Theorem sgoldbeven3prm 41671

Description: If the binary Goldbach conjecture is valid, then an even integer greater than 5 can be expressed as the sum of three primes: Since ( N - 2 ) is even iff N is even, there would be primes p and q with ( N - 2 ) = ( p + q ), and therefore N = ( ( p + q ) + 2 ). (Contributed by AV, 24-Dec-2021.)

Assertion

Ref	Expression
sgoldbeven3prm	|- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( ( N e. Even /\ 6 <_ N ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )

Distinct variable group:   n, N, p, q, r

Proof of Theorem sgoldbeven3prm

Step	Hyp	Ref	Expression
1		sbgoldbb 41670	. 2 |- ( 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 ) ) )
2		2p2e4 11144	. . . . 5 |- ( 2 + 2 ) = 4
3		evenz 41543	. . . . . . . 8 |- ( N e. Even -> N e. ZZ )
4	3	zred 11482	. . . . . . 7 |- ( N e. Even -> N e. RR )
5		4lt6 11205	. . . . . . . 8 |- 4 < 6
6		4re 11097	. . . . . . . . 9 |- 4 e. RR
7		6re 11101	. . . . . . . . 9 |- 6 e. RR
8		ltletr 10129	. . . . . . . . 9 |- ( ( 4 e. RR /\ 6 e. RR /\ N e. RR ) -> ( ( 4 < 6 /\ 6 <_ N ) -> 4 < N ) )
9	6, 7, 8	mp3an12 1414	. . . . . . . 8 |- ( N e. RR -> ( ( 4 < 6 /\ 6 <_ N ) -> 4 < N ) )
10	5, 9	mpani 712	. . . . . . 7 |- ( N e. RR -> ( 6 <_ N -> 4 < N ) )
11	4, 10	syl 17	. . . . . 6 |- ( N e. Even -> ( 6 <_ N -> 4 < N ) )
12	11	imp 445	. . . . 5 |- ( ( N e. Even /\ 6 <_ N ) -> 4 < N )
13	2, 12	syl5eqbr 4688	. . . 4 |- ( ( N e. Even /\ 6 <_ N ) -> ( 2 + 2 ) < N )
14		2re 11090	. . . . . 6 |- 2 e. RR
15	14	a1i 11	. . . . 5 |- ( ( N e. Even /\ 6 <_ N ) -> 2 e. RR )
16	4	adantr 481	. . . . 5 |- ( ( N e. Even /\ 6 <_ N ) -> N e. RR )
17	15, 15, 16	ltaddsub2d 10628	. . . 4 |- ( ( N e. Even /\ 6 <_ N ) -> ( ( 2 + 2 ) < N <-> 2 < ( N - 2 ) ) )
18	13, 17	mpbid 222	. . 3 |- ( ( N e. Even /\ 6 <_ N ) -> 2 < ( N - 2 ) )
19		2evenALTV 41603	. . . . . 6 |- 2 e. Even
20		emee 41615	. . . . . 6 |- ( ( N e. Even /\ 2 e. Even ) -> ( N - 2 ) e. Even )
21	19, 20	mpan2 707	. . . . 5 |- ( N e. Even -> ( N - 2 ) e. Even )
22		breq2 4657	. . . . . . . 8 |- ( n = ( N - 2 ) -> ( 2 < n <-> 2 < ( N - 2 ) ) )
23		eqeq1 2626	. . . . . . . . 9 |- ( n = ( N - 2 ) -> ( n = ( p + q ) <-> ( N - 2 ) = ( p + q ) ) )
24	23	2rexbidv 3057	. . . . . . . 8 |- ( n = ( N - 2 ) -> ( E. p e. Prime E. q e. Prime n = ( p + q ) <-> E. p e. Prime E. q e. Prime ( N - 2 ) = ( p + q ) ) )
25	22, 24	imbi12d 334	. . . . . . 7 |- ( n = ( N - 2 ) -> ( ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) <-> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime ( N - 2 ) = ( p + q ) ) ) )
26	25	rspcv 3305	. . . . . 6 |- ( ( N - 2 ) e. Even -> ( A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime ( N - 2 ) = ( p + q ) ) ) )
27		2prm 15405	. . . . . . . . . . . 12 |- 2 e. Prime
28	27	a1i 11	. . . . . . . . . . 11 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> 2 e. Prime )
29		oveq2 6658	. . . . . . . . . . . . 13 |- ( r = 2 -> ( ( p + q ) + r ) = ( ( p + q ) + 2 ) )
30	29	eqeq2d 2632	. . . . . . . . . . . 12 |- ( r = 2 -> ( N = ( ( p + q ) + r ) <-> N = ( ( p + q ) + 2 ) ) )
31	30	adantl 482	. . . . . . . . . . 11 |- ( ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) /\ r = 2 ) -> ( N = ( ( p + q ) + r ) <-> N = ( ( p + q ) + 2 ) ) )
32	3	zcnd 11483	. . . . . . . . . . . . . 14 |- ( N e. Even -> N e. CC )
33		2cnd 11093	. . . . . . . . . . . . . 14 |- ( N e. Even -> 2 e. CC )
34		npcan 10290	. . . . . . . . . . . . . . 15 |- ( ( N e. CC /\ 2 e. CC ) -> ( ( N - 2 ) + 2 ) = N )
35	34	eqcomd 2628	. . . . . . . . . . . . . 14 |- ( ( N e. CC /\ 2 e. CC ) -> N = ( ( N - 2 ) + 2 ) )
36	32, 33, 35	syl2anc 693	. . . . . . . . . . . . 13 |- ( N e. Even -> N = ( ( N - 2 ) + 2 ) )
37	36	adantr 481	. . . . . . . . . . . 12 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> N = ( ( N - 2 ) + 2 ) )
38		simpr 477	. . . . . . . . . . . . 13 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> ( N - 2 ) = ( p + q ) )
39	38	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> ( ( N - 2 ) + 2 ) = ( ( p + q ) + 2 ) )
40	37, 39	eqtrd 2656	. . . . . . . . . . 11 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> N = ( ( p + q ) + 2 ) )
41	28, 31, 40	rspcedvd 3317	. . . . . . . . . 10 |- ( ( N e. Even /\ ( N - 2 ) = ( p + q ) ) -> E. r e. Prime N = ( ( p + q ) + r ) )
42	41	ex 450	. . . . . . . . 9 |- ( N e. Even -> ( ( N - 2 ) = ( p + q ) -> E. r e. Prime N = ( ( p + q ) + r ) ) )
43	42	reximdv 3016	. . . . . . . 8 |- ( N e. Even -> ( E. q e. Prime ( N - 2 ) = ( p + q ) -> E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )
44	43	reximdv 3016	. . . . . . 7 |- ( N e. Even -> ( E. p e. Prime E. q e. Prime ( N - 2 ) = ( p + q ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )
45	44	imim2d 57	. . . . . 6 |- ( N e. Even -> ( ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime ( N - 2 ) = ( p + q ) ) -> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) ) )
46	26, 45	syl9r 78	. . . . 5 |- ( N e. Even -> ( ( N - 2 ) e. Even -> ( A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) ) ) )
47	21, 46	mpd 15	. . . 4 |- ( N e. Even -> ( A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) ) )
48	47	adantr 481	. . 3 |- ( ( N e. Even /\ 6 <_ N ) -> ( A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> ( 2 < ( N - 2 ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) ) )
49	18, 48	mpid 44	. 2 |- ( ( N e. Even /\ 6 <_ N ) -> ( A. n e. Even ( 2 < n -> E. p e. Prime E. q e. Prime n = ( p + q ) ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )
50	1, 49	syl5com 31	1 |- ( A. n e. Even ( 4 < n -> n e. GoldbachEven ) -> ( ( N e. Even /\ 6 <_ N ) -> E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = 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   <_ cle 10075   - cmin 10266   2c2 11070   4c4 11072   6c6 11074   Primecprime 15385   Even ceven 41537   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-5 11082  df-6 11083  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:  sbgoldbm  41672