Theorem bgoldbnnsum3prm 42521

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 11736	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 11454	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 11728	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 11424	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 11455	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 11562	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 10478	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 11973	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1447	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 12797	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) → (ℤ≥‘2) = ((2..^9) ∪ (ℤ≥‘9)))
11	10	eleq2d 2891	. . . . . 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 3979	. . . . 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 12767	. . . . . . . 8 ⊢ (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16		simp1 1172	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ≥‘2))
17		df-9 11420	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 4880	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12007	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 11450	. . . . . . . . . . . . . . 15 ⊢ 8 ∈ ℕ
21	19, 20	jctir 518	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
22	21	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23		nnleltp1 11759	. . . . . . . . . . . . 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 1151	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
28	16, 27	jca 509	. . . . . . . 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		nnsum3primesle9 42511	. . . . . . 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 4876	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34		eleq1w 2888	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
35	33, 34	imbi12d 336	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((4 < 𝑚 → 𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
36	35	rspcv 3521	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
37		4re 11435	. . . . . . . . . . . . . . 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 11978	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℝ)
41	38, 39, 40	3jca 1164	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
42	41	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43		eluzle 11980	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 9 ≤ 𝑛)
44	43	adantl 475	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → 9 ≤ 𝑛)
45		4lt9 11560	. . . . . . . . . . . . 13 ⊢ 4 < 9
46	44, 45	jctil 517	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47		ltletr 10447	. . . . . . . . . . . 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 403	. . . . . . . . 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 398	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54		nnsum3primesgbe 42509	. . . . . . 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 11429	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
57	56	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58		oveq2 6912	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
59	58	oveq2d 6920	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...3)))
60		breq1 4875	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
61	58	sumeq1d 14807	. . . . . . . . . . . . 13 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
62	61	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
63	60, 62	anbi12d 626	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
64	59, 63	rexeqbidv 3364	. . . . . . . . . 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 475	. . . . . . . . 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 11430	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
67	66	leidi 10885	. . . . . . . . . . 11 ⊢ 3 ≤ 3
68	67	a1i 11	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69		6nn 11442	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℕ
70	69	nnzi 11728	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℤ
71		6re 11443	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℝ
72		6lt9 11558	. . . . . . . . . . . . . 14 ⊢ 6 < 9
73	71, 5, 72	ltleii 10478	. . . . . . . . . . . . 13 ⊢ 6 ≤ 9
74		eluzuzle 11976	. . . . . . . . . . . . 13 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
75	70, 73, 74	mp2an 685	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
76	75	anim1i 610	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
77		nnsum4primesodd 42513	. . . . . . . . . . 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 504	. . . . . . . . . 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 3301	. . . . . . . . . 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 580	. . . . . . . . 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 3532	. . . . . . . 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 404	. . . . . . 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 42494	. . . . . . 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 11977	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
86		zeoALTV 42410	. . . . . . 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 988	. . . . 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 890	. . . 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 209	. . 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 398	. 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 3174	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 198   ∧ wa 386   ∨ wo 880   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3116  ∃wrex 3117   ∪ cun 3795   class class class wbr 4872  ‘cfv 6122  (class class class)co 6904   ↑_𝑚 cmap 8121  ℝcr 10250  1c1 10252   + caddc 10254   < clt 10390   ≤ cle 10391  ℕcn 11349  2c2 11405  3c3 11406  4c4 11407  5c5 11408  6c6 11409  8c8 11411  9c9 11412  ℤcz 11703  ℤ_≥cuz 11967  ...cfz 12618  ..^cfzo 12759  Σcsu 14792  ℙcprime 15756   Even ceven 42366   Odd codd 42367   GoldbachEven cgbe 42462   GoldbachOddW cgbow 42463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-inf2 8814  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328  ax-pre-sup 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-se 5301  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-isom 6131  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-2o 7826  df-oadd 7829  df-er 8008  df-map 8123  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-sup 8616  df-inf 8617  df-oi 8683  df-card 9077  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-div 11009  df-nn 11350  df-2 11413  df-3 11414  df-4 11415  df-5 11416  df-6 11417  df-7 11418  df-8 11419  df-9 11420  df-n0 11618  df-z 11704  df-dec 11821  df-uz 11968  df-rp 12112  df-fz 12619  df-fzo 12760  df-seq 13095  df-exp 13154  df-hash 13410  df-cj 14215  df-re 14216  df-im 14217  df-sqrt 14351  df-abs 14352  df-clim 14595  df-sum 14793  df-dvds 15357  df-prm 15757  df-even 42368  df-odd 42369  df-gbe 42465  df-gbow 42466

This theorem is referenced by: (None)