Theorem bgoldbnnsum3prm 44883

Description: If the binary Goldbach conjecture is valid, then every integer greater than 1 is the sum of at most 3 primes, showing that Schnirelmann's constant would be equal to 3. (Contributed by AV, 2-Aug-2020.)

Assertion

Ref	Expression
bgoldbnnsum3prm	⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Distinct variable group:   𝑓,𝑘,𝑚,𝑑,𝑛

Proof of Theorem bgoldbnnsum3prm

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2z 12192	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 11911	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 12184	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 11887	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 11912	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 12018	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 10938	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 12427	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1343	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 13260	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) → (ℤ≥‘2) = ((2..^9) ∪ (ℤ≥‘9)))
11	10	eleq2d 2819	. . . . . 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 4053	. . . . 5 ⊢ (𝑛 ∈ ((2..^9) ∪ (ℤ≥‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
14	12, 13	bitri 278	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)))
15		elfzo2 13229	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16		simp1 1138	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ≥‘2))
17		df-9 11883	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 5051	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12463	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 11908	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℕ
21	19, 20	jctir 524	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
22	21	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23		nnleltp1 12215	. . . . . . . . . . . . 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 251	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
26	18, 25	syl5bi 245	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27	26	3impia 1119	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
28	16, 27	jca 515	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
29	15, 28	sylbi 220	. . . . . . 7 ⊢ (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8))
30		nnsum3primesle9 44873	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 𝑛 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
31	29, 30	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (2..^9) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
32	31	a1d 25	. . . . 5 ⊢ (𝑛 ∈ (2..^9) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
33		breq2 5047	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34		eleq1w 2816	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
35	33, 34	imbi12d 348	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
36	35	rspcv 3525	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
37		4re 11897	. . . . . . . . . . . . . . 15 ⊢ 4 ∈ ℝ
38	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 4 ∈ ℝ)
39	5	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 9 ∈ ℝ)
40		eluzelre 12432	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℝ)
41	38, 39, 40	3jca 1130	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
42	41	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43		eluzle 12434	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 9 ≤ 𝑛)
44	43	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → 9 ≤ 𝑛)
45		4lt9 12016	. . . . . . . . . . . . 13 ⊢ 4 < 9
46	44, 45	jctil 523	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47		ltletr 10907	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((4 < 9 ∧ 9 ≤ 𝑛) → 4 < 𝑛))
48	42, 46, 47	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → 4 < 𝑛)
49		pm2.27 42	. . . . . . . . . . 11 ⊢ (4 < 𝑛 → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
50	48, 49	syl 17	. . . . . . . . . 10 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
51	50	ex 416	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (𝑛 ∈ (ℤ≥‘9) → ((4 < 𝑛 → 𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven )))
52	36, 51	syl5d 73	. . . . . . . 8 ⊢ (𝑛 ∈ Even → (𝑛 ∈ (ℤ≥‘9) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven )))
53	52	impcom 411	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54		nnsum3primesgbe 44871	. . . . . . 7 ⊢ (𝑛 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
55	53, 54	syl6 35	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
56		3nn 11892	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
57	56	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58		oveq2 7210	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
59	58	oveq2d 7218	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
60		breq1 5046	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
61	58	sumeq1d 15248	. . . . . . . . . . . . 13 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
62	61	eqeq2d 2745	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
63	60, 62	anbi12d 634	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
64	59, 63	rexeqbidv 3307	. . . . . . . . . 10 ⊢ (𝑑 = 3 → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
65	64	adantl 485	. . . . . . . . 9 ⊢ ((((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) ∧ 𝑑 = 3) → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
66		3re 11893	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
67	66	leidi 11349	. . . . . . . . . . 11 ⊢ 3 ≤ 3
68	67	a1i 11	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69		6nn 11902	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℕ
70	69	nnzi 12184	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℤ
71		6re 11903	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℝ
72		6lt9 12014	. . . . . . . . . . . . . 14 ⊢ 6 < 9
73	71, 5, 72	ltleii 10938	. . . . . . . . . . . . 13 ⊢ 6 ≤ 9
74		eluzuzle 12430	. . . . . . . . . . . . 13 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
75	70, 73, 74	mp2an 692	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
76	75	anim1i 618	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
77		nnsum4primesodd 44875	. . . . . . . . . . 11 ⊢ (∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
78	76, 77	mpan9 510	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
79		r19.42v 3256	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)) ↔ (3 ≤ 3 ∧ ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
80	68, 78, 79	sylanbrc 586	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
81	57, 65, 80	rspcedvd 3533	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
82	81	expcom 417	. . . . . . 7 ⊢ (∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
83		sbgoldbwt 44856	. . . . . . 7 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW ))
84	82, 83	syl11 33	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
85		eluzelz 12431	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
86		zeoALTV 44749	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
87	85, 86	syl 17	. . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
88	55, 84, 87	mpjaodan 959	. . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘9) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
89	32, 88	jaoi 857	. . . 4 ⊢ ((𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ≥‘9)) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
90	14, 89	sylbi 220	. . 3 ⊢ (𝑛 ∈ (ℤ≥‘2) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘))))
91	90	impcom 411	. 2 ⊢ ((∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ≥‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))
92	91	ralrimiva 3098	1 ⊢ (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ≥‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3054  ∃wrex 3055   ∪ cun 3855   class class class wbr 5043  ‘cfv 6369  (class class class)co 7202   ↑_m cmap 8497  ℝcr 10711  1c1 10713   + caddc 10715   < clt 10850   ≤ cle 10851  ℕcn 11813  2c2 11868  3c3 11869  4c4 11870  5c5 11871  6c6 11872  8c8 11874  9c9 11875  ℤcz 12159  ℤ_≥cuz 12421  ...cfz 13078  ..^cfzo 13221  Σcsu 15232  ℙcprime 16209   Even ceven 44703   Odd codd 44704   GoldbachEven cgbe 44824   GoldbachOddW cgbow 44825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-inf2 9245  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789  ax-pre-sup 10790

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-1o 8191  df-2o 8192  df-er 8380  df-map 8499  df-en 8616  df-dom 8617  df-sdom 8618  df-fin 8619  df-sup 9047  df-inf 9048  df-oi 9115  df-card 9538  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-div 11473  df-nn 11814  df-2 11876  df-3 11877  df-4 11878  df-5 11879  df-6 11880  df-7 11881  df-8 11882  df-9 11883  df-n0 12074  df-z 12160  df-dec 12277  df-uz 12422  df-rp 12570  df-fz 13079  df-fzo 13222  df-seq 13558  df-exp 13619  df-hash 13880  df-cj 14645  df-re 14646  df-im 14647  df-sqrt 14781  df-abs 14782  df-clim 15032  df-sum 15233  df-dvds 15797  df-prm 16210  df-even 44705  df-odd 44706  df-gbe 44827  df-gbow 44828

This theorem is referenced by: (None)