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Theorem bgoldbnnsum3prm 42521
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 11736 . . . . . . 7 2 ∈ ℤ
2 9nn 11454 . . . . . . . 8 9 ∈ ℕ
32nnzi 11728 . . . . . . 7 9 ∈ ℤ
4 2re 11424 . . . . . . . 8 2 ∈ ℝ
5 9re 11455 . . . . . . . 8 9 ∈ ℝ
6 2lt9 11562 . . . . . . . 8 2 < 9
74, 5, 6ltleii 10478 . . . . . . 7 2 ≤ 9
8 eluz2 11973 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1447 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 12797 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2891 . . . . . 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 3979 . . . . 5 (𝑛 ∈ ((2..^9) ∪ (ℤ‘9)) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
1412, 13bitri 267 . . . 4 (𝑛 ∈ (ℤ‘2) ↔ (𝑛 ∈ (2..^9) ∨ 𝑛 ∈ (ℤ‘9)))
15 elfzo2 12767 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1172 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 11420 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 4880 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12007 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 11450 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 518 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 474 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 11759 . . . . . . . . . . . . 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 240 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < (8 + 1) → 𝑛 ≤ 8))
2618, 25syl5bi 234 . . . . . . . . . 10 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 < 9 → 𝑛 ≤ 8))
27263impia 1151 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 509 . . . . . . . 8 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
2915, 28sylbi 209 . . . . . . 7 (𝑛 ∈ (2..^9) → (𝑛 ∈ (ℤ‘2) ∧ 𝑛 ≤ 8))
30 nnsum3primesle9 42511 . . . . . . 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 4876 . . . . . . . . . . 11 (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34 eleq1w 2888 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
3533, 34imbi12d 336 . . . . . . . . . 10 (𝑚 = 𝑛 → ((4 < 𝑚𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛𝑛 ∈ GoldbachEven )))
3635rspcv 3521 . . . . . . . . 9 (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → (4 < 𝑛𝑛 ∈ GoldbachEven )))
37 4re 11435 . . . . . . . . . . . . . . 15 4 ∈ ℝ
3837a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 4 ∈ ℝ)
395a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ∈ ℝ)
40 eluzelre 11978 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℝ)
4138, 39, 403jca 1164 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
4241adantl 475 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43 eluzle 11980 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ≤ 𝑛)
4443adantl 475 . . . . . . . . . . . . 13 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → 9 ≤ 𝑛)
45 4lt9 11560 . . . . . . . . . . . . 13 4 < 9
4644, 45jctil 517 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47 ltletr 10447 . . . . . . . . . . . 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 403 . . . . . . . . 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 398 . . . . . . 7 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54 nnsum3primesgbe 42509 . . . . . . 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 11429 . . . . . . . . . 10 3 ∈ ℕ
5756a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58 oveq2 6912 . . . . . . . . . . . 12 (𝑑 = 3 → (1...𝑑) = (1...3))
5958oveq2d 6920 . . . . . . . . . . 11 (𝑑 = 3 → (ℙ ↑𝑚 (1...𝑑)) = (ℙ ↑𝑚 (1...3)))
60 breq1 4875 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
6158sumeq1d 14807 . . . . . . . . . . . . 13 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
6261eqeq2d 2834 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
6360, 62anbi12d 626 . . . . . . . . . . 11 (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6459, 63rexeqbidv 3364 . . . . . . . . . 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 475 . . . . . . . . 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 11430 . . . . . . . . . . . 12 3 ∈ ℝ
6766leidi 10885 . . . . . . . . . . 11 3 ≤ 3
6867a1i 11 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69 6nn 11442 . . . . . . . . . . . . . 14 6 ∈ ℕ
7069nnzi 11728 . . . . . . . . . . . . 13 6 ∈ ℤ
71 6re 11443 . . . . . . . . . . . . . 14 6 ∈ ℝ
72 6lt9 11558 . . . . . . . . . . . . . 14 6 < 9
7371, 5, 72ltleii 10478 . . . . . . . . . . . . 13 6 ≤ 9
74 eluzuzle 11976 . . . . . . . . . . . . 13 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7570, 73, 74mp2an 685 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7675anim1i 610 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
77 nnsum4primesodd 42513 . . . . . . . . . . 11 (∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7876, 77mpan9 504 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑𝑚 (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
79 r19.42v 3301 . . . . . . . . . 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 580 . . . . . . . . 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 3532 . . . . . . . 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 404 . . . . . . 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 42494 . . . . . . 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 11977 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
86 zeoALTV 42410 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8785, 86syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8855, 84, 87mpjaodan 988 . . . . 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 890 . . . 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 209 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
9190impcom 398 . 2 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑𝑚 (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
9291ralrimiva 3174 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 198  wa 386  wo 880  w3a 1113   = wceq 1658  wcel 2166  wral 3116  wrex 3117  cun 3795   class class class wbr 4872  cfv 6122  (class class class)co 6904  𝑚 cmap 8121  cr 10250  1c1 10252   + caddc 10254   < clt 10390  cle 10391  cn 11349  2c2 11405  3c3 11406  4c4 11407  5c5 11408  6c6 11409  8c8 11411  9c9 11412  cz 11703  cuz 11967  ...cfz 12618  ..^cfzo 12759  Σcsu 14792  cprime 15756   Even ceven 42366   Odd codd 42367   GoldbachEven cgbe 42462   GoldbachOddW cgbow 42463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208  ax-inf2 8814  ax-cnex 10307  ax-resscn 10308  ax-1cn 10309  ax-icn 10310  ax-addcl 10311  ax-addrcl 10312  ax-mulcl 10313  ax-mulrcl 10314  ax-mulcom 10315  ax-addass 10316  ax-mulass 10317  ax-distr 10318  ax-i2m1 10319  ax-1ne0 10320  ax-1rid 10321  ax-rnegex 10322  ax-rrecex 10323  ax-cnre 10324  ax-pre-lttri 10325  ax-pre-lttrn 10326  ax-pre-ltadd 10327  ax-pre-mulgt0 10328  ax-pre-sup 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-nel 3102  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-se 5301  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-pred 5919  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-isom 6131  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-wrecs 7671  df-recs 7733  df-rdg 7771  df-1o 7825  df-2o 7826  df-oadd 7829  df-er 8008  df-map 8123  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-sup 8616  df-inf 8617  df-oi 8683  df-card 9077  df-pnf 10392  df-mnf 10393  df-xr 10394  df-ltxr 10395  df-le 10396  df-sub 10586  df-neg 10587  df-div 11009  df-nn 11350  df-2 11413  df-3 11414  df-4 11415  df-5 11416  df-6 11417  df-7 11418  df-8 11419  df-9 11420  df-n0 11618  df-z 11704  df-dec 11821  df-uz 11968  df-rp 12112  df-fz 12619  df-fzo 12760  df-seq 13095  df-exp 13154  df-hash 13410  df-cj 14215  df-re 14216  df-im 14217  df-sqrt 14351  df-abs 14352  df-clim 14595  df-sum 14793  df-dvds 15357  df-prm 15757  df-even 42368  df-odd 42369  df-gbe 42465  df-gbow 42466
This theorem is referenced by: (None)
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