Theorem sbgoldbalt 41669

Description: An alternate (related to the original) formulation of the binary Goldbach conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (Contributed by AV, 22-Jul-2020.)

Assertion

Ref	Expression
sbgoldbalt	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )

Distinct variable group:   n, p, q

Proof of Theorem sbgoldbalt

Step	Hyp	Ref	Expression
1		2z 11409	. . . . . 6 |- 2 e. ZZ
2		evenz 41543	. . . . . 6 |- ( n e. Even -> n e. ZZ )
3		zltp1le 11427	. . . . . 6 |- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 < n <-> ( 2 + 1 ) <_ n ) )
4	1, 2, 3	sylancr 695	. . . . 5 |- ( n e. Even -> ( 2 < n <-> ( 2 + 1 ) <_ n ) )
5		2p1e3 11151	. . . . . . 7 |- ( 2 + 1 ) = 3
6	5	breq1i 4660	. . . . . 6 |- ( ( 2 + 1 ) <_ n <-> 3 <_ n )
7		3re 11094	. . . . . . . . 9 |- 3 e. RR
8	7	a1i 11	. . . . . . . 8 |- ( n e. Even -> 3 e. RR )
9	2	zred 11482	. . . . . . . 8 |- ( n e. Even -> n e. RR )
10	8, 9	leloed 10180	. . . . . . 7 |- ( n e. Even -> ( 3 <_ n <-> ( 3 < n \/ 3 = n ) ) )
11		3z 11410	. . . . . . . . . . . 12 |- 3 e. ZZ
12		zltp1le 11427	. . . . . . . . . . . 12 |- ( ( 3 e. ZZ /\ n e. ZZ ) -> ( 3 < n <-> ( 3 + 1 ) <_ n ) )
13	11, 2, 12	sylancr 695	. . . . . . . . . . 11 |- ( n e. Even -> ( 3 < n <-> ( 3 + 1 ) <_ n ) )
14		3p1e4 11153	. . . . . . . . . . . . 13 |- ( 3 + 1 ) = 4
15	14	breq1i 4660	. . . . . . . . . . . 12 |- ( ( 3 + 1 ) <_ n <-> 4 <_ n )
16		4re 11097	. . . . . . . . . . . . . . 15 |- 4 e. RR
17	16	a1i 11	. . . . . . . . . . . . . 14 |- ( n e. Even -> 4 e. RR )
18	17, 9	leloed 10180	. . . . . . . . . . . . 13 |- ( n e. Even -> ( 4 <_ n <-> ( 4 < n \/ 4 = n ) ) )
19		pm3.35 611	. . . . . . . . . . . . . . . . . 18 |- ( ( 4 < n /\ ( 4 < n -> n e. GoldbachEven ) ) -> n e. GoldbachEven )
20		isgbe 41639	. . . . . . . . . . . . . . . . . . . 20 |- ( n e. GoldbachEven <-> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) )
21		simp3 1063	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> n = ( p + q ) )
22	21	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( n e. Even /\ p e. Prime ) /\ q e. Prime ) -> ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> n = ( p + q ) ) )
23	22	reximdva 3017	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( n e. Even /\ p e. Prime ) -> ( E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> E. q e. Prime n = ( p + q ) ) )
24	23	reximdva 3017	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. Even -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
25	24	imp 445	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) -> E. p e. Prime E. q e. Prime n = ( p + q ) )
26	20, 25	sylbi 207	. . . . . . . . . . . . . . . . . . 19 |- ( n e. GoldbachEven -> E. p e. Prime E. q e. Prime n = ( p + q ) )
27	26	a1d 25	. . . . . . . . . . . . . . . . . 18 |- ( n e. GoldbachEven -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
28	19, 27	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( 4 < n /\ ( 4 < n -> n e. GoldbachEven ) ) -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
29	28	ex 450	. . . . . . . . . . . . . . . 16 |- ( 4 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> ( n e. Even -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
30	29	com23 86	. . . . . . . . . . . . . . 15 |- ( 4 < n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
31		2prm 15405	. . . . . . . . . . . . . . . . . . 19 |- 2 e. Prime
32		2p2e4 11144	. . . . . . . . . . . . . . . . . . . 20 |- ( 2 + 2 ) = 4
33	32	eqcomi 2631	. . . . . . . . . . . . . . . . . . 19 |- 4 = ( 2 + 2 )
34		rspceov 6692	. . . . . . . . . . . . . . . . . . 19 |- ( ( 2 e. Prime /\ 2 e. Prime /\ 4 = ( 2 + 2 ) ) -> E. p e. Prime E. q e. Prime 4 = ( p + q ) )
35	31, 31, 33, 34	mp3an 1424	. . . . . . . . . . . . . . . . . 18 |- E. p e. Prime E. q e. Prime 4 = ( p + q )
36		eqeq1 2626	. . . . . . . . . . . . . . . . . . 19 |- ( 4 = n -> ( 4 = ( p + q ) <-> n = ( p + q ) ) )
37	36	2rexbidv 3057	. . . . . . . . . . . . . . . . . 18 |- ( 4 = n -> ( E. p e. Prime E. q e. Prime 4 = ( p + q ) <-> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
38	35, 37	mpbii 223	. . . . . . . . . . . . . . . . 17 |- ( 4 = n -> E. p e. Prime E. q e. Prime n = ( p + q ) )
39	38	a1d 25	. . . . . . . . . . . . . . . 16 |- ( 4 = n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )
40	39	a1d 25	. . . . . . . . . . . . . . 15 |- ( 4 = n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
41	30, 40	jaoi 394	. . . . . . . . . . . . . 14 |- ( ( 4 < n \/ 4 = n ) -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
42	41	com12 32	. . . . . . . . . . . . 13 |- ( n e. Even -> ( ( 4 < n \/ 4 = n ) -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
43	18, 42	sylbid 230	. . . . . . . . . . . 12 |- ( n e. Even -> ( 4 <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
44	15, 43	syl5bi 232	. . . . . . . . . . 11 |- ( n e. Even -> ( ( 3 + 1 ) <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
45	13, 44	sylbid 230	. . . . . . . . . 10 |- ( n e. Even -> ( 3 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
46	45	com12 32	. . . . . . . . 9 |- ( 3 < n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
47		3odd 41617	. . . . . . . . . . . 12 |- 3 e. Odd
48		eleq1 2689	. . . . . . . . . . . 12 |- ( 3 = n -> ( 3 e. Odd <-> n e. Odd ) )
49	47, 48	mpbii 223	. . . . . . . . . . 11 |- ( 3 = n -> n e. Odd )
50		oddneven 41557	. . . . . . . . . . 11 |- ( n e. Odd -> -. n e. Even )
51	49, 50	syl 17	. . . . . . . . . 10 |- ( 3 = n -> -. n e. Even )
52	51	pm2.21d 118	. . . . . . . . 9 |- ( 3 = n -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
53	46, 52	jaoi 394	. . . . . . . 8 |- ( ( 3 < n \/ 3 = n ) -> ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
54	53	com12 32	. . . . . . 7 |- ( n e. Even -> ( ( 3 < n \/ 3 = n ) -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
55	10, 54	sylbid 230	. . . . . 6 |- ( n e. Even -> ( 3 <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
56	6, 55	syl5bi 232	. . . . 5 |- ( n e. Even -> ( ( 2 + 1 ) <_ n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
57	4, 56	sylbid 230	. . . 4 |- ( n e. Even -> ( 2 < n -> ( ( 4 < n -> n e. GoldbachEven ) -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
58	57	com23 86	. . 3 |- ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) -> ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
59		2lt4 11198	. . . . . . . 8 |- 2 < 4
60		2re 11090	. . . . . . . . . 10 |- 2 e. RR
61	60	a1i 11	. . . . . . . . 9 |- ( n e. Even -> 2 e. RR )
62		lttr 10114	. . . . . . . . 9 |- ( ( 2 e. RR /\ 4 e. RR /\ n e. RR ) -> ( ( 2 < 4 /\ 4 < n ) -> 2 < n ) )
63	61, 17, 9, 62	syl3anc 1326	. . . . . . . 8 |- ( n e. Even -> ( ( 2 < 4 /\ 4 < n ) -> 2 < n ) )
64	59, 63	mpani 712	. . . . . . 7 |- ( n e. Even -> ( 4 < n -> 2 < n ) )
65	64	imp 445	. . . . . 6 |- ( ( n e. Even /\ 4 < n ) -> 2 < n )
66		simpll 790	. . . . . . . . 9 |- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. Even )
67		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) -> p e. Prime )
68	67	anim1i 592	. . . . . . . . . . . . . . . 16 |- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( p e. Prime /\ q e. Prime ) )
69	68	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Prime /\ q e. Prime ) )
70		simpll 790	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( n e. Even /\ 4 < n ) )
71	70	anim1i 592	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( ( n e. Even /\ 4 < n ) /\ n = ( p + q ) ) )
72		df-3an 1039	. . . . . . . . . . . . . . . 16 |- ( ( n e. Even /\ 4 < n /\ n = ( p + q ) ) <-> ( ( n e. Even /\ 4 < n ) /\ n = ( p + q ) ) )
73	71, 72	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( n e. Even /\ 4 < n /\ n = ( p + q ) ) )
74		sbgoldbaltlem2 41668	. . . . . . . . . . . . . . 15 |- ( ( p e. Prime /\ q e. Prime ) -> ( ( n e. Even /\ 4 < n /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd ) ) )
75	69, 73, 74	sylc 65	. . . . . . . . . . . . . 14 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd ) )
76		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> n = ( p + q ) )
77		df-3an 1039	. . . . . . . . . . . . . 14 |- ( ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) <-> ( ( p e. Odd /\ q e. Odd ) /\ n = ( p + q ) ) )
78	75, 76, 77	sylanbrc 698	. . . . . . . . . . . . 13 |- ( ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) /\ n = ( p + q ) ) -> ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) )
79	78	ex 450	. . . . . . . . . . . 12 |- ( ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) /\ q e. Prime ) -> ( n = ( p + q ) -> ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) )
80	79	reximdva 3017	. . . . . . . . . . 11 |- ( ( ( n e. Even /\ 4 < n ) /\ p e. Prime ) -> ( E. q e. Prime n = ( p + q ) -> E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) )
81	80	reximdva 3017	. . . . . . . . . 10 |- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) )
82	81	imp 445	. . . . . . . . 9 |- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) )
83	66, 82	jca 554	. . . . . . . 8 |- ( ( ( n e. Even /\ 4 < n ) /\ E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) )
84	83	ex 450	. . . . . . 7 |- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> ( n e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ n = ( p + q ) ) ) ) )
85	84, 20	syl6ibr 242	. . . . . 6 |- ( ( n e. Even /\ 4 < n ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) -> n e. GoldbachEven ) )
86	65, 85	embantd 59	. . . . 5 |- ( ( n e. Even /\ 4 < n ) -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. GoldbachEven ) )
87	86	ex 450	. . . 4 |- ( n e. Even -> ( 4 < n -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> n e. GoldbachEven ) ) )
88	87	com23 86	. . 3 |- ( n e. Even -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 4 < n -> n e. GoldbachEven ) ) )
89	58, 88	impbid 202	. 2 |- ( n e. Even -> ( ( 4 < n -> n e. GoldbachEven ) <-> ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) ) )
90	89	ralbiia 2979	1 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) <-> A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   3c3 11071   4c4 11072   ZZcz 11377   Primecprime 15385   Even ceven 41537   Odd codd 41538   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-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-1st 7168  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  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  df-gbe 41636

This theorem is referenced by:  sbgoldbb  41670  sbgoldbmb  41674