Theorem trireciplem 15912

Description: Lemma for trirecip 15913. 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.

Step	Hyp	Ref	Expression
1		nnuz 12896	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 12620	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		1cnd 11197	. . . . 5 ⊢ (⊤ → 1 ∈ ℂ)
4		nnex 12234	. . . . . . 7 ⊢ ℕ ∈ V
5	4	mptex 7221	. . . . . 6 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
6	5	a1i 11	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
7		oveq1 7417	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
8	7	oveq2d 7426	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
9		eqid 2763	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
10		ovex 7443	. . . . . . 7 ⊢ (1 / (𝑘 + 1)) ∈ V
11	8, 9, 10	fvmpt 6989	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
12	11	adantl 486	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
13	1, 2, 3, 2, 6, 12	divcnvshft 15905	. . . 4 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
14		seqex 14035	. . . . 5 ⊢ seq1( + , 𝐹) ∈ V
15	14	a1i 11	. . . 4 ⊢ (⊤ → seq1( + , 𝐹) ∈ V)
16		peano2nn 12240	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
17	16	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
18	17	nnrecred 12282	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
19	18	recnd 11232	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
20	12, 19	eqeltrd 2863	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
21	12	oveq2d 7426	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
22		elfznn 13577	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
23	22	adantl 486	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
24	23	nncnd 12244	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
25		peano2cn 11377	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
26	24, 25	syl 18	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
27		peano2nn 12240	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
28	23, 27	syl 18	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
29	23, 28	nnmulcld 12284	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
30	29	nncnd 12244	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
31	29	nnne0d 12281	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ≠ 0)
32	26, 24, 30, 31	divsubdird 12025	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
33		ax-1cn 11153	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
34		pncan2 11459	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
35	24, 33, 34	sylancl 597	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
36	35	oveq1d 7425	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
37	26	mulridd 11221	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
38	26, 24	mulcomd 11225	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
39	37, 38	oveq12d 7428	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
40		1cnd 11197	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
41	23	nnne0d 12281	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ≠ 0)
42	28	nnne0d 12281	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ≠ 0)
43	40, 24, 26, 41, 42	divcan5d 12012	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
44	39, 43	eqtr3d 2800	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
45	24	mulridd 11221	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
46	45	oveq1d 7425	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
47	40, 26, 24, 42, 41	divcan5d 12012	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
48	46, 47	eqtr3d 2800	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
49	44, 48	oveq12d 7428	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
50	32, 36, 49	3eqtr3d 2806	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
51	50	sumeq2dv 15749	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
52		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
53		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
54		oveq2 7418	. . . . . . . 8 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
55		1div1e1 11900	. . . . . . . 8 ⊢ (1 / 1) = 1
56	54, 55	eqtrdi 2814	. . . . . . 7 ⊢ (𝑛 = 1 → (1 / 𝑛) = 1)
57		oveq2 7418	. . . . . . 7 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
58		nnz 12607	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
59	58	adantl 486	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
60	17, 1	eleqtrdi 2873	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ≥‘1))
61		elfznn 13577	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
62	61	adantl 486	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
63	62	nnrecred 12282	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
64	63	recnd 11232	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
65	52, 53, 56, 57, 59, 60, 64	telfsum 15852	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
66	51, 65	eqtrd 2798	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
67		id 23	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → 𝑛 = 𝑗)
68		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
69	67, 68	oveq12d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
70	69	oveq2d 7426	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
71		trireciplem.1	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
72		ovex 7443	. . . . . . . 8 ⊢ (1 / (𝑗 · (𝑗 + 1))) ∈ V
73	70, 71, 72	fvmpt 6989	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
74	23, 73	syl 18	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
75		simpr 489	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
76	75, 1	eleqtrdi 2873	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
77	29	nnrecred 12282	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
78	77	recnd 11232	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
79	74, 76, 78	fsumser 15777	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
80	21, 66, 79	3eqtr2rd 2805	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
81	1, 2, 13, 3, 15, 20, 80	climsubc2 15686	. . 3 ⊢ (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
82	81	mptru 1577	. 2 ⊢ seq1( + , 𝐹) ⇝ (1 − 0)
83		1m0e1 12355	. 2 ⊢ (1 − 0) = 1
84	82, 83	breqtri 5136	1 ⊢ seq1( + , 𝐹) ⇝ 1

Syntax hints:   ∧ wa 400   = wceq 1570  ⊤wtru 1571   ∈ wcel 2143  Vcvv 3455   class class class wbr 5109   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  ℂcc 11093  0cc0 11095  1c1 11096   + caddc 11098   · cmul 11100   − cmin 11436   / cdiv 11866  ℕcn 12228  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  seqcseq 14033   ⇝ cli 15531  Σcsu 15733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-oi 9468  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-shft 15100  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734

This theorem is referenced by:  trirecip  15913  stirlinglem12  46819