Theorem trireciplem 14594

Description: Lemma for trirecip 14595. 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 11723	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 11408	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		1cnd 10056	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
4		divcnv 14585	. . . . . 6 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
5	3, 4	syl 17	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6		nnex 11026	. . . . . . . 8 ⊢ ℕ ∈ V
7	6	mptex 6486	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
8	7	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
9	6	mptex 6486	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
10	9	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11		peano2nn 11032	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
12	11	adantl 482	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
13		oveq2 6658	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
14		eqid 2622	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
15		ovex 6678	. . . . . . . . 9 ⊢ (1 / (𝑘 + 1)) ∈ V
16	13, 14, 15	fvmpt 6282	. . . . . . . 8 ⊢ ((𝑘 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
17	12, 16	syl 17	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
19	18	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
20		eqid 2622	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
21	19, 20, 15	fvmpt 6282	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
22	21	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
23	17, 22	eqtr4d 2659	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
24	1, 2, 2, 8, 10, 23	climshft2 14313	. . . . 5 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
25	5, 24	mpbird 247	. . . 4 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
26		seqex 12803	. . . . 5 ⊢ seq1( + , 𝐹) ∈ V
27	26	a1i 11	. . . 4 ⊢ (⊤ → seq1( + , 𝐹) ∈ V)
28	12	nnrecred 11066	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
29	28	recnd 10068	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
30	22, 29	eqeltrd 2701	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
31	22	oveq2d 6666	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32		elfznn 12370	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
33	32	adantl 482	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
34	33	nncnd 11036	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35		peano2cn 10208	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
36	34, 35	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37		peano2nn 11032	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
38	33, 37	syl 17	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
39	33, 38	nnmulcld 11068	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
40	39	nncnd 11036	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
41	39	nnne0d 11065	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ≠ 0)
42	36, 34, 40, 41	divsubdird 10840	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43		ax-1cn 9994	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
44		pncan2 10288	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
45	34, 43, 44	sylancl 694	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
46	45	oveq1d 6665	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
47	36	mulid1d 10057	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
48	36, 34	mulcomd 10061	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
49	47, 48	oveq12d 6668	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50		1cnd 10056	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
51	33	nnne0d 11065	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ≠ 0)
52	38	nnne0d 11065	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ≠ 0)
53	50, 34, 36, 51, 52	divcan5d 10827	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
54	49, 53	eqtr3d 2658	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
55	34	mulid1d 10057	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
56	55	oveq1d 6665	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
57	50, 36, 34, 52, 51	divcan5d 10827	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
58	56, 57	eqtr3d 2658	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
59	54, 58	oveq12d 6668	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
60	42, 46, 59	3eqtr3d 2664	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
61	60	sumeq2dv 14433	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63		oveq2 6658	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64		oveq2 6658	. . . . . . . 8 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65		1div1e1 10717	. . . . . . . 8 ⊢ (1 / 1) = 1
66	64, 65	syl6eq 2672	. . . . . . 7 ⊢ (𝑛 = 1 → (1 / 𝑛) = 1)
67		nnz 11399	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68	67	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
69	12, 1	syl6eleq 2711	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ≥‘1))
70		elfznn 12370	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
71	70	adantl 482	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
72	71	nnrecred 11066	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
73	72	recnd 10068	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
74	62, 63, 66, 13, 68, 69, 73	telfsum 14536	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
75	61, 74	eqtrd 2656	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76		id 22	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → 𝑛 = 𝑗)
77		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
78	76, 77	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
79	78	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
80		trireciplem.1	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
81		ovex 6678	. . . . . . . 8 ⊢ (1 / (𝑗 · (𝑗 + 1))) ∈ V
82	79, 80, 81	fvmpt 6282	. . . . . . 7 ⊢ (𝑗 ∈ ℕ → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
83	33, 82	syl 17	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
84		simpr 477	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
85	84, 1	syl6eleq 2711	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
86	39	nnrecred 11066	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℝ)
87	86	recnd 10068	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
88	83, 85, 87	fsumser 14461	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
89	31, 75, 88	3eqtr2rd 2663	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
90	1, 2, 25, 3, 27, 30, 89	climsubc2 14369	. . 3 ⊢ (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
91	90	trud 1493	. 2 ⊢ seq1( + , 𝐹) ⇝ (1 − 0)
92		1m0e1 11131	. 2 ⊢ (1 − 0) = 1
93	91, 92	breqtri 4678	1 ⊢ seq1( + , 𝐹) ⇝ 1

Syntax hints:   ∧ wa 384   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801   ⇝ cli 14215  Σcsu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417

This theorem is referenced by:  trirecip  14595  stirlinglem12  40302