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Theorem bccolsum 31360
Description: A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)
Assertion
Ref Expression
bccolsum ((𝑁 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))
Distinct variable groups:   𝑘,𝑁   𝐶,𝑘

Proof of Theorem bccolsum
Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6618 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
21sumeq1d 14372 . . . . 5 (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3 oveq1 6617 . . . . . . 7 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4 0p1e1 11083 . . . . . . 7 (0 + 1) = 1
53, 4syl6eq 2671 . . . . . 6 (𝑚 = 0 → (𝑚 + 1) = 1)
65oveq1d 6625 . . . . 5 (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
72, 6eqeq12d 2636 . . . 4 (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
87imbi2d 330 . . 3 (𝑚 = 0 → ((𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))))
9 oveq2 6618 . . . . . 6 (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
109sumeq1d 14372 . . . . 5 (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11 oveq1 6617 . . . . . 6 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
1211oveq1d 6625 . . . . 5 (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
1310, 12eqeq12d 2636 . . . 4 (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
1413imbi2d 330 . . 3 (𝑚 = 𝑛 → ((𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)))))
15 oveq2 6618 . . . . . 6 (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
1615sumeq1d 14372 . . . . 5 (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17 oveq1 6617 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
1817oveq1d 6625 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
1916, 18eqeq12d 2636 . . . 4 (𝑚 = (𝑛 + 1) → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1))))
2019imbi2d 330 . . 3 (𝑚 = (𝑛 + 1) → ((𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
21 oveq2 6618 . . . . . 6 (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
2221sumeq1d 14372 . . . . 5 (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23 oveq1 6617 . . . . . 6 (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
2423oveq1d 6625 . . . . 5 (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
2522, 24eqeq12d 2636 . . . 4 (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
2625imbi2d 330 . . 3 (𝑚 = 𝑁 → ((𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1))) ↔ (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))))
27 0z 11339 . . . . 5 0 ∈ ℤ
28 0nn0 11258 . . . . . . 7 0 ∈ ℕ0
29 nn0z 11351 . . . . . . 7 (𝐶 ∈ ℕ0𝐶 ∈ ℤ)
30 bccl 13056 . . . . . . 7 ((0 ∈ ℕ0𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ0)
3128, 29, 30sylancr 694 . . . . . 6 (𝐶 ∈ ℕ0 → (0C𝐶) ∈ ℕ0)
3231nn0cnd 11304 . . . . 5 (𝐶 ∈ ℕ0 → (0C𝐶) ∈ ℂ)
33 oveq1 6617 . . . . . 6 (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
3433fsum1 14413 . . . . 5 ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
3527, 32, 34sylancr 694 . . . 4 (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36 elnn0 11245 . . . . 5 (𝐶 ∈ ℕ0 ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37 1red 10006 . . . . . . . . . . 11 (𝐶 ∈ ℕ → 1 ∈ ℝ)
38 nnrp 11793 . . . . . . . . . . 11 (𝐶 ∈ ℕ → 𝐶 ∈ ℝ+)
3937, 38ltaddrp2d 11857 . . . . . . . . . 10 (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40 peano2nn 10983 . . . . . . . . . . . 12 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
4140nnred 10986 . . . . . . . . . . 11 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
4237, 41ltnled 10135 . . . . . . . . . 10 (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
4339, 42mpbid 222 . . . . . . . . 9 (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44 elfzle2 12294 . . . . . . . . 9 ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
4543, 44nsyl 135 . . . . . . . 8 (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
4645iffalsed 4074 . . . . . . 7 (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47 1nn0 11259 . . . . . . . 8 1 ∈ ℕ0
4840nnzd 11432 . . . . . . . 8 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49 bcval 13038 . . . . . . . 8 ((1 ∈ ℕ0 ∧ (𝐶 + 1) ∈ ℤ) → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
5047, 48, 49sylancr 694 . . . . . . 7 (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51 bc0k 13045 . . . . . . 7 (𝐶 ∈ ℕ → (0C𝐶) = 0)
5246, 50, 513eqtr4rd 2666 . . . . . 6 (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53 bcnn 13046 . . . . . . . . 9 (0 ∈ ℕ0 → (0C0) = 1)
5428, 53ax-mp 5 . . . . . . . 8 (0C0) = 1
55 bcnn 13046 . . . . . . . . 9 (1 ∈ ℕ0 → (1C1) = 1)
5647, 55ax-mp 5 . . . . . . . 8 (1C1) = 1
5754, 56eqtr4i 2646 . . . . . . 7 (0C0) = (1C1)
58 oveq2 6618 . . . . . . 7 (𝐶 = 0 → (0C𝐶) = (0C0))
59 oveq1 6617 . . . . . . . . 9 (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
6059, 4syl6eq 2671 . . . . . . . 8 (𝐶 = 0 → (𝐶 + 1) = 1)
6160oveq2d 6626 . . . . . . 7 (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
6257, 58, 613eqtr4a 2681 . . . . . 6 (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
6352, 62jaoi 394 . . . . 5 ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
6436, 63sylbi 207 . . . 4 (𝐶 ∈ ℕ0 → (0C𝐶) = (1C(𝐶 + 1)))
6535, 64eqtrd 2655 . . 3 (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66 elnn0uz 11676 . . . . . . . . . 10 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
6766biimpi 206 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
6867adantr 481 . . . . . . . 8 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
69 elfznn0 12381 . . . . . . . . . . 11 (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ0)
7069adantl 482 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ0)
71 simplr 791 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ0)
7271nn0zd 11431 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73 bccl 13056 . . . . . . . . . 10 ((𝑘 ∈ ℕ0𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ0)
7470, 72, 73syl2anc 692 . . . . . . . . 9 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ0)
7574nn0cnd 11304 . . . . . . . 8 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76 oveq1 6617 . . . . . . . 8 (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
7768, 75, 76fsump1 14422 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
7877adantr 481 . . . . . 6 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
79 id 22 . . . . . . 7 𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)) → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)))
80 nn0cn 11253 . . . . . . . . . . 11 (𝐶 ∈ ℕ0𝐶 ∈ ℂ)
8180adantl 482 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
82 1cnd 10007 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 1 ∈ ℂ)
8381, 82pncand 10344 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 + 1) − 1) = 𝐶)
8483oveq2d 6626 . . . . . . . 8 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
8584eqcomd 2627 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
8679, 85oveqan12rd 6630 . . . . . 6 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)) = (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))))
87 peano2nn0 11284 . . . . . . . 8 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
88 peano2nn0 11284 . . . . . . . . 9 (𝐶 ∈ ℕ0 → (𝐶 + 1) ∈ ℕ0)
8988nn0zd 11431 . . . . . . . 8 (𝐶 ∈ ℕ0 → (𝐶 + 1) ∈ ℤ)
90 bcpasc 13055 . . . . . . . 8 (((𝑛 + 1) ∈ ℕ0 ∧ (𝐶 + 1) ∈ ℤ) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
9187, 89, 90syl2an 494 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
9291adantr 481 . . . . . 6 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
9378, 86, 923eqtrd 2659 . . . . 5 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
9493exp31 629 . . . 4 (𝑛 ∈ ℕ0 → (𝐶 ∈ ℕ0 → (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1)) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
9594a2d 29 . . 3 (𝑛 ∈ ℕ0 → ((𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))) → (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (((𝑛 + 1) + 1)C(𝐶 + 1)))))
968, 14, 20, 26, 65, 95nn0ind 11423 . 2 (𝑁 ∈ ℕ0 → (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
9796imp 445 1 ((𝑁 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1480  wcel 1987  ifcif 4063   class class class wbr 4618  cfv 5852  (class class class)co 6610  cc 9885  0cc0 9887  1c1 9888   + caddc 9890   · cmul 9892   < clt 10025  cle 10026  cmin 10217   / cdiv 10635  cn 10971  0cn0 11243  cz 11328  cuz 11638  ...cfz 12275  !cfa 13007  Ccbc 13036  Σcsu 14357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-fz 12276  df-fzo 12414  df-seq 12749  df-exp 12808  df-fac 13008  df-bc 13037  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358
This theorem is referenced by: (None)
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