Theorem evengpoap3 38894

Description: If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of an odd Goldbach number and 3. (Contributed by AV, 27-Jul-2020.)

Assertion

Ref	Expression
evengpoap3	|- ( A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) -> ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> E. o e. GoldbachOddALTV N = ( o + 3 ) ) )

Distinct variable groups:   m, N   o, N

Proof of Theorem evengpoap3

Step	Hyp	Ref	Expression
1		3odd 38835	. . . . . . . 8 |- 3 e. Odd
2	1	a1i 11	. . . . . . 7 |- ( N e. ( ZZ>= ` ; 1 2 ) -> 3 e. Odd )
3	2	anim1i 572	. . . . . 6 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( 3 e. Odd /\ N e. Even ) )
4	3	ancomd 453	. . . . 5 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N e. Even /\ 3 e. Odd ) )
5		emoo 38831	. . . . 5 |- ( ( N e. Even /\ 3 e. Odd ) -> ( N - 3 ) e. Odd )
6	4, 5	syl 17	. . . 4 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N - 3 ) e. Odd )
7		breq2 4406	. . . . . 6 |- ( m = ( N - 3 ) -> ( 7 < m <-> 7 < ( N - 3 ) ) )
8		eleq1 2517	. . . . . 6 |- ( m = ( N - 3 ) -> ( m e. GoldbachOddALTV <-> ( N - 3 ) e. GoldbachOddALTV ) )
9	7, 8	imbi12d 322	. . . . 5 |- ( m = ( N - 3 ) -> ( ( 7 < m -> m e. GoldbachOddALTV ) <-> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddALTV ) ) )
10	9	adantl 468	. . . 4 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ m = ( N - 3 ) ) -> ( ( 7 < m -> m e. GoldbachOddALTV ) <-> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddALTV ) ) )
11	6, 10	rspcdv 3153	. . 3 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) -> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddALTV ) ) )
12		eluz2 11165	. . . . . 6 |- ( N e. ( ZZ>= ` ; 1 2 ) <-> (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) )
13		7p3e10 10755	. . . . . . . 8 |- ( 7 + 3 ) = 10
14		dec10p 11080	. . . . . . . . . . . . 13 |- ( 10 + 2 ) = ; 1 2
15	14	eqcomi 2460	. . . . . . . . . . . 12 |- ; 1 2 = ( 10 + 2 )
16	15	breq1i 4409	. . . . . . . . . . 11 |- (; 1 2 <_ N <-> ( 10 + 2 ) <_ N )
17		2pos 10701	. . . . . . . . . . . . 13 |- 0 < 2
18		2re 10679	. . . . . . . . . . . . . 14 |- 2 e. RR
19		10re 10698	. . . . . . . . . . . . . 14 |- 10 e. RR
20	18, 19	ltaddposi 10163	. . . . . . . . . . . . 13 |- ( 0 < 2 <-> 10 < ( 10 + 2 ) )
21	17, 20	mpbi 212	. . . . . . . . . . . 12 |- 10 < ( 10 + 2 )
22	19	a1i 11	. . . . . . . . . . . . 13 |- ( N e. ZZ -> 10 e. RR )
23	18	a1i 11	. . . . . . . . . . . . . 14 |- ( N e. ZZ -> 2 e. RR )
24	22, 23	readdcld 9670	. . . . . . . . . . . . 13 |- ( N e. ZZ -> ( 10 + 2 ) e. RR )
25		zre 10941	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. RR )
26		ltletr 9725	. . . . . . . . . . . . 13 |- ( ( 10 e. RR /\ ( 10 + 2 ) e. RR /\ N e. RR ) -> ( ( 10 < ( 10 + 2 ) /\ ( 10 + 2 ) <_ N ) -> 10 < N ) )
27	22, 24, 25, 26	syl3anc 1268	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( ( 10 < ( 10 + 2 ) /\ ( 10 + 2 ) <_ N ) -> 10 < N ) )
28	21, 27	mpani 682	. . . . . . . . . . 11 |- ( N e. ZZ -> ( ( 10 + 2 ) <_ N -> 10 < N ) )
29	16, 28	syl5bi 221	. . . . . . . . . 10 |- ( N e. ZZ -> (; 1 2 <_ N -> 10 < N ) )
30	29	imp 431	. . . . . . . . 9 |- ( ( N e. ZZ /\ ; 1 2 <_ N ) -> 10 < N )
31	30	3adant1 1026	. . . . . . . 8 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 10 < N )
32	13, 31	syl5eqbr 4436	. . . . . . 7 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> ( 7 + 3 ) < N )
33		7re 10692	. . . . . . . . 9 |- 7 e. RR
34	33	a1i 11	. . . . . . . 8 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 7 e. RR )
35		3re 10683	. . . . . . . . 9 |- 3 e. RR
36	35	a1i 11	. . . . . . . 8 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 3 e. RR )
37	25	3ad2ant2 1030	. . . . . . . 8 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> N e. RR )
38	34, 36, 37	ltaddsubd 10213	. . . . . . 7 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> ( ( 7 + 3 ) < N <-> 7 < ( N - 3 ) ) )
39	32, 38	mpbid 214	. . . . . 6 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 7 < ( N - 3 ) )
40	12, 39	sylbi 199	. . . . 5 |- ( N e. ( ZZ>= ` ; 1 2 ) -> 7 < ( N - 3 ) )
41	40	adantr 467	. . . 4 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> 7 < ( N - 3 ) )
42		simpr 463	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddALTV ) -> ( N - 3 ) e. GoldbachOddALTV )
43		oveq1 6297	. . . . . . . 8 |- ( o = ( N - 3 ) -> ( o + 3 ) = ( ( N - 3 ) + 3 ) )
44	43	eqeq2d 2461	. . . . . . 7 |- ( o = ( N - 3 ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
45	44	adantl 468	. . . . . 6 |- ( ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddALTV ) /\ o = ( N - 3 ) ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
46		eluzelcn 11170	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` ; 1 2 ) -> N e. CC )
47		3cn 10684	. . . . . . . . . 10 |- 3 e. CC
48	46, 47	jctir 541	. . . . . . . . 9 |- ( N e. ( ZZ>= ` ; 1 2 ) -> ( N e. CC /\ 3 e. CC ) )
49	48	adantr 467	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N e. CC /\ 3 e. CC ) )
50	49	adantr 467	. . . . . . 7 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddALTV ) -> ( N e. CC /\ 3 e. CC ) )
51		npcan 9884	. . . . . . . 8 |- ( ( N e. CC /\ 3 e. CC ) -> ( ( N - 3 ) + 3 ) = N )
52	51	eqcomd 2457	. . . . . . 7 |- ( ( N e. CC /\ 3 e. CC ) -> N = ( ( N - 3 ) + 3 ) )
53	50, 52	syl 17	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddALTV ) -> N = ( ( N - 3 ) + 3 ) )
54	42, 45, 53	rspcedvd 3155	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOddALTV ) -> E. o e. GoldbachOddALTV N = ( o + 3 ) )
55	54	ex 436	. . . 4 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( ( N - 3 ) e. GoldbachOddALTV -> E. o e. GoldbachOddALTV N = ( o + 3 ) ) )
56	41, 55	embantd 56	. . 3 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOddALTV ) -> E. o e. GoldbachOddALTV N = ( o + 3 ) ) )
57	11, 56	syld 45	. 2 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) -> E. o e. GoldbachOddALTV N = ( o + 3 ) ) )
58	57	com12 32	1 |- ( A. m e. Odd ( 7 < m -> m e. GoldbachOddALTV ) -> ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> E. o e. GoldbachOddALTV N = ( o + 3 ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   E.wrex 2738   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   2c2 10659   3c3 10660   7c7 10664   10c10 10667   ZZcz 10937  ;cdc 11051   ZZ>=cuz 11159   Even ceven 38753   Odd codd 38754   GoldbachOddALTV cgboa 38848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-even 38755  df-odd 38756

This theorem is referenced by:  nnsum4primesevenALTV  38896