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Theorem wtgoldbnnsum4prm 47065
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 12616 . . . . . . 7 2 ∈ ℤ
2 9nn 12332 . . . . . . . 8 9 ∈ ℕ
32nnzi 12608 . . . . . . 7 9 ∈ ℤ
4 2re 12308 . . . . . . . 8 2 ∈ ℝ
5 9re 12333 . . . . . . . 8 9 ∈ ℝ
6 2lt9 12439 . . . . . . . 8 2 < 9
74, 5, 6ltleii 11359 . . . . . . 7 2 ≤ 9
8 eluz2 12850 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1339 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 13691 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2814 . . . . . 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 4144 . . . . 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 13659 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1134 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 12304 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 5150 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12890 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 12329 . . . . . . . . . . . . . . 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 12639 . . . . . . . . . . . . 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 1115 . . . . . . . . 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 216 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum4primesle9 47058 . . . . . . 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 12317 . . . . . . . . 9 4 ∈ ℕ
3433a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ∈ ℕ)
35 oveq2 7422 . . . . . . . . . . 11 (𝑑 = 4 → (1...𝑑) = (1...4))
3635oveq2d 7430 . . . . . . . . . 10 (𝑑 = 4 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...4)))
37 breq1 5145 . . . . . . . . . . 11 (𝑑 = 4 → (𝑑 ≤ 4 ↔ 4 ≤ 4))
3835sumeq1d 15671 . . . . . . . . . . . 12 (𝑑 = 4 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...4)(𝑓𝑘))
3938eqeq2d 2738 . . . . . . . . . . 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 3338 . . . . . . . . 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 12318 . . . . . . . . . . 11 4 ∈ ℝ
4443leidi 11770 . . . . . . . . . 10 4 ≤ 4
4544a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 4 ≤ 4)
46 nnsum4primeseven 47063 . . . . . . . . . 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 3185 . . . . . . . . 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 582 . . . . . . . 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 3609 . . . . . . 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 12313 . . . . . . . . 9 3 ∈ ℕ
5352a1i 11 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ∈ ℕ)
54 oveq2 7422 . . . . . . . . . . 11 (𝑑 = 3 → (1...𝑑) = (1...3))
5554oveq2d 7430 . . . . . . . . . 10 (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
56 breq1 5145 . . . . . . . . . . 11 (𝑑 = 3 → (𝑑 ≤ 4 ↔ 3 ≤ 4))
5754sumeq1d 15671 . . . . . . . . . . . 12 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
5857eqeq2d 2738 . . . . . . . . . . 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 3338 . . . . . . . . 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 12314 . . . . . . . . . . 11 3 ∈ ℝ
63 3lt4 12408 . . . . . . . . . . 11 3 < 4
6462, 43, 63ltleii 11359 . . . . . . . . . 10 3 ≤ 4
6564a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑚 ∈ Odd (5 < 𝑚𝑚 ∈ GoldbachOddW )) → 3 ≤ 4)
66 6nn 12323 . . . . . . . . . . . . 13 6 ∈ ℕ
6766nnzi 12608 . . . . . . . . . . . 12 6 ∈ ℤ
68 6re 12324 . . . . . . . . . . . . 13 6 ∈ ℝ
69 6lt9 12435 . . . . . . . . . . . . 13 6 < 9
7068, 5, 69ltleii 11359 . . . . . . . . . . . 12 6 ≤ 9
71 eluzuzle 12853 . . . . . . . . . . . 12 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7267, 70, 71mp2an 691 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7372anim1i 614 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
74 nnsum4primesodd 47059 . . . . . . . . . 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 3185 . . . . . . . . 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 582 . . . . . . . 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 3609 . . . . . . 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 12854 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
81 zeoALTV 46933 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8280, 81syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8351, 79, 82mpjaodan 957 . . . . 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 856 . . . 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 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 3141 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 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wrex 3065  cun 3942   class class class wbr 5142  cfv 6542  (class class class)co 7414  m cmap 8836  1c1 11131   + caddc 11133   < clt 11270  cle 11271  cn 12234  2c2 12289  3c3 12290  4c4 12291  5c5 12292  6c6 12293  8c8 12295  9c9 12296  cz 12580  cuz 12844  ...cfz 13508  ..^cfzo 13651  Σcsu 15656  cprime 16633   Even ceven 46887   Odd codd 46888   GoldbachOddW cgbow 47009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-sum 15657  df-dvds 16223  df-prm 16634  df-even 46889  df-odd 46890  df-gbe 47011  df-gbow 47012
This theorem is referenced by:  stgoldbnnsum4prm  47066
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