by:  sbgoldbmb  41674