Theorem trireciplem 11811

Description: Lemma for trirecip 11812. 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 9684	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 9399	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		1cnd 8088	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
4		divcnv 11808	. . . . . 6 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
5	3, 4	syl 14	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6		nnex 9042	. . . . . . . 8 ⊢ ℕ ∈ V
7	6	mptex 5810	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
8	7	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
9	6	mptex 5810	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11		peano2nn 9048	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
12	11	adantl 277	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
13	12	nnrecred 9083	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
14		oveq2 5952	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
15		eqid 2205	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
16	14, 15	fvmptg 5655	. . . . . . . 8 ⊢ (((𝑘 + 1) ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
17	12, 13, 16	syl2anc 411	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18		simpr 110	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19		oveq1 5951	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
20	19	oveq2d 5960	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
21		eqid 2205	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
22	20, 21	fvmptg 5655	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
23	18, 13, 22	syl2anc 411	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
24	17, 23	eqtr4d 2241	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
25	1, 2, 2, 8, 10, 24	climshft2 11617	. . . . 5 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
26	5, 25	mpbird 167	. . . 4 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
27		seqex 10594	. . . . 5 ⊢ seq1( + , 𝐹) ∈ V
28	27	a1i 9	. . . 4 ⊢ (⊤ → seq1( + , 𝐹) ∈ V)
29	13	recnd 8101	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
30	23, 29	eqeltrd 2282	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
31	23	oveq2d 5960	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32		elfznn 10176	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
33	32	adantl 277	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
34	33	nncnd 9050	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35		peano2cn 8207	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
36	34, 35	syl 14	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37		peano2nn 9048	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
38	33, 37	syl 14	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
39	33, 38	nnmulcld 9085	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
40	39	nncnd 9050	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
41	39	nnap0d 9082	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) # 0)
42	36, 34, 40, 41	divsubdirapd 8903	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43		ax-1cn 8018	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
44		pncan2 8279	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
45	34, 43, 44	sylancl 413	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
46	45	oveq1d 5959	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
47	36	mulridd 8089	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
48	36, 34	mulcomd 8094	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
49	47, 48	oveq12d 5962	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50		1cnd 8088	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
51	33	nnap0d 9082	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 # 0)
52	38	nnap0d 9082	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) # 0)
53	50, 34, 36, 51, 52	divcanap5d 8890	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
54	49, 53	eqtr3d 2240	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
55	34	mulridd 8089	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
56	55	oveq1d 5959	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
57	50, 36, 34, 52, 51	divcanap5d 8890	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
58	56, 57	eqtr3d 2240	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
59	54, 58	oveq12d 5962	. . . . . . . 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 2246	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
61	60	sumeq2dv 11679	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62		oveq2 5952	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63		oveq2 5952	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64		oveq2 5952	. . . . . . . 8 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65		1div1e1 8777	. . . . . . . 8 ⊢ (1 / 1) = 1
66	64, 65	eqtrdi 2254	. . . . . . 7 ⊢ (𝑛 = 1 → (1 / 𝑛) = 1)
67		nnz 9391	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68	67	adantl 277	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
69	12, 1	eleqtrdi 2298	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ≥‘1))
70		elfznn 10176	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
71	70	adantl 277	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
72	71	nnrecred 9083	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
73	72	recnd 8101	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
74	62, 63, 66, 14, 68, 69, 73	telfsum 11779	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
75	61, 74	eqtrd 2238	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76		elnnuz 9685	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1))
77	76	biimpri 133	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ≥‘1) → 𝑗 ∈ ℕ)
78	77	adantl 277	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℕ)
79		eluzelz 9657	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘1) → 𝑗 ∈ ℤ)
80	79	adantl 277	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℤ)
81	80	zcnd 9496	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℂ)
82	81, 35	syl 14	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 + 1) ∈ ℂ)
83	81, 82	mulcld 8093	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
84	78	nnap0d 9082	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 # 0)
85	78, 37	syl 14	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 + 1) ∈ ℕ)
86	85	nnap0d 9082	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 + 1) # 0)
87	81, 82, 84, 86	mulap0d 8731	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) # 0)
88	83, 87	recclapd 8854	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
89		id 19	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → 𝑛 = 𝑗)
90		oveq1 5951	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
91	89, 90	oveq12d 5962	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
92	91	oveq2d 5960	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → (1 / (𝑛 · (𝑛 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
93		trireciplem.1	. . . . . . . 8 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (1 / (𝑛 · (𝑛 + 1))))
94	92, 93	fvmptg 5655	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
95	78, 88, 94	syl2anc 411	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
96	18, 1	eleqtrdi 2298	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
97	95, 96, 88	fsum3ser 11708	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
98	31, 75, 97	3eqtr2rd 2245	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
99	1, 2, 26, 3, 28, 30, 98	climsubc2 11644	. . 3 ⊢ (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
100	99	mptru 1382	. 2 ⊢ seq1( + , 𝐹) ⇝ (1 − 0)
101		1m0e1 9149	. 2 ⊢ (1 − 0) = 1
102	100, 101	breqtri 4069	1 ⊢ seq1( + , 𝐹) ⇝ 1

Syntax hints:   ∧ wa 104   = wceq 1373  ⊤wtru 1374   ∈ wcel 2176  Vcvv 2772   class class class wbr 4044   ↦ cmpt 4105  ‘cfv 5271  (class class class)co 5944  ℂcc 7923  ℝcr 7924  0cc0 7925  1c1 7926   + caddc 7928   · cmul 7930   − cmin 8243   / cdiv 8745  ℕcn 9036  ℤcz 9372  ℤ_≥cuz 9648  ...cfz 10130  seqcseq 10592   ⇝ cli 11589  Σcsu 11664

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-shft 11126  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  trirecip  11812