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Theorem wtgoldbnnsum4prm 47284
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 12632 . . . . . . 7 2 ∈ ℤ
2 9nn 12348 . . . . . . . 8 9 ∈ ℕ
32nnzi 12624 . . . . . . 7 9 ∈ ℤ
4 2re 12324 . . . . . . . 8 2 ∈ ℝ
5 9re 12349 . . . . . . . 8 9 ∈ ℝ
6 2lt9 12455 . . . . . . . 8 2 < 9
74, 5, 6ltleii 11374 . . . . . . 7 2 ≤ 9
8 eluz2 12866 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1338 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 13707 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2811 . . . . . 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 4145 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 274 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 13675 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1133 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 12320 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 5157 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12906 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 12345 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 519 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 479 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 12655 . . . . . . . . . . . . 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 247 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25biimtrid 241 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1114 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 510 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 216 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum4primesle9 47277 . . . . . . 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 12333 . . . . . . . . 9 4 ∈ ℕ
3433a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ∈ ℕ)
35 oveq2 7427 . . . . . . . . . . 11 (𝑑 = 4 → (1...𝑑) = (1...4))
3635oveq2d 7435 . . . . . . . . . 10 (𝑑 = 4 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...4)))
37 breq1 5152 . . . . . . . . . . 11 (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
3835sumeq1d 15691 . . . . . . . . . . . 12 (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...4)(𝑓𝑘))
3938eqeq2d 2736 . . . . . . . . . . 11 (𝑑 = 4 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4037, 39anbi12d 630 . . . . . . . . . 10 (𝑑 = 4 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (4 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))))
4136, 40rexeqbidv 3330 . . . . . . . . 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 ∈ 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 12334 . . . . . . . . . . 11 4 ∈ ℝ
4443leidi 11785 . . . . . . . . . 10 4 ≤ 4
4544a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ≤ 4)
46 nnsum4primeseven 47282 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘)))
4746impcom 406 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...4))𝑛 = Σ𝑘 ∈ (1...4)(𝑓𝑘))
48 r19.42v 3180 . . . . . . . . 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 581 . . . . . . . 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 3608 . . . . . . 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 411 . . . . . 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 12329 . . . . . . . . 9 3 ∈ ℕ
5352a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ∈ ℕ)
54 oveq2 7427 . . . . . . . . . . 11 (𝑑 = 3 → (1...𝑑) = (1...3))
5554oveq2d 7435 . . . . . . . . . 10 (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
56 breq1 5152 . . . . . . . . . . 11 (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
5754sumeq1d 15691 . . . . . . . . . . . 12 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
5857eqeq2d 2736 . . . . . . . . . . 11 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
5956, 58anbi12d 630 . . . . . . . . . 10 (𝑑 = 3 → ((𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6055, 59rexeqbidv 3330 . . . . . . . . 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 ∈ 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 12330 . . . . . . . . . . 11 3 ∈ ℝ
63 3lt4 12424 . . . . . . . . . . 11 3 < 4
6462, 43, 63ltleii 11374 . . . . . . . . . 10 3 ≤ 4
6564a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ≤ 4)
66 6nn 12339 . . . . . . . . . . . . 13 6 ∈ ℕ
6766nnzi 12624 . . . . . . . . . . . 12 6 ∈ ℤ
68 6re 12340 . . . . . . . . . . . . 13 6 ∈ ℝ
69 6lt9 12451 . . . . . . . . . . . . 13 6 < 9
7068, 5, 69ltleii 11374 . . . . . . . . . . . 12 6 ≤ 9
71 eluzuzle 12869 . . . . . . . . . . . 12 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7267, 70, 71mp2an 690 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7372anim1i 613 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
74 nnsum4primesodd 47278 . . . . . . . . . 10 (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7573, 74mpan9 505 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
76 r19.42v 3180 . . . . . . . . 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 581 . . . . . . . 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 3608 . . . . . . 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 411 . . . . . 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 12870 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
81 zeoALTV 47152 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8280, 81syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8351, 79, 82mpjaodan 956 . . . . 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 855 . . . 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 216 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8685impcom 406 . 2 ((∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 4 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
8786ralrimiva 3135 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 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  cun 3942   class class class wbr 5149  cfv 6549  (class class class)co 7419  m cmap 8845  1c1 11146   + caddc 11148   < clt 11285  cle 11286  cn 12250  2c2 12305  3c3 12306  4c4 12307  5c5 12308  6c6 12309  8c8 12311  9c9 12312  cz 12596  cuz 12860  ...cfz 13524  ..^cfzo 13667  Σcsu 15676  cprime 16658   Even ceven 47106   Odd codd 47107   GoldbachOddW cgbow 47228
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9671  ax-cnex 11201  ax-resscn 11202  ax-1cn 11203  ax-icn 11204  ax-addcl 11205  ax-addrcl 11206  ax-mulcl 11207  ax-mulrcl 11208  ax-mulcom 11209  ax-addass 11210  ax-mulass 11211  ax-distr 11212  ax-i2m1 11213  ax-1ne0 11214  ax-1rid 11215  ax-rnegex 11216  ax-rrecex 11217  ax-cnre 11218  ax-pre-lttri 11219  ax-pre-lttrn 11220  ax-pre-ltadd 11221  ax-pre-mulgt0 11222  ax-pre-sup 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9472  df-inf 9473  df-oi 9540  df-card 9969  df-pnf 11287  df-mnf 11288  df-xr 11289  df-ltxr 11290  df-le 11291  df-sub 11483  df-neg 11484  df-div 11909  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-seq 14008  df-exp 14068  df-hash 14334  df-cj 15090  df-re 15091  df-im 15092  df-sqrt 15226  df-abs 15227  df-clim 15476  df-sum 15677  df-dvds 16243  df-prm 16659  df-even 47108  df-odd 47109  df-gbe 47230  df-gbow 47231
This theorem is referenced by:  stgoldbnnsum4prm  47285
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