Theorem bgoldbst 38879

Description: If the binary Goldbach conjecture is valid, then the (strong) ternary Goldbach conjecture holds, too. (Contributed by AV, 26-Jul-2020.)

Assertion

Ref	Expression
bgoldbst	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) )

Distinct variable group:   m, n

Proof of Theorem bgoldbst

Dummy variables p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 459	. . . . . . 7 |- ( ( m e. Odd /\ 7 < m ) -> m e. Odd )
2		3odd 38835	. . . . . . 7 |- 3 e. Odd
3	1, 2	jctir 541	. . . . . 6 |- ( ( m e. Odd /\ 7 < m ) -> ( m e. Odd /\ 3 e. Odd ) )
4		omoeALTV 38814	. . . . . 6 |- ( ( m e. Odd /\ 3 e. Odd ) -> ( m - 3 ) e. Even )
5		breq2 4406	. . . . . . . 8 |- ( n = ( m - 3 ) -> ( 4 < n <-> 4 < ( m - 3 ) ) )
6		eleq1 2517	. . . . . . . 8 |- ( n = ( m - 3 ) -> ( n e. GoldbachEven <-> ( m - 3 ) e. GoldbachEven ) )
7	5, 6	imbi12d 322	. . . . . . 7 |- ( n = ( m - 3 ) -> ( ( 4 < n -> n e. GoldbachEven ) <-> ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) ) )
8	7	rspcv 3146	. . . . . 6 |- ( ( m - 3 ) e. Even -> ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) ) )
9	3, 4, 8	3syl 18	. . . . 5 |- ( ( m e. Odd /\ 7 < m ) -> ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) ) )
10		4p3e7 10745	. . . . . . . . 9 |- ( 4 + 3 ) = 7
11	10	breq1i 4409	. . . . . . . 8 |- ( ( 4 + 3 ) < m <-> 7 < m )
12		4re 10686	. . . . . . . . . . 11 |- 4 e. RR
13	12	a1i 11	. . . . . . . . . 10 |- ( m e. Odd -> 4 e. RR )
14		3re 10683	. . . . . . . . . . 11 |- 3 e. RR
15	14	a1i 11	. . . . . . . . . 10 |- ( m e. Odd -> 3 e. RR )
16		oddz 38760	. . . . . . . . . . 11 |- ( m e. Odd -> m e. ZZ )
17	16	zred 11040	. . . . . . . . . 10 |- ( m e. Odd -> m e. RR )
18	13, 15, 17	ltaddsubd 10213	. . . . . . . . 9 |- ( m e. Odd -> ( ( 4 + 3 ) < m <-> 4 < ( m - 3 ) ) )
19	18	biimpd 211	. . . . . . . 8 |- ( m e. Odd -> ( ( 4 + 3 ) < m -> 4 < ( m - 3 ) ) )
20	11, 19	syl5bir 222	. . . . . . 7 |- ( m e. Odd -> ( 7 < m -> 4 < ( m - 3 ) ) )
21	20	imp 431	. . . . . 6 |- ( ( m e. Odd /\ 7 < m ) -> 4 < ( m - 3 ) )
22		pm2.27 40	. . . . . 6 |- ( 4 < ( m - 3 ) -> ( ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) -> ( m - 3 ) e. GoldbachEven ) )
23	21, 22	syl 17	. . . . 5 |- ( ( m e. Odd /\ 7 < m ) -> ( ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) -> ( m - 3 ) e. GoldbachEven ) )
24		isgbe 38852	. . . . . 6 |- ( ( m - 3 ) e. GoldbachEven <-> ( ( m - 3 ) e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) )
25		3prm 14641	. . . . . . . . . . . . . 14 |- 3 e. Prime
26	25	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> 3 e. Prime )
27		eleq1 2517	. . . . . . . . . . . . . . . 16 |- ( r = 3 -> ( r e. Odd <-> 3 e. Odd ) )
28	27	3anbi3d 1345	. . . . . . . . . . . . . . 15 |- ( r = 3 -> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) ) )
29		oveq2 6298	. . . . . . . . . . . . . . . 16 |- ( r = 3 -> ( ( p + q ) + r ) = ( ( p + q ) + 3 ) )
30	29	eqeq2d 2461	. . . . . . . . . . . . . . 15 |- ( r = 3 -> ( m = ( ( p + q ) + r ) <-> m = ( ( p + q ) + 3 ) ) )
31	28, 30	anbi12d 717	. . . . . . . . . . . . . 14 |- ( r = 3 -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) <-> ( ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) /\ m = ( ( p + q ) + 3 ) ) ) )
32	31	adantl 468	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) /\ r = 3 ) -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) <-> ( ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) /\ m = ( ( p + q ) + 3 ) ) ) )
33		simp1 1008	. . . . . . . . . . . . . . . 16 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> p e. Odd )
34		simp2 1009	. . . . . . . . . . . . . . . 16 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> q e. Odd )
35	2	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> 3 e. Odd )
36	33, 34, 35	3jca 1188	. . . . . . . . . . . . . . 15 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) )
37	36	adantl 468	. . . . . . . . . . . . . 14 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) )
38	16	zcnd 11041	. . . . . . . . . . . . . . . . . . 19 |- ( m e. Odd -> m e. CC )
39	38	ad3antrrr 736	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> m e. CC )
40		3cn 10684	. . . . . . . . . . . . . . . . . . 19 |- 3 e. CC
41	40	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> 3 e. CC )
42		prmz 14626	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. Prime -> p e. ZZ )
43		prmz 14626	. . . . . . . . . . . . . . . . . . . . 21 |- ( q e. Prime -> q e. ZZ )
44		zaddcl 10977	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( p e. ZZ /\ q e. ZZ ) -> ( p + q ) e. ZZ )
45	42, 43, 44	syl2an 480	. . . . . . . . . . . . . . . . . . . 20 |- ( ( p e. Prime /\ q e. Prime ) -> ( p + q ) e. ZZ )
46	45	zcnd 11041	. . . . . . . . . . . . . . . . . . 19 |- ( ( p e. Prime /\ q e. Prime ) -> ( p + q ) e. CC )
47	46	adantll 720	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> ( p + q ) e. CC )
48	39, 41, 47	subadd2d 10005	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> ( ( m - 3 ) = ( p + q ) <-> ( ( p + q ) + 3 ) = m ) )
49	48	biimpa 487	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( m - 3 ) = ( p + q ) ) -> ( ( p + q ) + 3 ) = m )
50	49	eqcomd 2457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( m - 3 ) = ( p + q ) ) -> m = ( ( p + q ) + 3 ) )
51	50	3ad2antr3 1175	. . . . . . . . . . . . . 14 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> m = ( ( p + q ) + 3 ) )
52	37, 51	jca 535	. . . . . . . . . . . . 13 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> ( ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) /\ m = ( ( p + q ) + 3 ) ) )
53	26, 32, 52	rspcedvd 3155	. . . . . . . . . . . 12 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) )
54	53	ex 436	. . . . . . . . . . 11 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) ) )
55	54	reximdva 2862	. . . . . . . . . 10 |- ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) -> ( E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) ) )
56	55	reximdva 2862	. . . . . . . . 9 |- ( ( m e. Odd /\ 7 < m ) -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) ) )
57	56, 1	jctild 546	. . . . . . . 8 |- ( ( m e. Odd /\ 7 < m ) -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> ( m e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) ) ) )
58		isgboa 38854	. . . . . . . 8 |- ( m e. GoldbachOddALTV <-> ( m e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ m = ( ( p + q ) + r ) ) ) )
59	57, 58	syl6ibr 231	. . . . . . 7 |- ( ( m e. Odd /\ 7 < m ) -> ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> m e. GoldbachOddALTV ) )
60	59	adantld 469	. . . . . 6 |- ( ( m e. Odd /\ 7 < m ) -> ( ( ( m - 3 ) e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) ) -> m e. GoldbachOddALTV ) )
61	24, 60	syl5bi 221	. . . . 5 |- ( ( m e. Odd /\ 7 < m ) -> ( ( m - 3 ) e. GoldbachEven -> m e. GoldbachOddALTV ) )
62	9, 23, 61	3syld 57	. . . 4 |- ( ( m e. Odd /\ 7 < m ) -> ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> m e. GoldbachOddALTV ) )
63	62	com12 32	. . 3 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( ( m e. Odd /\ 7 < m ) -> m e. GoldbachOddALTV ) )
64	63	expd 438	. 2 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( m e. Odd -> ( 7 < m -> m e. GoldbachOddALTV ) ) )
65	64	ralrimiv 2800	1 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   class class class wbr 4402  (class class class)co 6290   CCcc 9537   RRcr 9538   + caddc 9542   < clt 9675   - cmin 9860   3c3 10660   4c4 10661   7c7 10664   ZZcz 10937   Primecprime 14622   Even ceven 38753   Odd codd 38754   GoldbachEven cgbe 38846   GoldbachOddALTV cgboa 38848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-dvds 14306  df-prm 14623  df-even 38755  df-odd 38756  df-gbe 38849  df-gboa 38851

This theorem is referenced by: (None)