Theorem bgoldbnnsum3prm 48117

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 12527	. . . . . . 7 ⊢ 2 ∈ ℤ
2		9nn 12247	. . . . . . . 8 ⊢ 9 ∈ ℕ
3	2	nnzi 12519	. . . . . . 7 ⊢ 9 ∈ ℤ
4		2re 12223	. . . . . . . 8 ⊢ 2 ∈ ℝ
5		9re 12248	. . . . . . . 8 ⊢ 9 ∈ ℝ
6		2lt9 12349	. . . . . . . 8 ⊢ 2 < 9
7	4, 5, 6	ltleii 11260	. . . . . . 7 ⊢ 2 ≤ 9
8		eluz2 12761	. . . . . . 7 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
9	1, 3, 7, 8	mpbir3an 1343	. . . . . 6 ⊢ 9 ∈ (ℤ≥‘2)
10		fzouzsplit 13614	. . . . . . 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 13582	. . . . . . . 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 12219	. . . . . . . . . . . 12 ⊢ 9 = (8 + 1)
18	17	breq2i 5107	. . . . . . . . . . 11 ⊢ (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19		eluz2nn 12805	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℕ)
20		8nn 12244	. . . . . . . . . . . . . . 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 12551	. . . . . . . . . . . . 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		nnsum3primesle9 48107	. . . . . . 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 5103	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34		eleq1w 2820	. . . . . . . . . . 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 3573	. . . . . . . . 9 ⊢ (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚 → 𝑚 ∈ GoldbachEven ) → (4 < 𝑛 → 𝑛 ∈ GoldbachEven )))
37		4re 12233	. . . . . . . . . . . . . . 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 12766	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℝ)
41	38, 39, 40	3jca 1129	. . . . . . . . . . . . 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 12768	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘9) → 9 ≤ 𝑛)
44	43	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → 9 ≤ 𝑛)
45		4lt9 12347	. . . . . . . . . . . . 13 ⊢ 4 < 9
46	44, 45	jctil 519	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ≥‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47		ltletr 11229	. . . . . . . . . . . 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 48105	. . . . . . 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 12228	. . . . . . . . . 10 ⊢ 3 ∈ ℕ
57	56	a1i 11	. . . . . . . . 9 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (1...𝑑) = (1...3))
59	58	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
60		breq1 5102	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
61	58	sumeq1d 15627	. . . . . . . . . . . . 13 ⊢ (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))
62	61	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘)))
63	60, 62	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓‘𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓‘𝑘))))
64	59, 63	rexeqbidv 3318	. . . . . . . . . 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 12229	. . . . . . . . . . . 12 ⊢ 3 ∈ ℝ
67	66	leidi 11675	. . . . . . . . . . 11 ⊢ 3 ≤ 3
68	67	a1i 11	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜 → 𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69		6nn 12238	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℕ
70	69	nnzi 12519	. . . . . . . . . . . . 13 ⊢ 6 ∈ ℤ
71		6re 12239	. . . . . . . . . . . . . 14 ⊢ 6 ∈ ℝ
72		6lt9 12345	. . . . . . . . . . . . . 14 ⊢ 6 < 9
73	71, 5, 72	ltleii 11260	. . . . . . . . . . . . 13 ⊢ 6 ≤ 9
74		eluzuzle 12764	. . . . . . . . . . . . 13 ⊢ ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6)))
75	70, 73, 74	mp2an 693	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ (ℤ≥‘6))
76	75	anim1i 616	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ≥‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ≥‘6) ∧ 𝑛 ∈ Odd ))
77		nnsum4primesodd 48109	. . . . . . . . . . 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 3169	. . . . . . . . . 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 584	. . . . . . . . 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 3579	. . . . . . . 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 48090	. . . . . . 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 12765	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘9) → 𝑛 ∈ ℤ)
86		zeoALTV 47983	. . . . . . 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 961	. . . . 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 858	. . . 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 3129	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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   ∪ cun 3900   class class class wbr 5099  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767  ℝcr 11029  1c1 11031   + caddc 11033   < clt 11170   ≤ cle 11171  ℕcn 12149  2c2 12204  3c3 12205  4c4 12206  5c5 12207  6c6 12208  8c8 12210  9c9 12211  ℤcz 12492  ℤ_≥cuz 12755  ...cfz 13427  ..^cfzo 13574  Σcsu 15613  ℙcprime 16602   Even ceven 47937   Odd codd 47938   GoldbachEven cgbe 48058   GoldbachOddW cgbow 48059

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 7682  ax-inf2 9554  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107  ax-pre-sup 11108

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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12406  df-z 12493  df-dec 12612  df-uz 12756  df-rp 12910  df-fz 13428  df-fzo 13575  df-seq 13929  df-exp 13989  df-hash 14258  df-cj 15026  df-re 15027  df-im 15028  df-sqrt 15162  df-abs 15163  df-clim 15415  df-sum 15614  df-dvds 16184  df-prm 16603  df-even 47939  df-odd 47940  df-gbe 48061  df-gbow 48062

This theorem is referenced by: (None)