Theorem evengpoap3 41687

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.) (Proof shortened by AV, 15-Sep-2021.)

Assertion

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

Distinct variable groups:   m, N   o, N

Proof of Theorem evengpoap3

Step	Hyp	Ref	Expression
1		3odd 41617	. . . . . . 7 |- 3 e. Odd
2	1	a1i 11	. . . . . 6 |- ( N e. ( ZZ>= ` ; 1 2 ) -> 3 e. Odd )
3	2	anim1i 592	. . . . 5 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( 3 e. Odd /\ N e. Even ) )
4	3	ancomd 467	. . . 4 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N e. Even /\ 3 e. Odd ) )
5		emoo 41613	. . . 4 |- ( ( N e. Even /\ 3 e. Odd ) -> ( N - 3 ) e. Odd )
6	4, 5	syl 17	. . 3 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N - 3 ) e. Odd )
7		breq2 4657	. . . . 5 |- ( m = ( N - 3 ) -> ( 7 < m <-> 7 < ( N - 3 ) ) )
8		eleq1 2689	. . . . 5 |- ( m = ( N - 3 ) -> ( m e. GoldbachOdd <-> ( N - 3 ) e. GoldbachOdd ) )
9	7, 8	imbi12d 334	. . . 4 |- ( m = ( N - 3 ) -> ( ( 7 < m -> m e. GoldbachOdd ) <-> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOdd ) ) )
10	9	adantl 482	. . 3 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ m = ( N - 3 ) ) -> ( ( 7 < m -> m e. GoldbachOdd ) <-> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOdd ) ) )
11	6, 10	rspcdv 3312	. 2 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( A. m e. Odd ( 7 < m -> m e. GoldbachOdd ) -> ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOdd ) ) )
12		eluz2 11693	. . . . 5 |- ( N e. ( ZZ>= ` ; 1 2 ) <-> (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) )
13		7p3e10 11603	. . . . . . . . . 10 |- ( 7 + 3 ) = ; 1 0
14		1nn0 11308	. . . . . . . . . . 11 |- 1 e. NN0
15		0nn0 11307	. . . . . . . . . . 11 |- 0 e. NN0
16		2nn 11185	. . . . . . . . . . 11 |- 2 e. NN
17		2pos 11112	. . . . . . . . . . 11 |- 0 < 2
18	14, 15, 16, 17	declt 11530	. . . . . . . . . 10 |- ; 1 0 < ; 1 2
19	13, 18	eqbrtri 4674	. . . . . . . . 9 |- ( 7 + 3 ) < ; 1 2
20		7re 11103	. . . . . . . . . . 11 |- 7 e. RR
21		3re 11094	. . . . . . . . . . 11 |- 3 e. RR
22	20, 21	readdcli 10053	. . . . . . . . . 10 |- ( 7 + 3 ) e. RR
23		2nn0 11309	. . . . . . . . . . . 12 |- 2 e. NN0
24	14, 23	deccl 11512	. . . . . . . . . . 11 |- ; 1 2 e. NN0
25	24	nn0rei 11303	. . . . . . . . . 10 |- ; 1 2 e. RR
26		zre 11381	. . . . . . . . . 10 |- ( N e. ZZ -> N e. RR )
27		ltletr 10129	. . . . . . . . . 10 |- ( ( ( 7 + 3 ) e. RR /\ ; 1 2 e. RR /\ N e. RR ) -> ( ( ( 7 + 3 ) < ; 1 2 /\ ; 1 2 <_ N ) -> ( 7 + 3 ) < N ) )
28	22, 25, 26, 27	mp3an12i 1428	. . . . . . . . 9 |- ( N e. ZZ -> ( ( ( 7 + 3 ) < ; 1 2 /\ ; 1 2 <_ N ) -> ( 7 + 3 ) < N ) )
29	19, 28	mpani 712	. . . . . . . 8 |- ( N e. ZZ -> (; 1 2 <_ N -> ( 7 + 3 ) < N ) )
30	29	imp 445	. . . . . . 7 |- ( ( N e. ZZ /\ ; 1 2 <_ N ) -> ( 7 + 3 ) < N )
31	30	3adant1 1079	. . . . . 6 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> ( 7 + 3 ) < N )
32	20	a1i 11	. . . . . . 7 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 7 e. RR )
33	21	a1i 11	. . . . . . 7 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 3 e. RR )
34	26	3ad2ant2 1083	. . . . . . 7 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> N e. RR )
35	32, 33, 34	ltaddsubd 10627	. . . . . 6 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> ( ( 7 + 3 ) < N <-> 7 < ( N - 3 ) ) )
36	31, 35	mpbid 222	. . . . 5 |- ( (; 1 2 e. ZZ /\ N e. ZZ /\ ; 1 2 <_ N ) -> 7 < ( N - 3 ) )
37	12, 36	sylbi 207	. . . 4 |- ( N e. ( ZZ>= ` ; 1 2 ) -> 7 < ( N - 3 ) )
38	37	adantr 481	. . 3 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> 7 < ( N - 3 ) )
39		simpr 477	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOdd ) -> ( N - 3 ) e. GoldbachOdd )
40		oveq1 6657	. . . . . . 7 |- ( o = ( N - 3 ) -> ( o + 3 ) = ( ( N - 3 ) + 3 ) )
41	40	eqeq2d 2632	. . . . . 6 |- ( o = ( N - 3 ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
42	41	adantl 482	. . . . 5 |- ( ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOdd ) /\ o = ( N - 3 ) ) -> ( N = ( o + 3 ) <-> N = ( ( N - 3 ) + 3 ) ) )
43		eluzelcn 11699	. . . . . . . . 9 |- ( N e. ( ZZ>= ` ; 1 2 ) -> N e. CC )
44		3cn 11095	. . . . . . . . 9 |- 3 e. CC
45	43, 44	jctir 561	. . . . . . . 8 |- ( N e. ( ZZ>= ` ; 1 2 ) -> ( N e. CC /\ 3 e. CC ) )
46	45	adantr 481	. . . . . . 7 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( N e. CC /\ 3 e. CC ) )
47	46	adantr 481	. . . . . 6 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOdd ) -> ( N e. CC /\ 3 e. CC ) )
48		npcan 10290	. . . . . . 7 |- ( ( N e. CC /\ 3 e. CC ) -> ( ( N - 3 ) + 3 ) = N )
49	48	eqcomd 2628	. . . . . 6 |- ( ( N e. CC /\ 3 e. CC ) -> N = ( ( N - 3 ) + 3 ) )
50	47, 49	syl 17	. . . . 5 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOdd ) -> N = ( ( N - 3 ) + 3 ) )
51	39, 42, 50	rspcedvd 3317	. . . 4 |- ( ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) /\ ( N - 3 ) e. GoldbachOdd ) -> E. o e. GoldbachOdd N = ( o + 3 ) )
52	51	ex 450	. . 3 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( ( N - 3 ) e. GoldbachOdd -> E. o e. GoldbachOdd N = ( o + 3 ) ) )
53	38, 52	embantd 59	. 2 |- ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> ( ( 7 < ( N - 3 ) -> ( N - 3 ) e. GoldbachOdd ) -> E. o e. GoldbachOdd N = ( o + 3 ) ) )
54	11, 53	syldc 48	1 |- ( A. m e. Odd ( 7 < m -> m e. GoldbachOdd ) -> ( ( N e. ( ZZ>= ` ; 1 2 ) /\ N e. Even ) -> E. o e. GoldbachOdd N = ( o + 3 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   3c3 11071   7c7 11075   ZZcz 11377  ;cdc 11493   ZZ>=cuz 11687   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

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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-even 41539  df-odd 41540

This theorem is referenced by:  nnsum4primesevenALTV  41689