Theorem 11gbo 41663

Description: 11 is an odd Goldbach number. (Contributed by AV, 29-Jul-2020.)

Assertion

Ref	Expression
11gbo	|- ; 1 1 e. GoldbachOdd

Proof of Theorem 11gbo

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

Step	Hyp	Ref	Expression
1		6p5e11 11600	. . 3 |- ( 6 + 5 ) = ; 1 1
2		6even 41620	. . . 4 |- 6 e. Even
3		5odd 41619	. . . 4 |- 5 e. Odd
4		epoo 41612	. . . 4 |- ( ( 6 e. Even /\ 5 e. Odd ) -> ( 6 + 5 ) e. Odd )
5	2, 3, 4	mp2an 708	. . 3 |- ( 6 + 5 ) e. Odd
6	1, 5	eqeltrri 2698	. 2 |- ; 1 1 e. Odd
7		3prm 15406	. . 3 |- 3 e. Prime
8		5prm 15815	. . . 4 |- 5 e. Prime
9		3odd 41617	. . . . . 6 |- 3 e. Odd
10	9, 9, 3	3pm3.2i 1239	. . . . 5 |- ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd )
11		gbpart11 41658	. . . . 5 |- ; 1 1 = ( ( 3 + 3 ) + 5 )
12	10, 11	pm3.2i 471	. . . 4 |- ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) )
13		eleq1 2689	. . . . . . 7 |- ( r = 5 -> ( r e. Odd <-> 5 e. Odd ) )
14	13	3anbi3d 1405	. . . . . 6 |- ( r = 5 -> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) ) )
15		oveq2 6658	. . . . . . 7 |- ( r = 5 -> ( ( 3 + 3 ) + r ) = ( ( 3 + 3 ) + 5 ) )
16	15	eqeq2d 2632	. . . . . 6 |- ( r = 5 -> (; 1 1 = ( ( 3 + 3 ) + r ) <-> ; 1 1 = ( ( 3 + 3 ) + 5 ) ) )
17	14, 16	anbi12d 747	. . . . 5 |- ( r = 5 -> ( ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) ) ) )
18	17	rspcev 3309	. . . 4 |- ( ( 5 e. Prime /\ ( ( 3 e. Odd /\ 3 e. Odd /\ 5 e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + 5 ) ) ) -> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) )
19	8, 12, 18	mp2an 708	. . 3 |- E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) )
20		eleq1 2689	. . . . . . 7 |- ( p = 3 -> ( p e. Odd <-> 3 e. Odd ) )
21	20	3anbi1d 1403	. . . . . 6 |- ( p = 3 -> ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) ) )
22		oveq1 6657	. . . . . . . 8 |- ( p = 3 -> ( p + q ) = ( 3 + q ) )
23	22	oveq1d 6665	. . . . . . 7 |- ( p = 3 -> ( ( p + q ) + r ) = ( ( 3 + q ) + r ) )
24	23	eqeq2d 2632	. . . . . 6 |- ( p = 3 -> (; 1 1 = ( ( p + q ) + r ) <-> ; 1 1 = ( ( 3 + q ) + r ) ) )
25	21, 24	anbi12d 747	. . . . 5 |- ( p = 3 -> ( ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) <-> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) ) )
26	25	rexbidv 3052	. . . 4 |- ( p = 3 -> ( E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) ) )
27		eleq1 2689	. . . . . . 7 |- ( q = 3 -> ( q e. Odd <-> 3 e. Odd ) )
28	27	3anbi2d 1404	. . . . . 6 |- ( q = 3 -> ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) <-> ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) ) )
29		oveq2 6658	. . . . . . . 8 |- ( q = 3 -> ( 3 + q ) = ( 3 + 3 ) )
30	29	oveq1d 6665	. . . . . . 7 |- ( q = 3 -> ( ( 3 + q ) + r ) = ( ( 3 + 3 ) + r ) )
31	30	eqeq2d 2632	. . . . . 6 |- ( q = 3 -> (; 1 1 = ( ( 3 + q ) + r ) <-> ; 1 1 = ( ( 3 + 3 ) + r ) ) )
32	28, 31	anbi12d 747	. . . . 5 |- ( q = 3 -> ( ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) <-> ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) )
33	32	rexbidv 3052	. . . 4 |- ( q = 3 -> ( E. r e. Prime ( ( 3 e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + q ) + r ) ) <-> E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) )
34	26, 33	rspc2ev 3324	. . 3 |- ( ( 3 e. Prime /\ 3 e. Prime /\ E. r e. Prime ( ( 3 e. Odd /\ 3 e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( 3 + 3 ) + r ) ) ) -> E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) )
35	7, 7, 19, 34	mp3an 1424	. 2 |- E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) )
36		isgbo 41641	. 2 |- (; 1 1 e. GoldbachOdd <-> (; 1 1 e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime ( ( p e. Odd /\ q e. Odd /\ r e. Odd ) /\ ; 1 1 = ( ( p + q ) + r ) ) ) )
37	6, 35, 36	mpbir2an 955	1 |- ; 1 1 e. GoldbachOdd

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   1c1 9937   + caddc 9939   3c3 11071   5c5 11073   6c6 11074  ;cdc 11493   Primecprime 15385   Even ceven 41537   Odd codd 41538   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-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  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-gbo 41638

This theorem is referenced by:  bgoldbtbndlem1  41693