Theorem bgoldbnnsum3prm 47678

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 12675	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 12391	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 12667	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 12367	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 12392	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 12498	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 11413	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 12909	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1341	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 13751	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) → (ℤ≥‘2) = ((2..^9) ∪ (ℤ≥‘9)))
11	10	eleq2d 2830	. . . . . 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 4176	. . . . 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 13719	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16		simp1 1136	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ≥‘2))
17		df-9 12363	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 5174	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12949	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 12388	. . . . . . . . . . . . . . 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 12698	. . . . . . . . . . . . 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 1117	. . . . . . . . 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		nnsum3primesle9 47668	. . . . . . 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 5170	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34		eleq1w 2827	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
35	33, 34	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
36	35	rspcv 3631	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
37		4re 12377	. . . . . . . . . . . . . . 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 12914	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℝ)
41	38, 39, 40	3jca 1128	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
42	41	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43		eluzle 12916	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 9 ≤ 𝑛)
44	43	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → 9 ≤ 𝑛)
45		4lt9 12496	. . . . . . . . . . . . 13 ⊢ 4 < 9
46	44, 45	jctil 519	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47		ltletr 11382	. . . . . . . . . . . 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 412	. . . . . . . . 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 407	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54		nnsum3primesgbe 47666	. . . . . . 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 12372	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
57	56	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
59	58	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
60		breq1 5169	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
61	58	sumeq1d 15748	. . . . . . . . . . . . 13 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
62	61	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
63	60, 62	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
64	59, 63	rexeqbidv 3355	. . . . . . . . . 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 481	. . . . . . . . 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 12373	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
67	66	leidi 11824	. . . . . . . . . . 11 ⊢ 3 ≤ 3
68	67	a1i 11	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69		6nn 12382	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℕ
70	69	nnzi 12667	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℤ
71		6re 12383	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℝ
72		6lt9 12494	. . . . . . . . . . . . . 14 ⊢ 6 < 9
73	71, 5, 72	ltleii 11413	. . . . . . . . . . . . 13 ⊢ 6 ≤ 9
74		eluzuzle 12912	. . . . . . . . . . . . 13 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
75	70, 73, 74	mp2an 691	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
76	75	anim1i 614	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
77		nnsum4primesodd 47670	. . . . . . . . . . 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 506	. . . . . . . . . 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 3197	. . . . . . . . . 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 582	. . . . . . . . 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 3637	. . . . . . . 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 413	. . . . . . 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 47651	. . . . . . 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 12913	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
86		zeoALTV 47544	. . . . . . 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 856	. . . 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 217	. . 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 407	. 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 3152	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 206   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∪ cun 3974   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  ℝcr 11183  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325  ℕcn 12293  2c2 12348  3c3 12349  4c4 12350  5c5 12351  6c6 12352  8c8 12354  9c9 12355  ℤcz 12639  ℤ_≥cuz 12903  ...cfz 13567  ..^cfzo 13711  Σcsu 15734  ℙcprime 16718   Even ceven 47498   Odd codd 47499   GoldbachEven cgbe 47619   GoldbachOddW cgbow 47620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-rp 13058  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-dvds 16303  df-prm 16719  df-even 47500  df-odd 47501  df-gbe 47622  df-gbow 47623

This theorem is referenced by: (None)