Theorem wtgoldbnnsum4prm 48125

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 12528	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 12248	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 12520	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 12224	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 12249	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 12350	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 11261	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 12762	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1343	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 13615	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) → (ℤ≥‘2) = ((2..^9) ∪ (ℤ≥‘9)))
11	10	eleq2d 2823	. . . . . 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 4106	. . . . 5 ⊢ (𝑛 ∈ ((2..^9) ∪ (ℤ≥‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
14	12, 13	bitri 275	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
15		elfzo2 13583	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16		simp1 1137	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ≥‘2))
17		df-9 12220	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 5107	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12806	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 12245	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℕ
21	19, 20	jctir 520	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
22	21	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23		nnleltp1 12552	. . . . . . . . . . . . 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 248	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
26	18, 25	biimtrid 242	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27	26	3impia 1118	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
28	16, 27	jca 511	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
29	15, 28	sylbi 217	. . . . . . 7 ⊢ (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
30		nnsum4primesle9 48118	. . . . . . 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 12233	. . . . . . . . 9 ⊢ 4 ∈ ℕ
34	33	a1i 11	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 4 ∈ ℕ)
35		oveq2 7369	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (1...𝑑) = (1...4))
36	35	oveq2d 7377	. . . . . . . . . 10 ⊢ (𝑑 = 4 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...4)))
37		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
38	35	sumeq1d 15628	. . . . . . . . . . . 12 ⊢ (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))
39	38	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑑 = 4 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘)))
40	37, 39	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑑 = 4 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓‘𝑘))))
41	36, 40	rexeqbidv 3318	. . . . . . . . 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 481	. . . . . . . 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 12234	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
44	43	leidi 11676	. . . . . . . . . 10 ⊢ 4 ≤ 4
45	44	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 4 ≤ 4)
46		nnsum4primeseven 48123	. . . . . . . . . 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 407	. . . . . . . . 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 3169	. . . . . . . . 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 584	. . . . . . . 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 3579	. . . . . . 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 412	. . . . . 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 12229	. . . . . . . . 9 ⊢ 3 ∈ ℕ
53	52	a1i 11	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 3 ∈ ℕ)
54		oveq2 7369	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
55	54	oveq2d 7377	. . . . . . . . . 10 ⊢ (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
56		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
57	54	sumeq1d 15628	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
58	57	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
59	56, 58	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑑 = 3 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
60	55, 59	rexeqbidv 3318	. . . . . . . . 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 481	. . . . . . . 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 12230	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
63		3lt4 12319	. . . . . . . . . . 11 ⊢ 3 < 4
64	62, 43, 63	ltleii 11261	. . . . . . . . . 10 ⊢ 3 ≤ 4
65	64	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚 → 𝑚 ∈ GoldbachOddW )) → 3 ≤ 4)
66		6nn 12239	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℕ
67	66	nnzi 12520	. . . . . . . . . . . 12 ⊢ 6 ∈ ℤ
68		6re 12240	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℝ
69		6lt9 12346	. . . . . . . . . . . . 13 ⊢ 6 < 9
70	68, 5, 69	ltleii 11261	. . . . . . . . . . . 12 ⊢ 6 ≤ 9
71		eluzuzle 12765	. . . . . . . . . . . 12 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
72	67, 70, 71	mp2an 693	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
73	72	anim1i 616	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
74		nnsum4primesodd 48119	. . . . . . . . . 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 506	. . . . . . . . 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 3169	. . . . . . . . 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 584	. . . . . . . 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 3579	. . . . . . 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 412	. . . . . 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 12766	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
81		zeoALTV 47993	. . . . . . 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 961	. . . . 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 858	. . . 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 217	. . 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 407	. 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 3129	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 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   ∪ cun 3900   class class class wbr 5099  ‘cfv 6493  (class class class)co 7361   ↑_m cmap 8768  1c1 11032   + caddc 11034   < clt 11171   ≤ cle 11172  ℕcn 12150  2c2 12205  3c3 12206  4c4 12207  5c5 12208  6c6 12209  8c8 12211  9c9 12212  ℤcz 12493  ℤ_≥cuz 12756  ...cfz 13428  ..^cfzo 13575  Σcsu 15614  ℙcprime 16603   Even ceven 47947   Odd codd 47948   GoldbachOddW cgbow 48069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-inf2 9555  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108  ax-pre-sup 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-oi 9420  df-card 9856  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12151  df-2 12213  df-3 12214  df-4 12215  df-5 12216  df-6 12217  df-7 12218  df-8 12219  df-9 12220  df-n0 12407  df-z 12494  df-dec 12613  df-uz 12757  df-rp 12911  df-fz 13429  df-fzo 13576  df-seq 13930  df-exp 13990  df-hash 14259  df-cj 15027  df-re 15028  df-im 15029  df-sqrt 15163  df-abs 15164  df-clim 15416  df-sum 15615  df-dvds 16185  df-prm 16604  df-even 47949  df-odd 47950  df-gbe 48071  df-gbow 48072

This theorem is referenced by:  stgoldbnnsum4prm  48126