Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bgoldbnnsum3prm Structured version   Visualization version   GIF version

Theorem bgoldbnnsum3prm 44319
 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 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
Distinct variable group:   𝑓,𝑘,𝑚,𝑑,𝑛

Proof of Theorem bgoldbnnsum3prm
Dummy variable 𝑜 is distinct from all other variables.
StepHypRef Expression
1 2z 12006 . . . . . . 7 2 ∈ ℤ
2 9nn 11727 . . . . . . . 8 9 ∈ ℕ
32nnzi 11998 . . . . . . 7 9 ∈ ℤ
4 2re 11703 . . . . . . . 8 2 ∈ ℝ
5 9re 11728 . . . . . . . 8 9 ∈ ℝ
6 2lt9 11834 . . . . . . . 8 2 < 9
74, 5, 6ltleii 10756 . . . . . . 7 2 ≤ 9
8 eluz2 12241 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1338 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 13071 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2878 . . . . . 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 4079 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 278 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 13040 . . . . . . . 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 11699 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 5041 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12276 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 11724 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 524 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 484 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 12029 . . . . . . . . . . . . 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 251 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25syl5bi 245 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1114 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 515 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 220 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum3primesle9 44309 . . . . . . 7 ((𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
3129, 30syl 17 . . . . . 6 (𝑛 ∈ (2..^9) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
3231a1d 25 . . . . 5 (𝑛 ∈ (2..^9) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
33 breq2 5037 . . . . . . . . . . 11 (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34 eleq1w 2875 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
3533, 34imbi12d 348 . . . . . . . . . 10 (𝑚 = 𝑛 → ((4 < 𝑚𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛𝑛 ∈ GoldbachEven )))
3635rspcv 3569 . . . . . . . . 9 (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → (4 < 𝑛𝑛 ∈ GoldbachEven )))
37 4re 11713 . . . . . . . . . . . . . . 15 4 ∈ ℝ
3837a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 4 ∈ ℝ)
395a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ∈ ℝ)
40 eluzelre 12246 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℝ)
4138, 39, 403jca 1125 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
4241adantl 485 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43 eluzle 12248 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ≤ 𝑛)
4443adantl 485 . . . . . . . . . . . . 13 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → 9 ≤ 𝑛)
45 4lt9 11832 . . . . . . . . . . . . 13 4 < 9
4644, 45jctil 523 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47 ltletr 10725 . . . . . . . . . . . 12 ((4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ) → ((4 < 9 ∧ 9 ≤ 𝑛) → 4 < 𝑛))
4842, 46, 47sylc 65 . . . . . . . . . . 11 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → 4 < 𝑛)
49 pm2.27 42 . . . . . . . . . . 11 (4 < 𝑛 → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
5048, 49syl 17 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
5150ex 416 . . . . . . . . 9 (𝑛 ∈ Even → (𝑛 ∈ (ℤ‘9) → ((4 < 𝑛𝑛 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven )))
5236, 51syl5d 73 . . . . . . . 8 (𝑛 ∈ Even → (𝑛 ∈ (ℤ‘9) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven )))
5352impcom 411 . . . . . . 7 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54 nnsum3primesgbe 44307 . . . . . . 7 (𝑛 ∈ GoldbachEven → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
5553, 54syl6 35 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
56 3nn 11708 . . . . . . . . . 10 3 ∈ ℕ
5756a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58 oveq2 7147 . . . . . . . . . . . 12 (𝑑 = 3 → (1...𝑑) = (1...3))
5958oveq2d 7155 . . . . . . . . . . 11 (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
60 breq1 5036 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
6158sumeq1d 15054 . . . . . . . . . . . . 13 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
6261eqeq2d 2812 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
6360, 62anbi12d 633 . . . . . . . . . . 11 (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6459, 63rexeqbidv 3358 . . . . . . . . . 10 (𝑑 = 3 → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6564adantl 485 . . . . . . . . 9 ((((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) ∧ 𝑑 = 3) → (∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
66 3re 11709 . . . . . . . . . . . 12 3 ∈ ℝ
6766leidi 11167 . . . . . . . . . . 11 3 ≤ 3
6867a1i 11 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69 6nn 11718 . . . . . . . . . . . . . 14 6 ∈ ℕ
7069nnzi 11998 . . . . . . . . . . . . 13 6 ∈ ℤ
71 6re 11719 . . . . . . . . . . . . . 14 6 ∈ ℝ
72 6lt9 11830 . . . . . . . . . . . . . 14 6 < 9
7371, 5, 72ltleii 10756 . . . . . . . . . . . . 13 6 ≤ 9
74 eluzuzle 12244 . . . . . . . . . . . . 13 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7570, 73, 74mp2an 691 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7675anim1i 617 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
77 nnsum4primesodd 44311 . . . . . . . . . . 11 (∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7876, 77mpan9 510 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
79 r19.42v 3306 . . . . . . . . . 10 (∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)) ↔ (3 ≤ 3 ∧ ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
8068, 78, 79sylanbrc 586 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))(3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
8157, 65, 80rspcedvd 3577 . . . . . . . 8 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
8281expcom 417 . . . . . . 7 (∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
83 sbgoldbwt 44292 . . . . . . 7 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ))
8482, 83syl11 33 . . . . . 6 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
85 eluzelz 12245 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
86 zeoALTV 44185 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8785, 86syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8855, 84, 87mpjaodan 956 . . . . 5 (𝑛 ∈ (ℤ‘9) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
8932, 88jaoi 854 . . . 4 ((𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
9014, 89sylbi 220 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
9190impcom 411 . 2 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
9291ralrimiva 3152 1 (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∀𝑛 ∈ (ℤ‘2)∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110   ∪ cun 3882   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  ℝcr 10529  1c1 10531   + caddc 10533   < clt 10668   ≤ cle 10669  ℕcn 11629  2c2 11684  3c3 11685  4c4 11686  5c5 11687  6c6 11688  8c8 11690  9c9 11691  ℤcz 11973  ℤ≥cuz 12235  ...cfz 12889  ..^cfzo 13032  Σcsu 15038  ℙcprime 16009   Even ceven 44139   Odd codd 44140   GoldbachEven cgbe 44260   GoldbachOddW cgbow 44261 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039  df-dvds 15604  df-prm 16010  df-even 44141  df-odd 44142  df-gbe 44263  df-gbow 44264 This theorem is referenced by: (None)
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