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Theorem trireciplem 11665
Description: Lemma for trirecip 11666. Show that the sum converges. (Contributed by Scott Fenton, 22-Apr-2014.) (Revised by Mario Carneiro, 22-May-2014.)
Hypothesis
Ref Expression
trireciplem.1 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
Assertion
Ref Expression
trireciplem seq1( + , 𝐹) ⇝ 1

Proof of Theorem trireciplem
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 9637 . . . 4 ℕ = (ℤ‘1)
2 1zzd 9353 . . . 4 (⊤ → 1 ∈ ℤ)
3 1cnd 8042 . . . . . 6 (⊤ → 1 ∈ ℂ)
4 divcnv 11662 . . . . . 6 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
53, 4syl 14 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6 nnex 8996 . . . . . . . 8 ℕ ∈ V
76mptex 5788 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
87a1i 9 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
96mptex 5788 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
109a1i 9 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11 peano2nn 9002 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1211adantl 277 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
1312nnrecred 9037 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
14 oveq2 5930 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
15 eqid 2196 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
1614, 15fvmptg 5637 . . . . . . . 8 (((𝑘 + 1) ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
1712, 13, 16syl2anc 411 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18 simpr 110 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19 oveq1 5929 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
2019oveq2d 5938 . . . . . . . . 9 (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
21 eqid 2196 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
2220, 21fvmptg 5637 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2318, 13, 22syl2anc 411 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2417, 23eqtr4d 2232 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
251, 2, 2, 8, 10, 24climshft2 11471 . . . . 5 (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
265, 25mpbird 167 . . . 4 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
27 seqex 10541 . . . . 5 seq1( + , 𝐹) ∈ V
2827a1i 9 . . . 4 (⊤ → seq1( + , 𝐹) ∈ V)
2913recnd 8055 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
3023, 29eqeltrd 2273 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
3123oveq2d 5938 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32 elfznn 10129 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
3332adantl 277 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
3433nncnd 9004 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35 peano2cn 8161 . . . . . . . . . 10 (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
3634, 35syl 14 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37 peano2nn 9002 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
3833, 37syl 14 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
3933, 38nnmulcld 9039 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
4039nncnd 9004 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
4139nnap0d 9036 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) # 0)
4236, 34, 40, 41divsubdirapd 8857 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43 ax-1cn 7972 . . . . . . . . . 10 1 ∈ ℂ
44 pncan2 8233 . . . . . . . . . 10 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
4534, 43, 44sylancl 413 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
4645oveq1d 5937 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
4736mulridd 8043 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
4836, 34mulcomd 8048 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
4947, 48oveq12d 5940 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50 1cnd 8042 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
5133nnap0d 9036 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 # 0)
5238nnap0d 9036 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) # 0)
5350, 34, 36, 51, 52divcanap5d 8844 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
5449, 53eqtr3d 2231 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
5534mulridd 8043 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
5655oveq1d 5937 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
5750, 36, 34, 52, 51divcanap5d 8844 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5856, 57eqtr3d 2231 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5954, 58oveq12d 5940 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6042, 46, 593eqtr3d 2237 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6160sumeq2dv 11533 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62 oveq2 5930 . . . . . . 7 (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63 oveq2 5930 . . . . . . 7 (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64 oveq2 5930 . . . . . . . 8 (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65 1div1e1 8731 . . . . . . . 8 (1 / 1) = 1
6664, 65eqtrdi 2245 . . . . . . 7 (𝑛 = 1 → (1 / 𝑛) = 1)
67 nnz 9345 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
6867adantl 277 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
6912, 1eleqtrdi 2289 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
70 elfznn 10129 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
7170adantl 277 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
7271nnrecred 9037 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
7372recnd 8055 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
7462, 63, 66, 14, 68, 69, 73telfsum 11633 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
7561, 74eqtrd 2229 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76 elnnuz 9638 . . . . . . . . 9 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
7776biimpri 133 . . . . . . . 8 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ ℕ)
7877adantl 277 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℕ)
79 eluzelz 9610 . . . . . . . . . . 11 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ ℤ)
8079adantl 277 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℤ)
8180zcnd 9449 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℂ)
8281, 35syl 14 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) ∈ ℂ)
8381, 82mulcld 8047 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
8478nnap0d 9036 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 # 0)
8578, 37syl 14 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) ∈ ℕ)
8685nnap0d 9036 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) # 0)
8781, 82, 84, 86mulap0d 8685 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 · (𝑗 + 1)) # 0)
8883, 87recclapd 8808 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
89 id 19 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 = 𝑗)
90 oveq1 5929 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
9189, 90oveq12d 5940 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
9291oveq2d 5938 . . . . . . . 8 (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
93 trireciplem.1 . . . . . . . 8 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
9492, 93fvmptg 5637 . . . . . . 7 ((𝑗 ∈ ℕ ∧ (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
9578, 88, 94syl2anc 411 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
9618, 1eleqtrdi 2289 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
9795, 96, 88fsum3ser 11562 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
9831, 75, 973eqtr2rd 2236 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
991, 2, 26, 3, 28, 30, 98climsubc2 11498 . . 3 (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
10099mptru 1373 . 2 seq1( + , 𝐹) ⇝ (1 − 0)
101 1m0e1 9103 . 2 (1 − 0) = 1
102100, 101breqtri 4058 1 seq1( + , 𝐹) ⇝ 1
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wtru 1365  wcel 2167  Vcvv 2763   class class class wbr 4033  cmpt 4094  cfv 5258  (class class class)co 5922  cc 7877  cr 7878  0cc0 7879  1c1 7880   + caddc 7882   · cmul 7884  cmin 8197   / cdiv 8699  cn 8990  cz 9326  cuz 9601  ...cfz 10083  seqcseq 10539  cli 11443  Σcsu 11518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-ihash 10868  df-shft 10980  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519
This theorem is referenced by:  trirecip  11666
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