Theorem nnsum3primesgbe 38897

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 38862	. 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 10774	. . . . . . . 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 6303	. . . . . . . . . . 11 |- ( d = 2 -> ( 1 ... d ) = ( 1 ... 2 ) )
5		df-2 10675	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
6	5	oveq2i 6306	. . . . . . . . . . . 12 |- ( 1 ... 2 ) = ( 1 ... ( 1 + 1 ) )
7		1z 10974	. . . . . . . . . . . . 13 |- 1 e. ZZ
8		fzpr 11858	. . . . . . . . . . . . 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 10730	. . . . . . . . . . . . 13 |- ( 1 + 1 ) = 2
11	10	preq2i 4058	. . . . . . . . . . . 12 |- { 1 , ( 1 + 1 ) } = { 1 , 2 }
12	6, 9, 11	3eqtri 2479	. . . . . . . . . . 11 |- ( 1 ... 2 ) = { 1 , 2 }
13	4, 12	syl6eq 2503	. . . . . . . . . 10 |- ( d = 2 -> ( 1 ... d ) = { 1 , 2 } )
14	13	oveq2d 6311	. . . . . . . . 9 |- ( d = 2 -> ( Prime ^m ( 1 ... d ) ) = ( Prime ^m { 1 , 2 } ) )
15		breq1 4408	. . . . . . . . . 10 |- ( d = 2 -> ( d <_ 3 <-> 2 <_ 3 ) )
16	13	sumeq1d 13779	. . . . . . . . . . 11 |- ( d = 2 -> sum_ k e. ( 1 ... d ) ( f ` k ) = sum_ k e. { 1 , 2 } ( f ` k ) )
17	16	eqeq2d 2463	. . . . . . . . . 10 |- ( d = 2 -> ( N = sum_ k e. ( 1 ... d ) ( f ` k ) <-> N = sum_ k e. { 1 , 2 } ( f ` k ) ) )
18	15, 17	anbi12d 718	. . . . . . . . 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 3004	. . . . . . . 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 468	. . . . . . 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 10829	. . . . . . . . . . . . 13 |- 1 =/= 2
22		1ex 9643	. . . . . . . . . . . . . 14 |- 1 e. _V
23		2ex 10688	. . . . . . . . . . . . . 14 |- 2 e. _V
24		vex 3050	. . . . . . . . . . . . . 14 |- p e. _V
25		vex 3050	. . . . . . . . . . . . . 14 |- q e. _V
26	22, 23, 24, 25	fpr 6077	. . . . . . . . . . . . 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 4131	. . . . . . . . . . . 12 |- ( ( p e. Prime /\ q e. Prime ) -> { p , q } C_ Prime )
29	27, 28	fssd 5743	. . . . . . . . . . 11 |- ( ( p e. Prime /\ q e. Prime ) -> { <. 1 , p >. , <. 2 , q >. } : { 1 , 2 } --> Prime )
30		prmex 14640	. . . . . . . . . . . . 13 |- Prime e. _V
31		prex 4645	. . . . . . . . . . . . 13 |- { 1 , 2 } e. _V
32	30, 31	pm3.2i 457	. . . . . . . . . . . 12 |- ( Prime e. _V /\ { 1 , 2 } e. _V )
33		elmapg 7490	. . . . . . . . . . . 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 236	. . . . . . . . . 10 |- ( ( p e. Prime /\ q e. Prime ) -> { <. 1 , p >. , <. 2 , q >. } e. ( Prime ^m { 1 , 2 } ) )
36		fveq1 5869	. . . . . . . . . . . . . . 15 |- ( f = { <. 1 , p >. , <. 2 , q >. } -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` k ) )
37	36	adantr 467	. . . . . . . . . . . . . 14 |- ( ( f = { <. 1 , p >. , <. 2 , q >. } /\ k e. { 1 , 2 } ) -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` k ) )
38	37	sumeq2dv 13781	. . . . . . . . . . . . 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 2455	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 468	. . . . . . . . . 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 14638	. . . . . . . . . . . 12 |- ( p e. Prime -> p e. ZZ )
43		prmz 14638	. . . . . . . . . . . 12 |- ( q e. Prime -> q e. ZZ )
44		fveq2 5870	. . . . . . . . . . . . . 14 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` 1 ) )
45	22, 24	fvpr1 6112	. . . . . . . . . . . . . . 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 2503	. . . . . . . . . . . . 13 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = p )
48		fveq2 5870	. . . . . . . . . . . . . 14 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. } ` 2 ) )
49	23, 25	fvpr2 6113	. . . . . . . . . . . . . . 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 2503	. . . . . . . . . . . . 13 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. } ` k ) = q )
52		zcn 10949	. . . . . . . . . . . . . 14 |- ( p e. ZZ -> p e. CC )
53		zcn 10949	. . . . . . . . . . . . . 14 |- ( q e. ZZ -> q e. CC )
54	52, 53	anim12i 570	. . . . . . . . . . . . 13 |- ( ( p e. ZZ /\ q e. ZZ ) -> ( p e. CC /\ q e. CC ) )
55	7, 2	pm3.2i 457	. . . . . . . . . . . . . 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 13821	. . . . . . . . . . . 12 |- ( ( p e. ZZ /\ q e. ZZ ) -> sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) )
59	42, 43, 58	syl2an 480	. . . . . . . . . . 11 |- ( ( p e. Prime /\ q e. Prime ) -> sum_ k e. { 1 , 2 } ( { <. 1 , p >. , <. 2 , q >. } ` k ) = ( p + q ) )
60		2re 10686	. . . . . . . . . . . 12 |- 2 e. RR
61		3re 10690	. . . . . . . . . . . 12 |- 3 e. RR
62		2lt3 10784	. . . . . . . . . . . 12 |- 2 < 3
63	60, 61, 62	ltleii 9762	. . . . . . . . . . 11 |- 2 <_ 3
64	59, 63	jctil 540	. . . . . . . . . 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 3157	. . . . . . . . 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 467	. . . . . . . 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 2457	. . . . . . . . . . . . 13 |- ( N = ( p + q ) -> ( N = sum_ k e. { 1 , 2 } ( f ` k ) <-> ( p + q ) = sum_ k e. { 1 , 2 } ( f ` k ) ) )
68		eqcom 2460	. . . . . . . . . . . . 13 |- ( ( p + q ) = sum_ k e. { 1 , 2 } ( f ` k ) <-> sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) )
69	67, 68	syl6bb 265	. . . . . . . . . . . 12 |- ( N = ( p + q ) -> ( N = sum_ k e. { 1 , 2 } ( f ` k ) <-> sum_ k e. { 1 , 2 } ( f ` k ) = ( p + q ) ) )
70	69	anbi2d 711	. . . . . . . . . . 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 2903	. . . . . . . . . 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 1032	. . . . . . . . 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 468	. . . . . . . 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 236	. . . . . . 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 3157	. . . . . 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 26	. . . . 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 436	. . . 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 2886	. . 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 432	. 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 199	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 188   /\ wa 371   /\ w3a 986   = wceq 1446   e. wcel 1889   =/= wne 2624   E.wrex 2740   _Vcvv 3047   {cpr 3972   <.cop 3976   class class class wbr 4405   -->wf 5581   ` cfv 5585  (class class class)co 6295   ^m cmap 7477   CCcc 9542   1c1 9545   + caddc 9547   <_ cle 9681   NNcn 10616   2c2 10666   3c3 10667   ZZcz 10944   ...cfz 11791   sum_csu 13764   Primecprime 14634   Even ceven 38763   Odd codd 38764   GoldbachEven cgbe 38856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-prm 14635  df-gbe 38859

This theorem is referenced by:  nnsum4primesgbe  38898  nnsum3primesle9  38899  bgoldbnnsum3prm  38909