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Theorem wtgoldbnnsum4prm 40016
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 ∈ GoldbachOdd ) → ∀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 11242 . . . . . . 7 2 ∈ ℤ
2 9nn 11039 . . . . . . . 8 9 ∈ ℕ
32nnzi 11234 . . . . . . 7 9 ∈ ℤ
4 2re 10937 . . . . . . . 8 2 ∈ ℝ
5 9re 10954 . . . . . . . 8 9 ∈ ℝ
6 2lt9 11075 . . . . . . . 8 2 < 9
74, 5, 6ltleii 10011 . . . . . . 7 2 ≤ 9
8 eluz2 11525 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1236 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 12327 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2672 . . . . . 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 3714 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 262 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 12297 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1053 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 10933 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 4585 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 11558 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 11038 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 558 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 479 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 11265 . . . . . . . . . . . . 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 236 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25syl5bi 230 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1252 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 552 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 205 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum4primesle9 40009 . . . . . . 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 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
33 4nn 11034 . . . . . . . . 9 4 ∈ ℕ
3433a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → 4 ∈ ℕ)
35 oveq2 6535 . . . . . . . . . . 11 (𝑑 = 4 → (1...𝑑) = (1...4))
3635oveq2d 6543 . . . . . . . . . 10 (𝑑 = 4 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...4)))
37 breq1 4580 . . . . . . . . . . 11 (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
3835sumeq1d 14225 . . . . . . . . . . . 12 (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...4)(𝑓𝑘))
3938eqeq2d 2619 . . . . . . . . . . 11 (𝑑 = 4 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4037, 39anbi12d 742 . . . . . . . . . 10 (𝑑 = 4 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))))
4136, 40rexeqbidv 3129 . . . . . . . . 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 480 . . . . . . . 8 ((((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) ∧ 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 10944 . . . . . . . . . . 11 4 ∈ ℝ
4443leidi 10411 . . . . . . . . . 10 4 ≤ 4
4544a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → 4 ≤ 4)
46 nnsum4primeseven 40014 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4746impcom 444 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))
48 r19.42v 3072 . . . . . . . . 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 694 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...4))(4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
5034, 42, 49rspcedvd 3288 . . . . . . 7 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
5150ex 448 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
52 3nn 11033 . . . . . . . . 9 3 ∈ ℕ
5352a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → 3 ∈ ℕ)
54 oveq2 6535 . . . . . . . . . . 11 (𝑑 = 3 → (1...𝑑) = (1...3))
5554oveq2d 6543 . . . . . . . . . 10 (𝑑 = 3 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...3)))
56 breq1 4580 . . . . . . . . . . 11 (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
5754sumeq1d 14225 . . . . . . . . . . . 12 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
5857eqeq2d 2619 . . . . . . . . . . 11 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
5956, 58anbi12d 742 . . . . . . . . . 10 (𝑑 = 3 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6055, 59rexeqbidv 3129 . . . . . . . . 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 480 . . . . . . . 8 ((((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) ∧ 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 10941 . . . . . . . . . . 11 3 ∈ ℝ
63 3lt4 11044 . . . . . . . . . . 11 3 < 4
6462, 43, 63ltleii 10011 . . . . . . . . . 10 3 ≤ 4
6564a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → 3 ≤ 4)
66 6nn 11036 . . . . . . . . . . . . 13 6 ∈ ℕ
6766nnzi 11234 . . . . . . . . . . . 12 6 ∈ ℤ
68 6re 10948 . . . . . . . . . . . . 13 6 ∈ ℝ
69 6lt9 11071 . . . . . . . . . . . . 13 6 < 9
7068, 5, 69ltleii 10011 . . . . . . . . . . . 12 6 ≤ 9
71 eluzuzle 11528 . . . . . . . . . . . 12 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7267, 70, 71mp2an 703 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7372anim1i 589 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
74 nnsum4primesodd 40010 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7573, 74mpan9 484 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
76 r19.42v 3072 . . . . . . . . 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 694 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))(3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7853, 61, 77rspcedvd 3288 . . . . . . 7 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
7978ex 448 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
80 eluzelz 11529 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
81 zeoALTV 39917 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8280, 81syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8351, 79, 82mpjaodan 822 . . . . 5 (𝑛 ∈ (ℤ‘9) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8432, 83jaoi 392 . . . 4 ((𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8514, 84sylbi 205 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8685impcom 444 . 2 ((∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
8786ralrimiva 2948 1 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOdd ) → ∀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 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wrex 2896  cun 3537   class class class wbr 4577  cfv 5790  (class class class)co 6527  𝑚 cmap 7721  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cn 10867  2c2 10917  3c3 10918  4c4 10919  5c5 10920  6c6 10921  8c8 10923  9c9 10924  cz 11210  cuz 11519  ...cfz 12152  ..^cfzo 12289  Σcsu 14210  cprime 15169   Even ceven 39873   Odd codd 39874   GoldbachOdd cgbo 39966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-rp 11665  df-fz 12153  df-fzo 12290  df-seq 12619  df-exp 12678  df-hash 12935  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-clim 14013  df-sum 14211  df-dvds 14768  df-prm 15170  df-even 39875  df-odd 39876  df-gbe 39968  df-gbo 39969
This theorem is referenced by:  stgoldbnnsum4prm  40017
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