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Theorem trireciplem 11300
Description: Lemma for trirecip 11301. 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 9384 . . . 4 ℕ = (ℤ‘1)
2 1zzd 9104 . . . 4 (⊤ → 1 ∈ ℤ)
3 1cnd 7805 . . . . . 6 (⊤ → 1 ∈ ℂ)
4 divcnv 11297 . . . . . 6 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
53, 4syl 14 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6 nnex 8749 . . . . . . . 8 ℕ ∈ V
76mptex 5653 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
87a1i 9 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
96mptex 5653 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
109a1i 9 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11 peano2nn 8755 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1211adantl 275 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
1312nnrecred 8790 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
14 oveq2 5789 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
15 eqid 2140 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
1614, 15fvmptg 5504 . . . . . . . 8 (((𝑘 + 1) ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
1712, 13, 16syl2anc 409 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18 simpr 109 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19 oveq1 5788 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
2019oveq2d 5797 . . . . . . . . 9 (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
21 eqid 2140 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
2220, 21fvmptg 5504 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2318, 13, 22syl2anc 409 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2417, 23eqtr4d 2176 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
251, 2, 2, 8, 10, 24climshft2 11106 . . . . 5 (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
265, 25mpbird 166 . . . 4 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
27 seqex 10250 . . . . 5 seq1( + , 𝐹) ∈ V
2827a1i 9 . . . 4 (⊤ → seq1( + , 𝐹) ∈ V)
2913recnd 7817 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
3023, 29eqeltrd 2217 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
3123oveq2d 5797 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32 elfznn 9864 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
3332adantl 275 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
3433nncnd 8757 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35 peano2cn 7920 . . . . . . . . . 10 (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
3634, 35syl 14 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37 peano2nn 8755 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
3833, 37syl 14 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
3933, 38nnmulcld 8792 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
4039nncnd 8757 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
4139nnap0d 8789 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) # 0)
4236, 34, 40, 41divsubdirapd 8613 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43 ax-1cn 7736 . . . . . . . . . 10 1 ∈ ℂ
44 pncan2 7992 . . . . . . . . . 10 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
4534, 43, 44sylancl 410 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
4645oveq1d 5796 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
4736mulid1d 7806 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
4836, 34mulcomd 7810 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
4947, 48oveq12d 5799 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50 1cnd 7805 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
5133nnap0d 8789 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 # 0)
5238nnap0d 8789 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) # 0)
5350, 34, 36, 51, 52divcanap5d 8600 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
5449, 53eqtr3d 2175 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
5534mulid1d 7806 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
5655oveq1d 5796 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
5750, 36, 34, 52, 51divcanap5d 8600 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5856, 57eqtr3d 2175 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5954, 58oveq12d 5799 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6042, 46, 593eqtr3d 2181 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6160sumeq2dv 11168 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62 oveq2 5789 . . . . . . 7 (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63 oveq2 5789 . . . . . . 7 (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64 oveq2 5789 . . . . . . . 8 (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65 1div1e1 8487 . . . . . . . 8 (1 / 1) = 1
6664, 65eqtrdi 2189 . . . . . . 7 (𝑛 = 1 → (1 / 𝑛) = 1)
67 nnz 9096 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
6867adantl 275 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
6912, 1eleqtrdi 2233 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
70 elfznn 9864 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
7170adantl 275 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
7271nnrecred 8790 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
7372recnd 7817 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
7462, 63, 66, 14, 68, 69, 73telfsum 11268 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
7561, 74eqtrd 2173 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76 elnnuz 9385 . . . . . . . . 9 (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ‘1))
7776biimpri 132 . . . . . . . 8 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ ℕ)
7877adantl 275 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℕ)
79 eluzelz 9358 . . . . . . . . . . 11 (𝑗 ∈ (ℤ‘1) → 𝑗 ∈ ℤ)
8079adantl 275 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℤ)
8180zcnd 9197 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 ∈ ℂ)
8281, 35syl 14 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) ∈ ℂ)
8381, 82mulcld 7809 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
8478nnap0d 8789 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → 𝑗 # 0)
8578, 37syl 14 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) ∈ ℕ)
8685nnap0d 8789 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 + 1) # 0)
8781, 82, 84, 86mulap0d 8442 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝑗 · (𝑗 + 1)) # 0)
8883, 87recclapd 8564 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
89 id 19 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 = 𝑗)
90 oveq1 5788 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
9189, 90oveq12d 5799 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
9291oveq2d 5797 . . . . . . . 8 (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
93 trireciplem.1 . . . . . . . 8 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
9492, 93fvmptg 5504 . . . . . . 7 ((𝑗 ∈ ℕ ∧ (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
9578, 88, 94syl2anc 409 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ‘1)) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
9618, 1eleqtrdi 2233 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
9795, 96, 88fsum3ser 11197 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
9831, 75, 973eqtr2rd 2180 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
991, 2, 26, 3, 28, 30, 98climsubc2 11133 . . 3 (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
10099mptru 1341 . 2 seq1( + , 𝐹) ⇝ (1 − 0)
101 1m0e1 8856 . 2 (1 − 0) = 1
102100, 101breqtri 3960 1 seq1( + , 𝐹) ⇝ 1
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wtru 1333  wcel 1481  Vcvv 2689   class class class wbr 3936  cmpt 3996  cfv 5130  (class class class)co 5781  cc 7641  cr 7642  0cc0 7643  1c1 7644   + caddc 7646   · cmul 7648  cmin 7956   / cdiv 8455  cn 8743  cz 9077  cuz 9349  ...cfz 9820  seqcseq 10248  cli 11078  Σcsu 11153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-frec 6295  df-1o 6320  df-oadd 6324  df-er 6436  df-en 6642  df-dom 6643  df-fin 6644  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fz 9821  df-fzo 9950  df-seqfrec 10249  df-exp 10323  df-ihash 10553  df-shft 10618  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-clim 11079  df-sumdc 11154
This theorem is referenced by:  trirecip  11301
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