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Theorem bccolsum 31853
 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 6773 . . . . . 6 (𝑚 = 0 → (0...𝑚) = (0...0))
21sumeq1d 14551 . . . . 5 (𝑚 = 0 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...0)(𝑘C𝐶))
3 oveq1 6772 . . . . . . 7 (𝑚 = 0 → (𝑚 + 1) = (0 + 1))
4 0p1e1 11245 . . . . . . 7 (0 + 1) = 1
53, 4syl6eq 2774 . . . . . 6 (𝑚 = 0 → (𝑚 + 1) = 1)
65oveq1d 6780 . . . . 5 (𝑚 = 0 → ((𝑚 + 1)C(𝐶 + 1)) = (1C(𝐶 + 1)))
72, 6eqeq12d 2739 . . . 4 (𝑚 = 0 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1))))
87imbi2d 329 . . 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 6773 . . . . . 6 (𝑚 = 𝑛 → (0...𝑚) = (0...𝑛))
109sumeq1d 14551 . . . . 5 (𝑚 = 𝑛 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶))
11 oveq1 6772 . . . . . 6 (𝑚 = 𝑛 → (𝑚 + 1) = (𝑛 + 1))
1211oveq1d 6780 . . . . 5 (𝑚 = 𝑛 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑛 + 1)C(𝐶 + 1)))
1310, 12eqeq12d 2739 . . . 4 (𝑚 = 𝑛 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) = ((𝑛 + 1)C(𝐶 + 1))))
1413imbi2d 329 . . 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 6773 . . . . . 6 (𝑚 = (𝑛 + 1) → (0...𝑚) = (0...(𝑛 + 1)))
1615sumeq1d 14551 . . . . 5 (𝑚 = (𝑛 + 1) → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶))
17 oveq1 6772 . . . . . 6 (𝑚 = (𝑛 + 1) → (𝑚 + 1) = ((𝑛 + 1) + 1))
1817oveq1d 6780 . . . . 5 (𝑚 = (𝑛 + 1) → ((𝑚 + 1)C(𝐶 + 1)) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
1916, 18eqeq12d 2739 . . . 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 329 . . 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 6773 . . . . . 6 (𝑚 = 𝑁 → (0...𝑚) = (0...𝑁))
2221sumeq1d 14551 . . . . 5 (𝑚 = 𝑁 → Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶))
23 oveq1 6772 . . . . . 6 (𝑚 = 𝑁 → (𝑚 + 1) = (𝑁 + 1))
2423oveq1d 6780 . . . . 5 (𝑚 = 𝑁 → ((𝑚 + 1)C(𝐶 + 1)) = ((𝑁 + 1)C(𝐶 + 1)))
2522, 24eqeq12d 2739 . . . 4 (𝑚 = 𝑁 → (Σ𝑘 ∈ (0...𝑚)(𝑘C𝐶) = ((𝑚 + 1)C(𝐶 + 1)) ↔ Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
2625imbi2d 329 . . 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 11501 . . . . 5 0 ∈ ℤ
28 0nn0 11420 . . . . . . 7 0 ∈ ℕ0
29 nn0z 11513 . . . . . . 7 (𝐶 ∈ ℕ0𝐶 ∈ ℤ)
30 bccl 13224 . . . . . . 7 ((0 ∈ ℕ0𝐶 ∈ ℤ) → (0C𝐶) ∈ ℕ0)
3128, 29, 30sylancr 698 . . . . . 6 (𝐶 ∈ ℕ0 → (0C𝐶) ∈ ℕ0)
3231nn0cnd 11466 . . . . 5 (𝐶 ∈ ℕ0 → (0C𝐶) ∈ ℂ)
33 oveq1 6772 . . . . . 6 (𝑘 = 0 → (𝑘C𝐶) = (0C𝐶))
3433fsum1 14596 . . . . 5 ((0 ∈ ℤ ∧ (0C𝐶) ∈ ℂ) → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
3527, 32, 34sylancr 698 . . . 4 (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (0C𝐶))
36 elnn0 11407 . . . . 5 (𝐶 ∈ ℕ0 ↔ (𝐶 ∈ ℕ ∨ 𝐶 = 0))
37 1red 10168 . . . . . . . . . . 11 (𝐶 ∈ ℕ → 1 ∈ ℝ)
38 nnrp 11956 . . . . . . . . . . 11 (𝐶 ∈ ℕ → 𝐶 ∈ ℝ+)
3937, 38ltaddrp2d 12020 . . . . . . . . . 10 (𝐶 ∈ ℕ → 1 < (𝐶 + 1))
40 peano2nn 11145 . . . . . . . . . . . 12 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℕ)
4140nnred 11148 . . . . . . . . . . 11 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℝ)
4237, 41ltnled 10297 . . . . . . . . . 10 (𝐶 ∈ ℕ → (1 < (𝐶 + 1) ↔ ¬ (𝐶 + 1) ≤ 1))
4339, 42mpbid 222 . . . . . . . . 9 (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ≤ 1)
44 elfzle2 12459 . . . . . . . . 9 ((𝐶 + 1) ∈ (0...1) → (𝐶 + 1) ≤ 1)
4543, 44nsyl 135 . . . . . . . 8 (𝐶 ∈ ℕ → ¬ (𝐶 + 1) ∈ (0...1))
4645iffalsed 4205 . . . . . . 7 (𝐶 ∈ ℕ → if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0) = 0)
47 1nn0 11421 . . . . . . . 8 1 ∈ ℕ0
4840nnzd 11594 . . . . . . . 8 (𝐶 ∈ ℕ → (𝐶 + 1) ∈ ℤ)
49 bcval 13206 . . . . . . . 8 ((1 ∈ ℕ0 ∧ (𝐶 + 1) ∈ ℤ) → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
5047, 48, 49sylancr 698 . . . . . . 7 (𝐶 ∈ ℕ → (1C(𝐶 + 1)) = if((𝐶 + 1) ∈ (0...1), ((!‘1) / ((!‘(1 − (𝐶 + 1))) · (!‘(𝐶 + 1)))), 0))
51 bc0k 13213 . . . . . . 7 (𝐶 ∈ ℕ → (0C𝐶) = 0)
5246, 50, 513eqtr4rd 2769 . . . . . 6 (𝐶 ∈ ℕ → (0C𝐶) = (1C(𝐶 + 1)))
53 bcnn 13214 . . . . . . . . 9 (0 ∈ ℕ0 → (0C0) = 1)
5428, 53ax-mp 5 . . . . . . . 8 (0C0) = 1
55 bcnn 13214 . . . . . . . . 9 (1 ∈ ℕ0 → (1C1) = 1)
5647, 55ax-mp 5 . . . . . . . 8 (1C1) = 1
5754, 56eqtr4i 2749 . . . . . . 7 (0C0) = (1C1)
58 oveq2 6773 . . . . . . 7 (𝐶 = 0 → (0C𝐶) = (0C0))
59 oveq1 6772 . . . . . . . . 9 (𝐶 = 0 → (𝐶 + 1) = (0 + 1))
6059, 4syl6eq 2774 . . . . . . . 8 (𝐶 = 0 → (𝐶 + 1) = 1)
6160oveq2d 6781 . . . . . . 7 (𝐶 = 0 → (1C(𝐶 + 1)) = (1C1))
6257, 58, 613eqtr4a 2784 . . . . . 6 (𝐶 = 0 → (0C𝐶) = (1C(𝐶 + 1)))
6352, 62jaoi 393 . . . . 5 ((𝐶 ∈ ℕ ∨ 𝐶 = 0) → (0C𝐶) = (1C(𝐶 + 1)))
6436, 63sylbi 207 . . . 4 (𝐶 ∈ ℕ0 → (0C𝐶) = (1C(𝐶 + 1)))
6535, 64eqtrd 2758 . . 3 (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...0)(𝑘C𝐶) = (1C(𝐶 + 1)))
66 elnn0uz 11839 . . . . . . . . . 10 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
6766biimpi 206 . . . . . . . . 9 (𝑛 ∈ ℕ0𝑛 ∈ (ℤ‘0))
6867adantr 472 . . . . . . . 8 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 𝑛 ∈ (ℤ‘0))
69 elfznn0 12547 . . . . . . . . . . 11 (𝑘 ∈ (0...(𝑛 + 1)) → 𝑘 ∈ ℕ0)
7069adantl 473 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝑘 ∈ ℕ0)
71 simplr 809 . . . . . . . . . . 11 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℕ0)
7271nn0zd 11593 . . . . . . . . . 10 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → 𝐶 ∈ ℤ)
73 bccl 13224 . . . . . . . . . 10 ((𝑘 ∈ ℕ0𝐶 ∈ ℤ) → (𝑘C𝐶) ∈ ℕ0)
7470, 72, 73syl2anc 696 . . . . . . . . 9 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℕ0)
7574nn0cnd 11466 . . . . . . . 8 (((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) ∧ 𝑘 ∈ (0...(𝑛 + 1))) → (𝑘C𝐶) ∈ ℂ)
76 oveq1 6772 . . . . . . . 8 (𝑘 = (𝑛 + 1) → (𝑘C𝐶) = ((𝑛 + 1)C𝐶))
7768, 75, 76fsump1 14607 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...(𝑛 + 1))(𝑘C𝐶) = (Σ𝑘 ∈ (0...𝑛)(𝑘C𝐶) + ((𝑛 + 1)C𝐶)))
7877adantr 472 . . . . . 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 11415 . . . . . . . . . . 11 (𝐶 ∈ ℕ0𝐶 ∈ ℂ)
8180adantl 473 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
82 1cnd 10169 . . . . . . . . . 10 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → 1 ∈ ℂ)
8381, 82pncand 10506 . . . . . . . . 9 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝐶 + 1) − 1) = 𝐶)
8483oveq2d 6781 . . . . . . . 8 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝑛 + 1)C((𝐶 + 1) − 1)) = ((𝑛 + 1)C𝐶))
8584eqcomd 2730 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → ((𝑛 + 1)C𝐶) = ((𝑛 + 1)C((𝐶 + 1) − 1)))
8679, 85oveqan12rd 6785 . . . . . 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 11446 . . . . . . . 8 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ0)
88 peano2nn0 11446 . . . . . . . . 9 (𝐶 ∈ ℕ0 → (𝐶 + 1) ∈ ℕ0)
8988nn0zd 11593 . . . . . . . 8 (𝐶 ∈ ℕ0 → (𝐶 + 1) ∈ ℤ)
90 bcpasc 13223 . . . . . . . 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 495 . . . . . . 7 ((𝑛 ∈ ℕ0𝐶 ∈ ℕ0) → (((𝑛 + 1)C(𝐶 + 1)) + ((𝑛 + 1)C((𝐶 + 1) − 1))) = (((𝑛 + 1) + 1)C(𝐶 + 1)))
9291adantr 472 . . . . . 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 2762 . . . . 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 631 . . . 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 11585 . 2 (𝑁 ∈ ℕ0 → (𝐶 ∈ ℕ0 → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))))
9796imp 444 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 382   ∧ wa 383   = wceq 1596   ∈ wcel 2103  ifcif 4194   class class class wbr 4760  ‘cfv 6001  (class class class)co 6765  ℂcc 10047  0cc0 10049  1c1 10050   + caddc 10052   · cmul 10054   < clt 10187   ≤ cle 10188   − cmin 10379   / cdiv 10797  ℕcn 11133  ℕ0cn0 11405  ℤcz 11490  ℤ≥cuz 11800  ...cfz 12440  !cfa 13175  Ccbc 13204  Σcsu 14536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-rp 11947  df-fz 12441  df-fzo 12581  df-seq 12917  df-exp 12976  df-fac 13176  df-bc 13205  df-hash 13233  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-clim 14339  df-sum 14537 This theorem is referenced by: (None)
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