Theorem wtgoldbnnsum4prm 43275

Description: If the (weak) ternary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 4 primes, showing that Schnirelmann's constant would be less than or equal to 4. See corollary 1.1 in [Helfgott] p. 4. (Contributed by AV, 25-Jul-2020.)

Assertion

Ref	Expression
wtgoldbnnsum4prm	⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Distinct variable group:   𝑓,𝑘,𝑚,𝑑,𝑛

Proof of Theorem wtgoldbnnsum4prm

Step	Hyp	Ref	Expression
1		2z 11820	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 11537	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 11812	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 11507	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 11538	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 11645	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 10555	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 12057	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1321	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 12880	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) → (ℤ≥‘2) = ((2..^9) ∪ (ℤ≥‘9)))
11	10	eleq2d 2845	. . . . . 6 ⊢ (9 ∈ (ℤ≥‘2) → (𝑛 ∈ (ℤ≥‘2) ↔ 𝑛 ∈ ((2..^9) ∪ (ℤ≥‘9))))
12	9, 11	ax-mp 5	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘2) ↔ 𝑛 ∈ ((2..^9) ∪ (ℤ≥‘9)))
13		elun 4010	. . . . 5 ⊢ (𝑛 ∈ ((2..^9) ∪ (ℤ≥‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
14	12, 13	bitri 267	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
15		elfzo2 12850	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16		simp1 1116	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ≥‘2))
17		df-9 11503	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 4931	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12091	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 11533	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℕ
21	19, 20	jctir 513	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
22	21	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23		nnleltp1 11843	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 8 ∈ ℕ) → (𝑛 ≤ 8 ↔ 𝑛 < (8 + 1)))
24	22, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 ≤ 8 ↔ 𝑛 < (8 + 1)))
25	24	biimprd 240	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
26	18, 25	syl5bi 234	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27	26	3impia 1097	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
28	16, 27	jca 504	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
29	15, 28	sylbi 209	. . . . . . 7 ⊢ (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
30		nnsum4primesle9 43268	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
31	29, 30	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (2..^9) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
32	31	a1d 25	. . . . 5 ⊢ (𝑛 ∈ (2..^9) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
33		4nn 11517	. . . . . . . . 9 ⊢ 4 ∈ ℕ
34	33	a1i 11	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 4 ∈ ℕ)
35		oveq2 6978	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (1...𝑑) = (1...4))
36	35	oveq2d 6986	. . . . . . . . . 10 ⊢ (𝑑 = 4 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...4)))
37		breq1 4926	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
38	35	sumeq1d 14908	. . . . . . . . . . . 12 ⊢ (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
39	38	eqeq2d 2782	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
40	37, 39	anbi12d 621	. . . . . . . . . 10 ⊢ (𝑑 = 4 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
41	36, 40	rexeqbidv 3336	. . . . . . . . 9 ⊢ (𝑑 = 4 → (∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
42	41	adantl 474	. . . . . . . 8 ⊢ ((((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) ∧ 𝑑 = 4) → (∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
43		4re 11518	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
44	43	leidi 10967	. . . . . . . . . 10 ⊢ 4 ≤ 4
45	44	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 4 ≤ 4)
46		nnsum4primeseven 43273	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
47	46	impcom 399	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
48		r19.42v 3285	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (ℙ ↑𝑚 (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)) ↔ (4 ≤ 4 ∧ ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
49	45, 47, 48	sylanbrc 575	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
50	34, 42, 49	rspcedvd 3536	. . . . . . 7 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
51	50	ex 405	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
52		3nn 11512	. . . . . . . . 9 ⊢ 3 ∈ ℕ
53	52	a1i 11	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 3 ∈ ℕ)
54		oveq2 6978	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
55	54	oveq2d 6986	. . . . . . . . . 10 ⊢ (𝑑 = 3 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...3)))
56		breq1 4926	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
57	54	sumeq1d 14908	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
58	57	eqeq2d 2782	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
59	56, 58	anbi12d 621	. . . . . . . . . 10 ⊢ (𝑑 = 3 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
60	55, 59	rexeqbidv 3336	. . . . . . . . 9 ⊢ (𝑑 = 3 → (∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
61	60	adantl 474	. . . . . . . 8 ⊢ ((((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) ∧ 𝑑 = 3) → (∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
62		3re 11513	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
63		3lt4 11614	. . . . . . . . . . 11 ⊢ 3 < 4
64	62, 43, 63	ltleii 10555	. . . . . . . . . 10 ⊢ 3 ≤ 4
65	64	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 3 ≤ 4)
66		6nn 11525	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ
67	66	nnzi 11812	. . . . . . . . . . . 12 ⊢ 6 ∈ ℤ
68		6re 11526	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℝ
69		6lt9 11641	. . . . . . . . . . . . 13 ⊢ 6 < 9
70	68, 5, 69	ltleii 10555	. . . . . . . . . . . 12 ⊢ 6 ≤ 9
71		eluzuzle 12060	. . . . . . . . . . . 12 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
72	67, 70, 71	mp2an 679	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
73	72	anim1i 605	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
74		nnsum4primesodd 43269	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
75	73, 74	mpan9 499	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
76		r19.42v 3285	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (ℙ ↑𝑚 (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)) ↔ (3 ≤ 4 ∧ ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
77	65, 75, 76	sylanbrc 575	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
78	53, 61, 77	rspcedvd 3536	. . . . . . 7 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
79	78	ex 405	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
80		eluzelz 12061	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
81		zeoALTV 43143	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
82	80, 81	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
83	51, 79, 82	mpjaodan 941	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘9) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
84	32, 83	jaoi 843	. . . 4 ⊢ ((𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
85	14, 84	sylbi 209	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘2) → (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
86	85	impcom 399	. 2 ⊢ ((∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) ∧ 𝑛 ∈ (ℤ≥‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
87	86	ralrimiva 3126	1 ⊢ (∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∨ wo 833   ∧ w3a 1068   = wceq 1507   ∈ wcel 2048  ∀wral 3082  ∃wrex 3083   ∪ cun 3823   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970   ↑_𝑚 cmap 8198  1c1 10328   + caddc 10330   < clt 10466   ≤ cle 10467  ℕcn 11431  2c2 11488  3c3 11489  4c4 11490  5c5 11491  6c6 11492  8c8 11494  9c9 11495  ℤcz 11786  ℤ_≥cuz 12051  ...cfz 12701  ..^cfzo 12842  Σcsu 14893  ℙcprime 15861   Even ceven 43097   Odd codd 43098   GoldbachOddW cgbow 43219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-inf2 8890  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-map 8200  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-sup 8693  df-inf 8694  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-z 11787  df-dec 11905  df-uz 12052  df-rp 12198  df-fz 12702  df-fzo 12843  df-seq 13178  df-exp 13238  df-hash 13499  df-cj 14309  df-re 14310  df-im 14311  df-sqrt 14445  df-abs 14446  df-clim 14696  df-sum 14894  df-dvds 15458  df-prm 15862  df-even 43099  df-odd 43100  df-gbe 43221  df-gbow 43222

This theorem is referenced by:  stgoldbnnsum4prm  43276