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 42017
 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 11447 . . . . . . 7 2 ∈ ℤ
2 9nn 11230 . . . . . . . 8 9 ∈ ℕ
32nnzi 11439 . . . . . . 7 9 ∈ ℤ
4 2re 11128 . . . . . . . 8 2 ∈ ℝ
5 9re 11145 . . . . . . . 8 9 ∈ ℝ
6 2lt9 11266 . . . . . . . 8 2 < 9
74, 5, 6ltleii 10198 . . . . . . 7 2 ≤ 9
8 eluz2 11731 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1263 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 12542 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2716 . . . . . 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 3786 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 264 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 12512 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1081 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 11124 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 4693 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 11764 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 11229 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 560 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 480 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 11470 . . . . . . . . . . . . 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 238 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25syl5bi 232 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1280 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 553 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 207 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum3primesle9 42007 . . . . . . 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 4689 . . . . . . . . . . 11 (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34 eleq1 2718 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
3533, 34imbi12d 333 . . . . . . . . . 10 (𝑚 = 𝑛 → ((4 < 𝑚𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛𝑛 ∈ GoldbachEven )))
3635rspcv 3336 . . . . . . . . 9 (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → (4 < 𝑛𝑛 ∈ GoldbachEven )))
37 4re 11135 . . . . . . . . . . . . . . 15 4 ∈ ℝ
3837a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 4 ∈ ℝ)
395a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ∈ ℝ)
40 eluzelre 11736 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℝ)
4138, 39, 403jca 1261 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
4241adantl 481 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43 eluzle 11738 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ≤ 𝑛)
4443adantl 481 . . . . . . . . . . . . 13 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → 9 ≤ 𝑛)
45 4lt9 11264 . . . . . . . . . . . . 13 4 < 9
4644, 45jctil 559 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47 ltletr 10167 . . . . . . . . . . . 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 449 . . . . . . . . 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 445 . . . . . . 7 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54 nnsum3primesgbe 42005 . . . . . . 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 11224 . . . . . . . . . 10 3 ∈ ℕ
5756a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58 oveq2 6698 . . . . . . . . . . . 12 (𝑑 = 3 → (1...𝑑) = (1...3))
5958oveq2d 6706 . . . . . . . . . . 11 (𝑑 = 3 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...3)))
60 breq1 4688 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
6158sumeq1d 14475 . . . . . . . . . . . . 13 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
6261eqeq2d 2661 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
6360, 62anbi12d 747 . . . . . . . . . . 11 (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6459, 63rexeqbidv 3183 . . . . . . . . . 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 481 . . . . . . . . 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 11132 . . . . . . . . . . . 12 3 ∈ ℝ
6766leidi 10600 . . . . . . . . . . 11 3 ≤ 3
6867a1i 11 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69 6nn 11227 . . . . . . . . . . . . . 14 6 ∈ ℕ
7069nnzi 11439 . . . . . . . . . . . . 13 6 ∈ ℤ
71 6re 11139 . . . . . . . . . . . . . 14 6 ∈ ℝ
72 6lt9 11262 . . . . . . . . . . . . . 14 6 < 9
7371, 5, 72ltleii 10198 . . . . . . . . . . . . 13 6 ≤ 9
74 eluzuzle 11734 . . . . . . . . . . . . 13 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7570, 73, 74mp2an 708 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7675anim1i 591 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
77 nnsum4primesodd 42009 . . . . . . . . . . 11 (∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7876, 77mpan9 485 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
79 r19.42v 3121 . . . . . . . . . 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 699 . . . . . . . . 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 3348 . . . . . . . 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 450 . . . . . . 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 41990 . . . . . . 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 11735 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
86 zeoALTV 41906 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8785, 86syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8855, 84, 87mpjaodan 844 . . . . 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 393 . . . 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 207 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
9190impcom 445 . 2 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
9291ralrimiva 2995 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 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∪ cun 3605   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899  ℝcr 9973  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113  ℕcn 11058  2c2 11108  3c3 11109  4c4 11110  5c5 11111  6c6 11112  8c8 11114  9c9 11115  ℤcz 11415  ℤ≥cuz 11725  ...cfz 12364  ..^cfzo 12504  Σcsu 14460  ℙcprime 15432   Even ceven 41862   Odd codd 41863   GoldbachEven cgbe 41958   GoldbachOddW cgbow 41959 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-prm 15433  df-even 41864  df-odd 41865  df-gbe 41961  df-gbow 41962 This theorem is referenced by: (None)
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