Theorem sbgoldbst 41666

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

Assertion

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

Distinct variable group:   m, n

Proof of Theorem sbgoldbst

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . 7 |- ( ( m e. Odd /\ 7 < m ) -> m e. Odd )
2		3odd 41617	. . . . . . 7 |- 3 e. Odd
3	1, 2	jctir 561	. . . . . 6 |- ( ( m e. Odd /\ 7 < m ) -> ( m e. Odd /\ 3 e. Odd ) )
4		omoeALTV 41596	. . . . . 6 |- ( ( m e. Odd /\ 3 e. Odd ) -> ( m - 3 ) e. Even )
5		breq2 4657	. . . . . . . 8 |- ( n = ( m - 3 ) -> ( 4 < n <-> 4 < ( m - 3 ) ) )
6		eleq1 2689	. . . . . . . 8 |- ( n = ( m - 3 ) -> ( n e. GoldbachEven <-> ( m - 3 ) e. GoldbachEven ) )
7	5, 6	imbi12d 334	. . . . . . 7 |- ( n = ( m - 3 ) -> ( ( 4 < n -> n e. GoldbachEven ) <-> ( 4 < ( m - 3 ) -> ( m - 3 ) e. GoldbachEven ) ) )
8	7	rspcv 3305	. . . . . 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 11163	. . . . . . . . 9 |- ( 4 + 3 ) = 7
11	10	breq1i 4660	. . . . . . . 8 |- ( ( 4 + 3 ) < m <-> 7 < m )
12		4re 11097	. . . . . . . . . . 11 |- 4 e. RR
13	12	a1i 11	. . . . . . . . . 10 |- ( m e. Odd -> 4 e. RR )
14		3re 11094	. . . . . . . . . . 11 |- 3 e. RR
15	14	a1i 11	. . . . . . . . . 10 |- ( m e. Odd -> 3 e. RR )
16		oddz 41544	. . . . . . . . . . 11 |- ( m e. Odd -> m e. ZZ )
17	16	zred 11482	. . . . . . . . . 10 |- ( m e. Odd -> m e. RR )
18	13, 15, 17	ltaddsubd 10627	. . . . . . . . 9 |- ( m e. Odd -> ( ( 4 + 3 ) < m <-> 4 < ( m - 3 ) ) )
19	18	biimpd 219	. . . . . . . 8 |- ( m e. Odd -> ( ( 4 + 3 ) < m -> 4 < ( m - 3 ) ) )
20	11, 19	syl5bir 233	. . . . . . 7 |- ( m e. Odd -> ( 7 < m -> 4 < ( m - 3 ) ) )
21	20	imp 445	. . . . . 6 |- ( ( m e. Odd /\ 7 < m ) -> 4 < ( m - 3 ) )
22		pm2.27 42	. . . . . 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 41639	. . . . . 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 15406	. . . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . . . 16 |- ( r = 3 -> ( r e. Odd <-> 3 e. Odd ) )
28	27	3anbi3d 1405	. . . . . . . . . . . . . . 15 |- ( r = 3 -> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) ) )
29		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( r = 3 -> ( ( p + q ) + r ) = ( ( p + q ) + 3 ) )
30	29	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( r = 3 -> ( m = ( ( p + q ) + r ) <-> m = ( ( p + q ) + 3 ) ) )
31	28, 30	anbi12d 747	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 1061	. . . . . . . . . . . . . . . 16 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> p e. Odd )
34		simp2 1062	. . . . . . . . . . . . . . . 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 1242	. . . . . . . . . . . . . . 15 |- ( ( p e. Odd /\ q e. Odd /\ ( m - 3 ) = ( p + q ) ) -> ( p e. Odd /\ q e. Odd /\ 3 e. Odd ) )
37	36	adantl 482	. . . . . . . . . . . . . 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 11483	. . . . . . . . . . . . . . . . . . 19 |- ( m e. Odd -> m e. CC )
39	38	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> m e. CC )
40		3cn 11095	. . . . . . . . . . . . . . . . . . 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 15389	. . . . . . . . . . . . . . . . . . . . 21 |- ( p e. Prime -> p e. ZZ )
43		prmz 15389	. . . . . . . . . . . . . . . . . . . . 21 |- ( q e. Prime -> q e. ZZ )
44		zaddcl 11417	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( p e. ZZ /\ q e. ZZ ) -> ( p + q ) e. ZZ )
45	42, 43, 44	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 |- ( ( p e. Prime /\ q e. Prime ) -> ( p + q ) e. ZZ )
46	45	zcnd 11483	. . . . . . . . . . . . . . . . . . 19 |- ( ( p e. Prime /\ q e. Prime ) -> ( p + q ) e. CC )
47	46	adantll 750	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> ( p + q ) e. CC )
48	39, 41, 47	subadd2d 10411	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) -> ( ( m - 3 ) = ( p + q ) <-> ( ( p + q ) + 3 ) = m ) )
49	48	biimpa 501	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( m - 3 ) = ( p + q ) ) -> ( ( p + q ) + 3 ) = m )
50	49	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( m e. Odd /\ 7 < m ) /\ p e. Prime ) /\ q e. Prime ) /\ ( m - 3 ) = ( p + q ) ) -> m = ( ( p + q ) + 3 ) )
51	50	3ad2antr3 1228	. . . . . . . . . . . . . 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 554	. . . . . . . . . . . . 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 3317	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 3017	. . . . . . . . . 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 3017	. . . . . . . . 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 566	. . . . . . . 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		isgbo 41641	. . . . . . . 8 |- ( m e. GoldbachOdd <-> ( 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 242	. . . . . . 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. GoldbachOdd ) )
60	59	adantld 483	. . . . . 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. GoldbachOdd ) )
61	24, 60	syl5bi 232	. . . . 5 |- ( ( m e. Odd /\ 7 < m ) -> ( ( m - 3 ) e. GoldbachEven -> m e. GoldbachOdd ) )
62	9, 23, 61	3syld 60	. . . 4 |- ( ( m e. Odd /\ 7 < m ) -> ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> m e. GoldbachOdd ) )
63	62	com12 32	. . 3 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( ( m e. Odd /\ 7 < m ) -> m e. GoldbachOdd ) )
64	63	expd 452	. 2 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( m e. Odd -> ( 7 < m -> m e. GoldbachOdd ) ) )
65	64	ralrimiv 2965	1 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> A. m e. Odd ( 7 < m -> m e. GoldbachOdd ) )

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  (class class class)co 6650   CCcc 9934   RRcr 9935   + caddc 9939   < clt 10074   - cmin 10266   3c3 11071   4c4 11072   7c7 11075   ZZcz 11377   Primecprime 15385   Even ceven 41537   Odd codd 41538   GoldbachEven cgbe 41633   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

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-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-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  df-gbo 41638

This theorem is referenced by: (None)