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Theorem trireciplem 14638
Description: Lemma for trirecip 14639. 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 11761 . . . 4 ℕ = (ℤ‘1)
2 1zzd 11446 . . . 4 (⊤ → 1 ∈ ℤ)
3 1cnd 10094 . . . . . 6 (⊤ → 1 ∈ ℂ)
4 divcnv 14629 . . . . . 6 (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
53, 4syl 17 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6 nnex 11064 . . . . . . . 8 ℕ ∈ V
76mptex 6527 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
87a1i 11 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
96mptex 6527 . . . . . . 7 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
109a1i 11 . . . . . 6 (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11 peano2nn 11070 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1211adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
13 oveq2 6698 . . . . . . . . 9 (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
14 eqid 2651 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
15 ovex 6718 . . . . . . . . 9 (1 / (𝑘 + 1)) ∈ V
1613, 14, 15fvmpt 6321 . . . . . . . 8 ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
1712, 16syl 17 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18 oveq1 6697 . . . . . . . . . 10 (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
1918oveq2d 6706 . . . . . . . . 9 (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
20 eqid 2651 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
2119, 20, 15fvmpt 6321 . . . . . . . 8 (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2221adantl 481 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
2317, 22eqtr4d 2688 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
241, 2, 2, 8, 10, 23climshft2 14357 . . . . 5 (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
255, 24mpbird 247 . . . 4 (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
26 seqex 12843 . . . . 5 seq1( + , 𝐹) ∈ V
2726a1i 11 . . . 4 (⊤ → seq1( + , 𝐹) ∈ V)
2812nnrecred 11104 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
2928recnd 10106 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
3022, 29eqeltrd 2730 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
3122oveq2d 6706 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32 elfznn 12408 . . . . . . . . . . . 12 (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
3332adantl 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
3433nncnd 11074 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35 peano2cn 10246 . . . . . . . . . 10 (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
3634, 35syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37 peano2nn 11070 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
3833, 37syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
3933, 38nnmulcld 11106 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
4039nncnd 11074 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
4139nnne0d 11103 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ≠ 0)
4236, 34, 40, 41divsubdird 10878 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43 ax-1cn 10032 . . . . . . . . . 10 1 ∈ ℂ
44 pncan2 10326 . . . . . . . . . 10 ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
4534, 43, 44sylancl 695 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
4645oveq1d 6705 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
4736mulid1d 10095 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
4836, 34mulcomd 10099 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
4947, 48oveq12d 6708 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50 1cnd 10094 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
5133nnne0d 11103 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ≠ 0)
5238nnne0d 11103 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ≠ 0)
5350, 34, 36, 51, 52divcan5d 10865 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
5449, 53eqtr3d 2687 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
5534mulid1d 10095 . . . . . . . . . . 11 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
5655oveq1d 6705 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
5750, 36, 34, 52, 51divcan5d 10865 . . . . . . . . . 10 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5856, 57eqtr3d 2687 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
5954, 58oveq12d 6708 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6042, 46, 593eqtr3d 2693 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
6160sumeq2dv 14477 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62 oveq2 6698 . . . . . . 7 (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63 oveq2 6698 . . . . . . 7 (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64 oveq2 6698 . . . . . . . 8 (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65 1div1e1 10755 . . . . . . . 8 (1 / 1) = 1
6664, 65syl6eq 2701 . . . . . . 7 (𝑛 = 1 → (1 / 𝑛) = 1)
67 nnz 11437 . . . . . . . 8 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
6867adantl 481 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
6912, 1syl6eleq 2740 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
70 elfznn 12408 . . . . . . . . . 10 (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
7170adantl 481 . . . . . . . . 9 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
7271nnrecred 11104 . . . . . . . 8 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
7372recnd 10106 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
7462, 63, 66, 13, 68, 69, 73telfsum 14580 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
7561, 74eqtrd 2685 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76 id 22 . . . . . . . . . 10 (𝑛 = 𝑗𝑛 = 𝑗)
77 oveq1 6697 . . . . . . . . . 10 (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
7876, 77oveq12d 6708 . . . . . . . . 9 (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
7978oveq2d 6706 . . . . . . . 8 (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
80 trireciplem.1 . . . . . . . 8 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
81 ovex 6718 . . . . . . . 8 (1 / (𝑗 · (𝑗 + 1))) ∈ V
8279, 80, 81fvmpt 6321 . . . . . . 7 (𝑗 ∈ ℕ → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
8333, 82syl 17 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝐹𝑗) = (1 / (𝑗 · (𝑗 + 1))))
84 simpr 476 . . . . . . 7 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
8584, 1syl6eleq 2740 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
8639nnrecred 11104 . . . . . . 7 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
8786recnd 10106 . . . . . 6 (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
8883, 85, 87fsumser 14505 . . . . 5 ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
8931, 75, 883eqtr2rd 2692 . . . 4 ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
901, 2, 25, 3, 27, 30, 89climsubc2 14413 . . 3 (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
9190trud 1533 . 2 seq1( + , 𝐹) ⇝ (1 − 0)
92 1m0e1 11169 . 2 (1 − 0) = 1
9391, 92breqtri 4710 1 seq1( + , 𝐹) ⇝ 1
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wtru 1524  wcel 2030  Vcvv 3231   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  cmin 10304   / cdiv 10722  cn 11058  cz 11415  cuz 11725  ...cfz 12364  seqcseq 12841  cli 14259  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461
This theorem is referenced by:  trirecip  14639  stirlinglem12  40620
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