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Theorem bgoldbnnsum3prm 46070
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 12542 . . . . . . 7 2 ∈ ℤ
2 9nn 12258 . . . . . . . 8 9 ∈ ℕ
32nnzi 12534 . . . . . . 7 9 ∈ ℤ
4 2re 12234 . . . . . . . 8 2 ∈ ℝ
5 9re 12259 . . . . . . . 8 9 ∈ ℝ
6 2lt9 12365 . . . . . . . 8 2 < 9
74, 5, 6ltleii 11285 . . . . . . 7 2 ≤ 9
8 eluz2 12776 . . . . . . 7 (9 ∈ (ℤ‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9))
91, 3, 7, 8mpbir3an 1342 . . . . . 6 9 ∈ (ℤ‘2)
10 fzouzsplit 13614 . . . . . . 7 (9 ∈ (ℤ‘2) → (ℤ‘2) = ((2..^9) ∪ (ℤ‘9)))
1110eleq2d 2824 . . . . . 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 4113 . . . . 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 13582 . . . . . . . 8 (𝑛 ∈ (2..^9) ↔ (𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9))
16 simp1 1137 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ∈ (ℤ‘2))
17 df-9 12230 . . . . . . . . . . . 12 9 = (8 + 1)
1817breq2i 5118 . . . . . . . . . . 11 (𝑛 < 9 ↔ 𝑛 < (8 + 1))
19 eluz2nn 12816 . . . . . . . . . . . . . . 15 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℕ)
20 8nn 12255 . . . . . . . . . . . . . . 15 8 ∈ ℕ
2119, 20jctir 522 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘2) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
2221adantr 482 . . . . . . . . . . . . 13 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ) → (𝑛 ∈ ℕ ∧ 8 ∈ ℕ))
23 nnleltp1 12565 . . . . . . . . . . . . 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 248 . . . . . . . . . . 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 1118 . . . . . . . . 9 ((𝑛 ∈ (ℤ‘2) ∧ 9 ∈ ℤ ∧ 𝑛 < 9) → 𝑛 ≤ 8)
2816, 27jca 513 . . . . . . . 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 nnsum3primesle9 46060 . . . . . . 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 5114 . . . . . . . . . . 11 (𝑚 = 𝑛 → (4 < 𝑚 ↔ 4 < 𝑛))
34 eleq1w 2821 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑚 ∈ GoldbachEven ↔ 𝑛 ∈ GoldbachEven ))
3533, 34imbi12d 345 . . . . . . . . . 10 (𝑚 = 𝑛 → ((4 < 𝑚𝑚 ∈ GoldbachEven ) ↔ (4 < 𝑛𝑛 ∈ GoldbachEven )))
3635rspcv 3580 . . . . . . . . 9 (𝑛 ∈ Even → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → (4 < 𝑛𝑛 ∈ GoldbachEven )))
37 4re 12244 . . . . . . . . . . . . . . 15 4 ∈ ℝ
3837a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 4 ∈ ℝ)
395a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ∈ ℝ)
40 eluzelre 12781 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℝ)
4138, 39, 403jca 1129 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ‘9) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
4241adantl 483 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 ∈ ℝ ∧ 9 ∈ ℝ ∧ 𝑛 ∈ ℝ))
43 eluzle 12783 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ‘9) → 9 ≤ 𝑛)
4443adantl 483 . . . . . . . . . . . . 13 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → 9 ≤ 𝑛)
45 4lt9 12363 . . . . . . . . . . . . 13 4 < 9
4644, 45jctil 521 . . . . . . . . . . . 12 ((𝑛 ∈ Even ∧ 𝑛 ∈ (ℤ‘9)) → (4 < 9 ∧ 9 ≤ 𝑛))
47 ltletr 11254 . . . . . . . . . . . 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 414 . . . . . . . . 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 409 . . . . . . 7 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Even ) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → 𝑛 ∈ GoldbachEven ))
54 nnsum3primesgbe 46058 . . . . . . 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 12239 . . . . . . . . . 10 3 ∈ ℕ
5756a1i 11 . . . . . . . . 9 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ∈ ℕ)
58 oveq2 7370 . . . . . . . . . . . 12 (𝑑 = 3 → (1...𝑑) = (1...3))
5958oveq2d 7378 . . . . . . . . . . 11 (𝑑 = 3 → (ℙ ↑m (1...𝑑)) = (ℙ ↑m (1...3)))
60 breq1 5113 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑑 ≤ 3 ↔ 3 ≤ 3))
6158sumeq1d 15593 . . . . . . . . . . . . 13 (𝑑 = 3 → Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) = Σ𝑘 ∈ (1...3)(𝑓𝑘))
6261eqeq2d 2748 . . . . . . . . . . . 12 (𝑑 = 3 → (𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘) ↔ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
6360, 62anbi12d 632 . . . . . . . . . . 11 (𝑑 = 3 → ((𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)) ↔ (3 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))))
6459, 63rexeqbidv 3323 . . . . . . . . . 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 483 . . . . . . . . 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 12240 . . . . . . . . . . . 12 3 ∈ ℝ
6766leidi 11696 . . . . . . . . . . 11 3 ≤ 3
6867a1i 11 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → 3 ≤ 3)
69 6nn 12249 . . . . . . . . . . . . . 14 6 ∈ ℕ
7069nnzi 12534 . . . . . . . . . . . . 13 6 ∈ ℤ
71 6re 12250 . . . . . . . . . . . . . 14 6 ∈ ℝ
72 6lt9 12361 . . . . . . . . . . . . . 14 6 < 9
7371, 5, 72ltleii 11285 . . . . . . . . . . . . 13 6 ≤ 9
74 eluzuzle 12779 . . . . . . . . . . . . 13 ((6 ∈ ℤ ∧ 6 ≤ 9) → (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6)))
7570, 73, 74mp2an 691 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ (ℤ‘6))
7675anim1i 616 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) → (𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ))
77 nnsum4primesodd 46062 . . . . . . . . . . 11 (∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW ) → ((𝑛 ∈ (ℤ‘6) ∧ 𝑛 ∈ Odd ) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘)))
7876, 77mpan9 508 . . . . . . . . . 10 (((𝑛 ∈ (ℤ‘9) ∧ 𝑛 ∈ Odd ) ∧ ∀𝑜 ∈ Odd (5 < 𝑜𝑜 ∈ GoldbachOddW )) → ∃𝑓 ∈ (ℙ ↑m (1...3))𝑛 = Σ𝑘 ∈ (1...3)(𝑓𝑘))
79 r19.42v 3188 . . . . . . . . . 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 584 . . . . . . . . 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 3586 . . . . . . . 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 415 . . . . . . 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 46043 . . . . . . 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 12780 . . . . . . 7 (𝑛 ∈ (ℤ‘9) → 𝑛 ∈ ℤ)
86 zeoALTV 45936 . . . . . . 7 (𝑛 ∈ ℤ → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8785, 86syl 17 . . . . . 6 (𝑛 ∈ (ℤ‘9) → (𝑛 ∈ Even ∨ 𝑛 ∈ Odd ))
8855, 84, 87mpjaodan 958 . . . . 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 856 . . . 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 216 . . 3 (𝑛 ∈ (ℤ‘2) → (∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘))))
9190impcom 409 . 2 ((∀𝑚 ∈ Even (4 < 𝑚𝑚 ∈ GoldbachEven ) ∧ 𝑛 ∈ (ℤ‘2)) → ∃𝑑 ∈ ℕ ∃𝑓 ∈ (ℙ ↑m (1...𝑑))(𝑑 ≤ 3 ∧ 𝑛 = Σ𝑘 ∈ (1...𝑑)(𝑓𝑘)))
9291ralrimiva 3144 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wrex 3074  cun 3913   class class class wbr 5110  cfv 6501  (class class class)co 7362  m cmap 8772  cr 11057  1c1 11059   + caddc 11061   < clt 11196  cle 11197  cn 12160  2c2 12215  3c3 12216  4c4 12217  5c5 12218  6c6 12219  8c8 12221  9c9 12222  cz 12506  cuz 12770  ...cfz 13431  ..^cfzo 13574  Σcsu 15577  cprime 16554   Even ceven 45890   Odd codd 45891   GoldbachEven cgbe 46011   GoldbachOddW cgbow 46012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-inf 9386  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-z 12507  df-dec 12626  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-dvds 16144  df-prm 16555  df-even 45892  df-odd 45893  df-gbe 46014  df-gbow 46015
This theorem is referenced by: (None)
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