Theorem nnsum3primesgbe 41680

Description: Any even Goldbach number is the sum of at most 3 (actually 2) primes. (Contributed by AV, 2-Aug-2020.)

Assertion

Ref	Expression
nnsum3primesgbe	|- ( N e. GoldbachEven -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) )

Distinct variable group:   N, d, f, k

Proof of Theorem nnsum3primesgbe

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

Step	Hyp	Ref	Expression
1		isgbe 41639	. 2 |- ( N e. GoldbachEven <-> ( N e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) )
2		2nn 11185	. . . . . . . 8 |- 2 e. NN
3	2	a1i 11	. . . . . . 7 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> 2 e. NN )
4		oveq2 6658	. . . . . . . . . . 11 |- ( d = 2 -> ( 1 ... d ) = ( 1 ... 2 ) )
5		df-2 11079	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
6	5	oveq2i 6661	. . . . . . . . . . . 12 |- ( 1 ... 2 ) = ( 1 ... ( 1 + 1 ) )
7		1z 11407	. . . . . . . . . . . . 13 |- 1 e. ZZ
8		fzpr 12396	. . . . . . . . . . . . 13 |- ( 1 e. ZZ -> ( 1 ... ( 1 + 1 ) ) = { 1 , ( 1 + 1 ) } )
9	7, 8	ax-mp 5	. . . . . . . . . . . 12 |- ( 1 ... ( 1 + 1 ) ) = { 1 , ( 1 + 1 ) }
10		1p1e2 11134	. . . . . . . . . . . . 13 |- ( 1 + 1 ) = 2
11	10	preq2i 4272	. . . . . . . . . . . 12 |- { 1 , ( 1 + 1 ) } = { 1 , 2 }
12	6, 9, 11	3eqtri 2648	. . . . . . . . . . 11 |- ( 1 ... 2 ) = { 1 , 2 }
13	4, 12	syl6eq 2672	. . . . . . . . . 10 |- ( d = 2 -> ( 1 ... d ) = { 1 , 2 } )
14	13	oveq2d 6666	. . . . . . . . 9 |- ( d = 2 -> ( Prime ^m ( 1 ... d ) ) = ( Prime ^m { 1 , 2 } ) )
15		breq1 4656	. . . . . . . . . 10 |- ( d = 2 -> ( d <_ 3 <-> 2 <_ 3 ) )
16	13	sumeq1d 14431	. . . . . . . . . . 11 |- ( d = 2 -> sum_ k e. ( 1 ... d ) ( f ` k ) = sum_ k e. { 1 , 2 } ( f ` k ) )
17	16	eqeq2d 2632	. . . . . . . . . 10 |- ( d = 2 -> ( N = sum_ k e. ( 1 ... d ) ( f ` k ) <-> N = sum_ k e. { 1 , 2 } ( f ` k ) ) )
18	15, 17	anbi12d 747	. . . . . . . . 9 |- ( d = 2 -> ( ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) <-> ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) ) )
19	14, 18	rexeqbidv 3153	. . . . . . . 8 |- ( d = 2 -> ( E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) <-> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) ) )
20	19	adantl 482	. . . . . . 7 |- ( ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) /\ d = 2 ) -> ( E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) <-> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) ) )
21		1ne2 11240	. . . . . . . . . . . . 13 |- 1 =/= 2
22		1ex 10035	. . . . . . . . . . . . . 14 |- 1 e. _V
23		2ex 11092	. . . . . . . . . . . . . 14 |- 2 e. _V
24		vex 3203	. . . . . . . . . . . . . 14 |- p e. _V
25		vex 3203	. . . . . . . . . . . . . 14 |- q e. _V
26	22, 23, 24, 25	fpr 6421	. . . . . . . . . . . . 13 |- ( 1 =/= 2 -> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> { p , q } )
27	21, 26	mp1i 13	. . . . . . . . . . . 12 |- ( ( p e. Prime /\ q e. Prime ) -> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> { p , q } )
28		prssi 4353	. . . . . . . . . . . 12 |- ( ( p e. Prime /\ q e. Prime ) -> { p , q } C_ Prime )
29	27, 28	fssd 6057	. . . . . . . . . . 11 |- ( ( p e. Prime /\ q e. Prime ) -> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> Prime )
30		prmex 15391	. . . . . . . . . . . . 13 |- Prime e. _V
31		prex 4909	. . . . . . . . . . . . 13 |- { 1 , 2 } e. _V
32	30, 31	pm3.2i 471	. . . . . . . . . . . 12 |- ( Prime e. _V /\ { 1 , 2 } e. _V )
33		elmapg 7870	. . . . . . . . . . . 12 |- ( ( Prime e. _V /\ { 1 , 2 } e. _V ) -> ( { <. 1 , p >. , <. 2 , q >. } e. ( Prime ^m { 1 , 2 } ) <-> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> Prime ) )
34	32, 33	mp1i 13	. . . . . . . . . . 11 |- ( ( p e. Prime /\ q e. Prime ) -> ( { <. 1 , p >. , <. 2 , q >. } e. ( Prime ^m { 1 , 2 } ) <-> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> Prime ) )
35	29, 34	mpbird 247	. . . . . . . . . 10 |- ( ( p e. Prime /\ q e. Prime ) -> { <. 1 , p >. , <. 2 , q >. } e. ( Prime ^m { 1 , 2 } ) )
36		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( f = { <. 1 , p >. , <. 2 , q >. } -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` k ) )
37	36	adantr 481	. . . . . . . . . . . . . 14 |- ( ( f = { <. 1 , p >. , <. 2 , q >. } /\ k e. { 1 , 2 } ) -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` k ) )
38	37	sumeq2dv 14433	. . . . . . . . . . . . 13 |- ( f = { <. 1 , p >. , <. 2 , q >. } -> sum_ k e. { 1 , 2 } ( f ` k ) = sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) )
39	38	eqeq1d 2624	. . . . . . . . . . . 12 |- ( f = { <. 1 , p >. , <. 2 , q >. } -> ( sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) <-> sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) ) )
40	39	anbi2d 740	. . . . . . . . . . 11 |- ( f = { <. 1 , p >. , <. 2 , q >. } -> ( ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) <-> ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) ) ) )
41	40	adantl 482	. . . . . . . . . 10 |- ( ( ( p e. Prime /\ q e. Prime ) /\ f = { <. 1 , p >. , <. 2 , q >. } ) -> ( ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) <-> ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) ) ) )
42		prmz 15389	. . . . . . . . . . . 12 |- ( p e. Prime -> p e. ZZ )
43		prmz 15389	. . . . . . . . . . . 12 |- ( q e. Prime -> q e. ZZ )
44		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` 1 ) )
45	22, 24	fvpr1 6456	. . . . . . . . . . . . . . 15 |- ( 1 =/= 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` 1 ) = p )
46	21, 45	ax-mp 5	. . . . . . . . . . . . . 14 |- ( { <. 1 , p >. , <. 2 , q >. } ` 1 ) = p
47	44, 46	syl6eq 2672	. . . . . . . . . . . . 13 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = p )
48		fveq2 6191	. . . . . . . . . . . . . 14 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` 2 ) )
49	23, 25	fvpr2 6457	. . . . . . . . . . . . . . 15 |- ( 1 =/= 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` 2 ) = q )
50	21, 49	ax-mp 5	. . . . . . . . . . . . . 14 |- ( { <. 1 , p >. , <. 2 , q >. } ` 2 ) = q
51	48, 50	syl6eq 2672	. . . . . . . . . . . . 13 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = q )
52		zcn 11382	. . . . . . . . . . . . . 14 |- ( p e. ZZ -> p e. CC )
53		zcn 11382	. . . . . . . . . . . . . 14 |- ( q e. ZZ -> q e. CC )
54	52, 53	anim12i 590	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ q e. ZZ ) -> ( p e. CC /\ q e. CC ) )
55	7, 2	pm3.2i 471	. . . . . . . . . . . . . 14 |- ( 1 e. ZZ /\ 2 e. NN )
56	55	a1i 11	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ q e. ZZ ) -> ( 1 e. ZZ /\ 2 e. NN ) )
57	21	a1i 11	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ q e. ZZ ) -> 1 =/= 2 )
58	47, 51, 54, 56, 57	sumpr 14477	. . . . . . . . . . . 12 |- ( ( p e. ZZ /\ q e. ZZ ) -> sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) )
59	42, 43, 58	syl2an 494	. . . . . . . . . . 11 |- ( ( p e. Prime /\ q e. Prime ) -> sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) )
60		2re 11090	. . . . . . . . . . . 12 |- 2 e. RR
61		3re 11094	. . . . . . . . . . . 12 |- 3 e. RR
62		2lt3 11195	. . . . . . . . . . . 12 |- 2 < 3
63	60, 61, 62	ltleii 10160	. . . . . . . . . . 11 |- 2 <_ 3
64	59, 63	jctil 560	. . . . . . . . . 10 |- ( ( p e. Prime /\ q e. Prime ) -> ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) ) )
65	35, 41, 64	rspcedvd 3317	. . . . . . . . 9 |- ( ( p e. Prime /\ q e. Prime ) -> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) )
66	65	adantr 481	. . . . . . . 8 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) )
67		eqeq1 2626	. . . . . . . . . . . . 13 |- ( N = ( p + q ) -> ( N = sum_ k e. { 1 , 2 } ( f ` k ) <-> ( p + q ) = sum_ k e. { 1 , 2 } ( f ` k ) ) )
68		eqcom 2629	. . . . . . . . . . . . 13 |- ( ( p + q ) = sum_ k e. { 1 , 2 } ( f ` k ) <-> sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) )
69	67, 68	syl6bb 276	. . . . . . . . . . . 12 |- ( N = ( p + q ) -> ( N = sum_ k e. { 1 , 2 } ( f ` k ) <-> sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) )
70	69	anbi2d 740	. . . . . . . . . . 11 |- ( N = ( p + q ) -> ( ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) <-> ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) ) )
71	70	rexbidv 3052	. . . . . . . . . 10 |- ( N = ( p + q ) -> ( E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) <-> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) ) )
72	71	3ad2ant3 1084	. . . . . . . . 9 |- ( ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) -> ( E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) <-> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) ) )
73	72	adantl 482	. . . . . . . 8 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> ( E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) <-> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) ) )
74	66, 73	mpbird 247	. . . . . . 7 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> E. f e. ( Prime ^m { 1 , 2 } ) ( 2 <_ 3 /\ N = sum_ k e. { 1 , 2 } ( f ` k ) ) )
75	3, 20, 74	rspcedvd 3317	. . . . . 6 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) )
76	75	a1d 25	. . . . 5 |- ( ( ( p e. Prime /\ q e. Prime ) /\ ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> ( N e. Even -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) ) )
77	76	ex 450	. . . 4 |- ( ( p e. Prime /\ q e. Prime ) -> ( ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) -> ( N e. Even -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) ) ) )
78	77	rexlimivv 3036	. . 3 |- ( E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) -> ( N e. Even -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) ) )
79	78	impcom 446	. 2 |- ( ( N e. Even /\ E. p e. Prime E. q e. Prime ( p e. Odd /\ q e. Odd /\ N = ( p + q ) ) ) -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) )
80	1, 79	sylbi 207	1 |- ( N e. GoldbachEven -> E. d e. NN E. f e. ( Prime ^m ( 1 ... d ) ) ( d <_ 3 /\ N = sum_ k e. ( 1 ... d ) ( f ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   {cpr 4179   <.cop 4183   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   1c1 9937   + caddc 9939   <_ cle 10075   NNcn 11020   2c2 11070   3c3 11071   ZZcz 11377   ...cfz 12326   sum_csu 14416   Primecprime 15385   Even ceven 41537   Odd codd 41538   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-fal 1489  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-int 4476  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-se 5074  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-isom 5897  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-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-prm 15386  df-gbe 41636

This theorem is referenced by:  nnsum4primesgbe  41681  nnsum3primesle9  41682  bgoldbnnsum3prm  41692