Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wtgoldbnnsum4prm Structured version   Visualization version   GIF version

Theorem wtgoldbnnsum4prm 47789
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 < 𝑚𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
Distinct variable group:   𝑓,𝑘,𝑚,𝑑,𝑛

Proof of Theorem wtgoldbnnsum4prm
StepHypRef Expression
1 2z 12649 . . . . . . 7 2 ∈ ℤ
2 9nn 12364 . . . . . . . 8 9 ∈ ℕ
32nnzi 12641 . . . . . . 7 9 ∈ ℤ
4 2re 12340 . . . . . . . 8 2 ∈ ℝ
5 9re 12365 . . . . . . . 8 9 ∈ ℝ
6 2lt9 12471 . . . . . . . 8 2 < 9
74, 5, 6ltleii 11384 . . . . . . 7 2 ≤ 9
8 eluz2 12884 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1342 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 13734 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2827 . . . . . 6 (9 ∈ (ℤ‘2) → (𝑛 ∈ (ℤ‘2) ↔ 𝑛 ∈ ((2..^9) ∪ (ℤ‘9))))
129, 11ax-mp 5 . . . . 5 (𝑛 ∈ (ℤ‘2) ↔ 𝑛 ∈ ((2..^9) ∪ (ℤ‘9)))
13 elun 4153 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 275 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 13702 . . . . . . . 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 12336 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 5151 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12924 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 12361 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 520 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 480 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 12673 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ ∧ 8 ∈ ℕ) → (𝑛 ≤ 8 ↔ 𝑛 < (8 + 1)))
2422, 23syl 17 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ≤ 8 ↔ 𝑛 < (8 + 1)))
2524biimprd 248 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25biimtrid 242 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1118 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 511 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 217 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum4primesle9 47782 . . . . . . 7 ((𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
3129, 30syl 17 . . . . . 6 (𝑛 ∈ (2..^9) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
3231a1d 25 . . . . 5 (𝑛 ∈ (2..^9) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
33 4nn 12349 . . . . . . . . 9 4 ∈ ℕ
3433a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ∈ ℕ)
35 oveq2 7439 . . . . . . . . . . 11 (𝑑 = 4 → (1...𝑑) = (1...4))
3635oveq2d 7447 . . . . . . . . . 10 (𝑑 = 4 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...4)))
37 breq1 5146 . . . . . . . . . . 11 (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
3835sumeq1d 15736 . . . . . . . . . . . 12 (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...4)(𝑓𝑘))
3938eqeq2d 2748 . . . . . . . . . . 11 (𝑑 = 4 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4037, 39anbi12d 632 . . . . . . . . . 10 (𝑑 = 4 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))))
4136, 40rexeqbidv 3347 . . . . . . . . 9 (𝑑 = 4 → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))))
4241adantl 481 . . . . . . . 8 ((((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) ∧ 𝑑 = 4) → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))))
43 4re 12350 . . . . . . . . . . 11 4 ∈ ℝ
4443leidi 11797 . . . . . . . . . 10 4 ≤ 4
4544a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ≤ 4)
46 nnsum4primeseven 47787 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4746impcom 407 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))
48 r19.42v 3191 . . . . . . . . 9 (∃𝑓 ∈ (ℙ ↑m (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)) ↔ (4 ≤ 4 ∧ ∃𝑓 ∈ (ℙ ↑m (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4945, 47, 48sylanbrc 583 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
5034, 42, 49rspcedvd 3624 . . . . . . 7 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
5150ex 412 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
52 3nn 12345 . . . . . . . . 9 3 ∈ ℕ
5352a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ∈ ℕ)
54 oveq2 7439 . . . . . . . . . . 11 (𝑑 = 3 → (1...𝑑) = (1...3))
5554oveq2d 7447 . . . . . . . . . 10 (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
56 breq1 5146 . . . . . . . . . . 11 (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
5754sumeq1d 15736 . . . . . . . . . . . 12 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
5857eqeq2d 2748 . . . . . . . . . . 11 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
5956, 58anbi12d 632 . . . . . . . . . 10 (𝑑 = 3 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6055, 59rexeqbidv 3347 . . . . . . . . 9 (𝑑 = 3 → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6160adantl 481 . . . . . . . 8 ((((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) ∧ 𝑑 = 3) → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
62 3re 12346 . . . . . . . . . . 11 3 ∈ ℝ
63 3lt4 12440 . . . . . . . . . . 11 3 < 4
6462, 43, 63ltleii 11384 . . . . . . . . . 10 3 ≤ 4
6564a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ≤ 4)
66 6nn 12355 . . . . . . . . . . . . 13 6 ∈ ℕ
6766nnzi 12641 . . . . . . . . . . . 12 6 ∈ ℤ
68 6re 12356 . . . . . . . . . . . . 13 6 ∈ ℝ
69 6lt9 12467 . . . . . . . . . . . . 13 6 < 9
7068, 5, 69ltleii 11384 . . . . . . . . . . . 12 6 ≤ 9
71 eluzuzle 12887 . . . . . . . . . . . 12 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7267, 70, 71mp2an 692 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7372anim1i 615 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
74 nnsum4primesodd 47783 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7573, 74mpan9 506 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
76 r19.42v 3191 . . . . . . . . 9 (∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)) ↔ (3 ≤ 4 ∧ ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7765, 75, 76sylanbrc 583 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7853, 61, 77rspcedvd 3624 . . . . . . 7 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
7978ex 412 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
80 eluzelz 12888 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
81 zeoALTV 47657 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8280, 81syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8351, 79, 82mpjaodan 961 . . . . 5 (𝑛 ∈ (ℤ‘9) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8432, 83jaoi 858 . . . 4 ((𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8514, 84sylbi 217 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8685impcom 407 . 2 ((∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
8786ralrimiva 3146 1 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cun 3949   class class class wbr 5143  cfv 6561  (class class class)co 7431  m cmap 8866  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cn 12266  2c2 12321  3c3 12322  4c4 12323  5c5 12324  6c6 12325  8c8 12327  9c9 12328  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  Σcsu 15722  cprime 16708   Even ceven 47611   Odd codd 47612   GoldbachOddW cgbow 47733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-prm 16709  df-even 47613  df-odd 47614  df-gbe 47735  df-gbow 47736
This theorem is referenced by:  stgoldbnnsum4prm  47790
  Copyright terms: Public domain W3C validator