Theorem even3prm2 41628

Description: If an even number is the sum of three prime numbers, one of the prime numbers must be 2. (Contributed by AV, 25-Dec-2021.)

Assertion

Ref	Expression
even3prm2	|- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( P = 2 \/ Q = 2 \/ R = 2 ) )

Proof of Theorem even3prm2

Step	Hyp	Ref	Expression
1		olc 399	. . . 4 |- ( R = 2 -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) )
2	1	a1d 25	. . 3 |- ( R = 2 -> ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) )
3		df-ne 2795	. . . . . . . . . . . 12 |- ( R =/= 2 <-> -. R = 2 )
4		eldifsn 4317	. . . . . . . . . . . . . 14 |- ( R e. ( Prime \ { 2 } ) <-> ( R e. Prime /\ R =/= 2 ) )
5		oddprmALTV 41598	. . . . . . . . . . . . . . 15 |- ( R e. ( Prime \ { 2 } ) -> R e. Odd )
6		emoo 41613	. . . . . . . . . . . . . . . 16 |- ( ( N e. Even /\ R e. Odd ) -> ( N - R ) e. Odd )
7	6	expcom 451	. . . . . . . . . . . . . . 15 |- ( R e. Odd -> ( N e. Even -> ( N - R ) e. Odd ) )
8	5, 7	syl 17	. . . . . . . . . . . . . 14 |- ( R e. ( Prime \ { 2 } ) -> ( N e. Even -> ( N - R ) e. Odd ) )
9	4, 8	sylbir 225	. . . . . . . . . . . . 13 |- ( ( R e. Prime /\ R =/= 2 ) -> ( N e. Even -> ( N - R ) e. Odd ) )
10	9	ex 450	. . . . . . . . . . . 12 |- ( R e. Prime -> ( R =/= 2 -> ( N e. Even -> ( N - R ) e. Odd ) ) )
11	3, 10	syl5bir 233	. . . . . . . . . . 11 |- ( R e. Prime -> ( -. R = 2 -> ( N e. Even -> ( N - R ) e. Odd ) ) )
12	11	com23 86	. . . . . . . . . 10 |- ( R e. Prime -> ( N e. Even -> ( -. R = 2 -> ( N - R ) e. Odd ) ) )
13	12	3ad2ant3 1084	. . . . . . . . 9 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( N e. Even -> ( -. R = 2 -> ( N - R ) e. Odd ) ) )
14	13	impcom 446	. . . . . . . 8 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( -. R = 2 -> ( N - R ) e. Odd ) )
15	14	3adant3 1081	. . . . . . 7 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( -. R = 2 -> ( N - R ) e. Odd ) )
16	15	impcom 446	. . . . . 6 |- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( N - R ) e. Odd )
17		3simpa 1058	. . . . . . . 8 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( P e. Prime /\ Q e. Prime ) )
18	17	3ad2ant2 1083	. . . . . . 7 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( P e. Prime /\ Q e. Prime ) )
19	18	adantl 482	. . . . . 6 |- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( P e. Prime /\ Q e. Prime ) )
20		eqcom 2629	. . . . . . . . 9 |- ( N = ( ( P + Q ) + R ) <-> ( ( P + Q ) + R ) = N )
21		evenz 41543	. . . . . . . . . . . . 13 |- ( N e. Even -> N e. ZZ )
22	21	zcnd 11483	. . . . . . . . . . . 12 |- ( N e. Even -> N e. CC )
23	22	adantr 481	. . . . . . . . . . 11 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> N e. CC )
24		prmz 15389	. . . . . . . . . . . . . 14 |- ( R e. Prime -> R e. ZZ )
25	24	zcnd 11483	. . . . . . . . . . . . 13 |- ( R e. Prime -> R e. CC )
26	25	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> R e. CC )
27	26	adantl 482	. . . . . . . . . . 11 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> R e. CC )
28		prmz 15389	. . . . . . . . . . . . . . 15 |- ( P e. Prime -> P e. ZZ )
29		prmz 15389	. . . . . . . . . . . . . . 15 |- ( Q e. Prime -> Q e. ZZ )
30		zaddcl 11417	. . . . . . . . . . . . . . 15 |- ( ( P e. ZZ /\ Q e. ZZ ) -> ( P + Q ) e. ZZ )
31	28, 29, 30	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. ZZ )
32	31	zcnd 11483	. . . . . . . . . . . . 13 |- ( ( P e. Prime /\ Q e. Prime ) -> ( P + Q ) e. CC )
33	32	3adant3 1081	. . . . . . . . . . . 12 |- ( ( P e. Prime /\ Q e. Prime /\ R e. Prime ) -> ( P + Q ) e. CC )
34	33	adantl 482	. . . . . . . . . . 11 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( P + Q ) e. CC )
35	23, 27, 34	subadd2d 10411	. . . . . . . . . 10 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( ( N - R ) = ( P + Q ) <-> ( ( P + Q ) + R ) = N ) )
36	35	biimprd 238	. . . . . . . . 9 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( ( ( P + Q ) + R ) = N -> ( N - R ) = ( P + Q ) ) )
37	20, 36	syl5bi 232	. . . . . . . 8 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) ) -> ( N = ( ( P + Q ) + R ) -> ( N - R ) = ( P + Q ) ) )
38	37	3impia 1261	. . . . . . 7 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( N - R ) = ( P + Q ) )
39	38	adantl 482	. . . . . 6 |- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( N - R ) = ( P + Q ) )
40		odd2prm2 41627	. . . . . 6 |- ( ( ( N - R ) e. Odd /\ ( P e. Prime /\ Q e. Prime ) /\ ( N - R ) = ( P + Q ) ) -> ( P = 2 \/ Q = 2 ) )
41	16, 19, 39, 40	syl3anc 1326	. . . . 5 |- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( P = 2 \/ Q = 2 ) )
42	41	orcd 407	. . . 4 |- ( ( -. R = 2 /\ ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) )
43	42	ex 450	. . 3 |- ( -. R = 2 -> ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) ) )
44	2, 43	pm2.61i 176	. 2 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) )
45		df-3or 1038	. 2 |- ( ( P = 2 \/ Q = 2 \/ R = 2 ) <-> ( ( P = 2 \/ Q = 2 ) \/ R = 2 ) )
46	44, 45	sylibr 224	1 |- ( ( N e. Even /\ ( P e. Prime /\ Q e. Prime /\ R e. Prime ) /\ N = ( ( P + Q ) + R ) ) -> ( P = 2 \/ Q = 2 \/ R = 2 ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177  (class class class)co 6650   CCcc 9934   + caddc 9939   - cmin 10266   2c2 11070   ZZcz 11377   Primecprime 15385   Even ceven 41537   Odd codd 41538

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-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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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

This theorem is referenced by:  mogoldbblem  41629