Theorem nnsum4primesodd 41684

Description: If the (weak) ternary Goldbach conjecture is valid, then every odd integer greater than 5 is the sum of 3 primes. (Contributed by AV, 2-Jul-2020.)

Assertion

Ref	Expression
nnsum4primesodd	|- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )

Distinct variable group:   f, N, k, m

Proof of Theorem nnsum4primesodd

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

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . . 6 |- ( m = N -> ( 5 < m <-> 5 < N ) )
2		eleq1 2689	. . . . . 6 |- ( m = N -> ( m e. GoldbachOddW <-> N e. GoldbachOddW ) )
3	1, 2	imbi12d 334	. . . . 5 |- ( m = N -> ( ( 5 < m -> m e. GoldbachOddW ) <-> ( 5 < N -> N e. GoldbachOddW ) ) )
4	3	rspcv 3305	. . . 4 |- ( N e. Odd -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( 5 < N -> N e. GoldbachOddW ) ) )
5	4	adantl 482	. . 3 |- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( 5 < N -> N e. GoldbachOddW ) ) )
6		eluz2 11693	. . . . . 6 |- ( N e. ( ZZ>= ` 6 ) <-> ( 6 e. ZZ /\ N e. ZZ /\ 6 <_ N ) )
7		5lt6 11204	. . . . . . . . 9 |- 5 < 6
8		5re 11099	. . . . . . . . . . 11 |- 5 e. RR
9	8	a1i 11	. . . . . . . . . 10 |- ( N e. ZZ -> 5 e. RR )
10		6re 11101	. . . . . . . . . . 11 |- 6 e. RR
11	10	a1i 11	. . . . . . . . . 10 |- ( N e. ZZ -> 6 e. RR )
12		zre 11381	. . . . . . . . . 10 |- ( N e. ZZ -> N e. RR )
13		ltletr 10129	. . . . . . . . . 10 |- ( ( 5 e. RR /\ 6 e. RR /\ N e. RR ) -> ( ( 5 < 6 /\ 6 <_ N ) -> 5 < N ) )
14	9, 11, 12, 13	syl3anc 1326	. . . . . . . . 9 |- ( N e. ZZ -> ( ( 5 < 6 /\ 6 <_ N ) -> 5 < N ) )
15	7, 14	mpani 712	. . . . . . . 8 |- ( N e. ZZ -> ( 6 <_ N -> 5 < N ) )
16	15	imp 445	. . . . . . 7 |- ( ( N e. ZZ /\ 6 <_ N ) -> 5 < N )
17	16	3adant1 1079	. . . . . 6 |- ( ( 6 e. ZZ /\ N e. ZZ /\ 6 <_ N ) -> 5 < N )
18	6, 17	sylbi 207	. . . . 5 |- ( N e. ( ZZ>= ` 6 ) -> 5 < N )
19	18	adantr 481	. . . 4 |- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> 5 < N )
20		pm2.27 42	. . . 4 |- ( 5 < N -> ( ( 5 < N -> N e. GoldbachOddW ) -> N e. GoldbachOddW ) )
21	19, 20	syl 17	. . 3 |- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( ( 5 < N -> N e. GoldbachOddW ) -> N e. GoldbachOddW ) )
22		isgbow 41640	. . . . 5 |- ( N e. GoldbachOddW <-> ( N e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) )
23		1ex 10035	. . . . . . . . . . . . . . 15 |- 1 e. _V
24		2ex 11092	. . . . . . . . . . . . . . 15 |- 2 e. _V
25		3ex 11096	. . . . . . . . . . . . . . 15 |- 3 e. _V
26		vex 3203	. . . . . . . . . . . . . . 15 |- p e. _V
27		vex 3203	. . . . . . . . . . . . . . 15 |- q e. _V
28		vex 3203	. . . . . . . . . . . . . . 15 |- r e. _V
29		1ne2 11240	. . . . . . . . . . . . . . 15 |- 1 =/= 2
30		1re 10039	. . . . . . . . . . . . . . . 16 |- 1 e. RR
31		1lt3 11196	. . . . . . . . . . . . . . . 16 |- 1 < 3
32	30, 31	ltneii 10150	. . . . . . . . . . . . . . 15 |- 1 =/= 3
33		2re 11090	. . . . . . . . . . . . . . . 16 |- 2 e. RR
34		2lt3 11195	. . . . . . . . . . . . . . . 16 |- 2 < 3
35	33, 34	ltneii 10150	. . . . . . . . . . . . . . 15 |- 2 =/= 3
36	23, 24, 25, 26, 27, 28, 29, 32, 35	ftp 6424	. . . . . . . . . . . . . 14 |- { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r }
37	36	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r } )
38		1p2e3 11152	. . . . . . . . . . . . . . . . 17 |- ( 1 + 2 ) = 3
39	38	eqcomi 2631	. . . . . . . . . . . . . . . 16 |- 3 = ( 1 + 2 )
40	39	oveq2i 6661	. . . . . . . . . . . . . . 15 |- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) )
41		1z 11407	. . . . . . . . . . . . . . . 16 |- 1 e. ZZ
42		fztp 12397	. . . . . . . . . . . . . . . 16 |- ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } )
43	41, 42	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) }
44		eqid 2622	. . . . . . . . . . . . . . . 16 |- 1 = 1
45		id 22	. . . . . . . . . . . . . . . . 17 |- ( 1 = 1 -> 1 = 1 )
46		1p1e2 11134	. . . . . . . . . . . . . . . . . 18 |- ( 1 + 1 ) = 2
47	46	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( 1 = 1 -> ( 1 + 1 ) = 2 )
48	38	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( 1 = 1 -> ( 1 + 2 ) = 3 )
49	45, 47, 48	tpeq123d 4283	. . . . . . . . . . . . . . . 16 |- ( 1 = 1 -> { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 } )
50	44, 49	ax-mp 5	. . . . . . . . . . . . . . 15 |- { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 }
51	40, 43, 50	3eqtri 2648	. . . . . . . . . . . . . 14 |- ( 1 ... 3 ) = { 1 , 2 , 3 }
52	51	feq2i 6037	. . . . . . . . . . . . 13 |- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> { p , q , r } <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r } )
53	37, 52	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> { p , q , r } )
54		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) <-> ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) )
55	26, 27, 28	tpss 4368	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) <-> { p , q , r } C_ Prime )
56	54, 55	sylbb1 227	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { p , q , r } C_ Prime )
57	53, 56	fssd 6057	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime )
58		prmex 15391	. . . . . . . . . . . . 13 |- Prime e. _V
59		ovex 6678	. . . . . . . . . . . . 13 |- ( 1 ... 3 ) e. _V
60	58, 59	pm3.2i 471	. . . . . . . . . . . 12 |- ( Prime e. _V /\ ( 1 ... 3 ) e. _V )
61		elmapg 7870	. . . . . . . . . . . 12 |- ( ( Prime e. _V /\ ( 1 ... 3 ) e. _V ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime ) )
62	60, 61	mp1i 13	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime ) )
63	57, 62	mpbird 247	. . . . . . . . . 10 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) )
64		fveq1 6190	. . . . . . . . . . . . 13 |- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) )
65	64	sumeq2sdv 14435	. . . . . . . . . . . 12 |- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> sum_ k e. ( 1 ... 3 ) ( f ` k ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) )
66	65	eqeq2d 2632	. . . . . . . . . . 11 |- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> ( ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) )
67	66	adantl 482	. . . . . . . . . 10 |- ( ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) /\ f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ) -> ( ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) )
68	51	a1i 11	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( 1 ... 3 ) = { 1 , 2 , 3 } )
69	68	sumeq1d 14431	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = sum_ k e. { 1 , 2 , 3 } ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) )
70		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) )
71	23, 26	fvtp1 6460	. . . . . . . . . . . . . 14 |- ( ( 1 =/= 2 /\ 1 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) = p )
72	29, 32, 71	mp2an 708	. . . . . . . . . . . . 13 |- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) = p
73	70, 72	syl6eq 2672	. . . . . . . . . . . 12 |- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = p )
74		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) )
75	24, 27	fvtp2 6461	. . . . . . . . . . . . . 14 |- ( ( 1 =/= 2 /\ 2 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) = q )
76	29, 35, 75	mp2an 708	. . . . . . . . . . . . 13 |- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) = q
77	74, 76	syl6eq 2672	. . . . . . . . . . . 12 |- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = q )
78		fveq2 6191	. . . . . . . . . . . . 13 |- ( k = 3 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) )
79	25, 28	fvtp3 6462	. . . . . . . . . . . . . 14 |- ( ( 1 =/= 3 /\ 2 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) = r )
80	32, 35, 79	mp2an 708	. . . . . . . . . . . . 13 |- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) = r
81	78, 80	syl6eq 2672	. . . . . . . . . . . 12 |- ( k = 3 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = r )
82		prmz 15389	. . . . . . . . . . . . . . 15 |- ( p e. Prime -> p e. ZZ )
83	82	zcnd 11483	. . . . . . . . . . . . . 14 |- ( p e. Prime -> p e. CC )
84		prmz 15389	. . . . . . . . . . . . . . 15 |- ( q e. Prime -> q e. ZZ )
85	84	zcnd 11483	. . . . . . . . . . . . . 14 |- ( q e. Prime -> q e. CC )
86		prmz 15389	. . . . . . . . . . . . . . 15 |- ( r e. Prime -> r e. ZZ )
87	86	zcnd 11483	. . . . . . . . . . . . . 14 |- ( r e. Prime -> r e. CC )
88	83, 85, 87	3anim123i 1247	. . . . . . . . . . . . 13 |- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) -> ( p e. CC /\ q e. CC /\ r e. CC ) )
89	88	3expa 1265	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( p e. CC /\ q e. CC /\ r e. CC ) )
90		2z 11409	. . . . . . . . . . . . . 14 |- 2 e. ZZ
91		3z 11410	. . . . . . . . . . . . . 14 |- 3 e. ZZ
92	41, 90, 91	3pm3.2i 1239	. . . . . . . . . . . . 13 |- ( 1 e. ZZ /\ 2 e. ZZ /\ 3 e. ZZ )
93	92	a1i 11	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( 1 e. ZZ /\ 2 e. ZZ /\ 3 e. ZZ ) )
94	29	a1i 11	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 1 =/= 2 )
95	32	a1i 11	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 1 =/= 3 )
96	35	a1i 11	. . . . . . . . . . . 12 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 2 =/= 3 )
97	73, 77, 81, 89, 93, 94, 95, 96	sumtp 14478	. . . . . . . . . . 11 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> sum_ k e. { 1 , 2 , 3 } ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( ( p + q ) + r ) )
98	69, 97	eqtr2d 2657	. . . . . . . . . 10 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) )
99	63, 67, 98	rspcedvd 3317	. . . . . . . . 9 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) )
100		eqeq1 2626	. . . . . . . . . 10 |- ( N = ( ( p + q ) + r ) -> ( N = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
101	100	rexbidv 3052	. . . . . . . . 9 |- ( N = ( ( p + q ) + r ) -> ( E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> E. f e. ( Prime ^m ( 1 ... 3 ) ) ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
102	99, 101	syl5ibrcom 237	. . . . . . . 8 |- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
103	102	rexlimdva 3031	. . . . . . 7 |- ( ( p e. Prime /\ q e. Prime ) -> ( E. r e. Prime N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
104	103	rexlimivv 3036	. . . . . 6 |- ( E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) )
105	104	adantl 482	. . . . 5 |- ( ( N e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) )
106	22, 105	sylbi 207	. . . 4 |- ( N e. GoldbachOddW -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) )
107	106	a1i 11	. . 3 |- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( N e. GoldbachOddW -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
108	5, 21, 107	3syld 60	. 2 |- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )
109	108	com12 32	1 |- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {ctp 4181   <.cop 4183   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   2c2 11070   3c3 11071   5c5 11073   6c6 11074   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   sum_csu 14416   Primecprime 15385   Odd codd 41538   GoldbachOddW cgbow 41634

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-4 11081  df-5 11082  df-6 11083  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-gbow 41637

This theorem is referenced by:  nnsum4primeseven  41688  wtgoldbnnsum4prm  41690  bgoldbnnsum3prm  41692