Theorem stgoldbwt 41664

Description: If the strong ternary Goldbach conjecture is valid, then the weak ternary Goldbach conjecture holds, too. (Contributed by AV, 27-Jul-2020.)

Assertion

Ref	Expression
stgoldbwt	|- ( A. n e. Odd ( 7 < n -> n e. GoldbachOdd ) -> A. n e. Odd ( 5 < n -> n e. GoldbachOddW ) )

Proof of Theorem stgoldbwt

Step	Hyp	Ref	Expression
1		pm3.35 611	. . . . . 6 |- ( ( 7 < n /\ ( 7 < n -> n e. GoldbachOdd ) ) -> n e. GoldbachOdd )
2		gbogbow 41644	. . . . . . 7 |- ( n e. GoldbachOdd -> n e. GoldbachOddW )
3	2	a1d 25	. . . . . 6 |- ( n e. GoldbachOdd -> ( 5 < n -> n e. GoldbachOddW ) )
4	1, 3	syl 17	. . . . 5 |- ( ( 7 < n /\ ( 7 < n -> n e. GoldbachOdd ) ) -> ( 5 < n -> n e. GoldbachOddW ) )
5	4	ex 450	. . . 4 |- ( 7 < n -> ( ( 7 < n -> n e. GoldbachOdd ) -> ( 5 < n -> n e. GoldbachOddW ) ) )
6	5	a1d 25	. . 3 |- ( 7 < n -> ( n e. Odd -> ( ( 7 < n -> n e. GoldbachOdd ) -> ( 5 < n -> n e. GoldbachOddW ) ) ) )
7		oddz 41544	. . . . . . . 8 |- ( n e. Odd -> n e. ZZ )
8	7	zred 11482	. . . . . . 7 |- ( n e. Odd -> n e. RR )
9		7re 11103	. . . . . . . 8 |- 7 e. RR
10	9	a1i 11	. . . . . . 7 |- ( n e. Odd -> 7 e. RR )
11	8, 10	lenltd 10183	. . . . . 6 |- ( n e. Odd -> ( n <_ 7 <-> -. 7 < n ) )
12	8, 10	leloed 10180	. . . . . . . 8 |- ( n e. Odd -> ( n <_ 7 <-> ( n < 7 \/ n = 7 ) ) )
13	7	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( n e. Odd /\ 5 < n ) -> n e. ZZ )
14		6nn 11189	. . . . . . . . . . . . . . . . 17 |- 6 e. NN
15	14	nnzi 11401	. . . . . . . . . . . . . . . 16 |- 6 e. ZZ
16	13, 15	jctir 561	. . . . . . . . . . . . . . 15 |- ( ( n e. Odd /\ 5 < n ) -> ( n e. ZZ /\ 6 e. ZZ ) )
17	16	adantl 482	. . . . . . . . . . . . . 14 |- ( ( n < 7 /\ ( n e. Odd /\ 5 < n ) ) -> ( n e. ZZ /\ 6 e. ZZ ) )
18		df-7 11084	. . . . . . . . . . . . . . . . 17 |- 7 = ( 6 + 1 )
19	18	breq2i 4661	. . . . . . . . . . . . . . . 16 |- ( n < 7 <-> n < ( 6 + 1 ) )
20	19	biimpi 206	. . . . . . . . . . . . . . 15 |- ( n < 7 -> n < ( 6 + 1 ) )
21		df-6 11083	. . . . . . . . . . . . . . . 16 |- 6 = ( 5 + 1 )
22		5nn 11188	. . . . . . . . . . . . . . . . . . 19 |- 5 e. NN
23	22	nnzi 11401	. . . . . . . . . . . . . . . . . 18 |- 5 e. ZZ
24		zltp1le 11427	. . . . . . . . . . . . . . . . . 18 |- ( ( 5 e. ZZ /\ n e. ZZ ) -> ( 5 < n <-> ( 5 + 1 ) <_ n ) )
25	23, 7, 24	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( n e. Odd -> ( 5 < n <-> ( 5 + 1 ) <_ n ) )
26	25	biimpa 501	. . . . . . . . . . . . . . . 16 |- ( ( n e. Odd /\ 5 < n ) -> ( 5 + 1 ) <_ n )
27	21, 26	syl5eqbr 4688	. . . . . . . . . . . . . . 15 |- ( ( n e. Odd /\ 5 < n ) -> 6 <_ n )
28	20, 27	anim12ci 591	. . . . . . . . . . . . . 14 |- ( ( n < 7 /\ ( n e. Odd /\ 5 < n ) ) -> ( 6 <_ n /\ n < ( 6 + 1 ) ) )
29		zgeltp1eq 41318	. . . . . . . . . . . . . 14 |- ( ( n e. ZZ /\ 6 e. ZZ ) -> ( ( 6 <_ n /\ n < ( 6 + 1 ) ) -> n = 6 ) )
30	17, 28, 29	sylc 65	. . . . . . . . . . . . 13 |- ( ( n < 7 /\ ( n e. Odd /\ 5 < n ) ) -> n = 6 )
31	30	orcd 407	. . . . . . . . . . . 12 |- ( ( n < 7 /\ ( n e. Odd /\ 5 < n ) ) -> ( n = 6 \/ n = 7 ) )
32	31	ex 450	. . . . . . . . . . 11 |- ( n < 7 -> ( ( n e. Odd /\ 5 < n ) -> ( n = 6 \/ n = 7 ) ) )
33		olc 399	. . . . . . . . . . . 12 |- ( n = 7 -> ( n = 6 \/ n = 7 ) )
34	33	a1d 25	. . . . . . . . . . 11 |- ( n = 7 -> ( ( n e. Odd /\ 5 < n ) -> ( n = 6 \/ n = 7 ) ) )
35	32, 34	jaoi 394	. . . . . . . . . 10 |- ( ( n < 7 \/ n = 7 ) -> ( ( n e. Odd /\ 5 < n ) -> ( n = 6 \/ n = 7 ) ) )
36	35	expd 452	. . . . . . . . 9 |- ( ( n < 7 \/ n = 7 ) -> ( n e. Odd -> ( 5 < n -> ( n = 6 \/ n = 7 ) ) ) )
37	36	com12 32	. . . . . . . 8 |- ( n e. Odd -> ( ( n < 7 \/ n = 7 ) -> ( 5 < n -> ( n = 6 \/ n = 7 ) ) ) )
38	12, 37	sylbid 230	. . . . . . 7 |- ( n e. Odd -> ( n <_ 7 -> ( 5 < n -> ( n = 6 \/ n = 7 ) ) ) )
39		eleq1 2689	. . . . . . . . . 10 |- ( n = 6 -> ( n e. Odd <-> 6 e. Odd ) )
40		6even 41620	. . . . . . . . . . 11 |- 6 e. Even
41		evennodd 41556	. . . . . . . . . . . 12 |- ( 6 e. Even -> -. 6 e. Odd )
42	41	pm2.21d 118	. . . . . . . . . . 11 |- ( 6 e. Even -> ( 6 e. Odd -> n e. GoldbachOddW ) )
43	40, 42	mp1i 13	. . . . . . . . . 10 |- ( n = 6 -> ( 6 e. Odd -> n e. GoldbachOddW ) )
44	39, 43	sylbid 230	. . . . . . . . 9 |- ( n = 6 -> ( n e. Odd -> n e. GoldbachOddW ) )
45		7gbow 41660	. . . . . . . . . . 11 |- 7 e. GoldbachOddW
46		eleq1 2689	. . . . . . . . . . 11 |- ( n = 7 -> ( n e. GoldbachOddW <-> 7 e. GoldbachOddW ) )
47	45, 46	mpbiri 248	. . . . . . . . . 10 |- ( n = 7 -> n e. GoldbachOddW )
48	47	a1d 25	. . . . . . . . 9 |- ( n = 7 -> ( n e. Odd -> n e. GoldbachOddW ) )
49	44, 48	jaoi 394	. . . . . . . 8 |- ( ( n = 6 \/ n = 7 ) -> ( n e. Odd -> n e. GoldbachOddW ) )
50	49	com12 32	. . . . . . 7 |- ( n e. Odd -> ( ( n = 6 \/ n = 7 ) -> n e. GoldbachOddW ) )
51	38, 50	syl6d 75	. . . . . 6 |- ( n e. Odd -> ( n <_ 7 -> ( 5 < n -> n e. GoldbachOddW ) ) )
52	11, 51	sylbird 250	. . . . 5 |- ( n e. Odd -> ( -. 7 < n -> ( 5 < n -> n e. GoldbachOddW ) ) )
53	52	com12 32	. . . 4 |- ( -. 7 < n -> ( n e. Odd -> ( 5 < n -> n e. GoldbachOddW ) ) )
54	53	a1dd 50	. . 3 |- ( -. 7 < n -> ( n e. Odd -> ( ( 7 < n -> n e. GoldbachOdd ) -> ( 5 < n -> n e. GoldbachOddW ) ) ) )
55	6, 54	pm2.61i 176	. 2 |- ( n e. Odd -> ( ( 7 < n -> n e. GoldbachOdd ) -> ( 5 < n -> n e. GoldbachOddW ) ) )
56	55	ralimia 2950	1 |- ( A. n e. Odd ( 7 < n -> n e. GoldbachOdd ) -> A. n e. Odd ( 5 < n -> n e. GoldbachOddW ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   5c5 11073   6c6 11074   7c7 11075   ZZcz 11377   Even ceven 41537   Odd codd 41538   GoldbachOddW cgbow 41634   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-gbow 41637  df-gbo 41638

This theorem is referenced by:  stgoldbnnsum4prm  41691