Theorem trireciplem 12051

Description: Lemma for trirecip 12052. 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 9782	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 9496	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		1cnd 8185	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
4		divcnv 12048	. . . . . 6 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
5	3, 4	syl 14	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6		nnex 9139	. . . . . . . 8 ⊢ ℕ ∈ V
7	6	mptex 5875	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
8	7	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
9	6	mptex 5875	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11		peano2nn 9145	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
12	11	adantl 277	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
13	12	nnrecred 9180	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
14		oveq2 6021	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
15		eqid 2229	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
16	14, 15	fvmptg 5718	. . . . . . . 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 6020	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
20	19	oveq2d 6029	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
21		eqid 2229	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
22	20, 21	fvmptg 5718	. . . . . . . 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 2265	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
25	1, 2, 2, 8, 10, 24	climshft2 11857	. . . . 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 10701	. . . . 5 ⊢ seq1( + , 𝐹) ∈ V
28	27	a1i 9	. . . 4 ⊢ (⊤ → seq1( + , 𝐹) ∈ V)
29	13	recnd 8198	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
30	23, 29	eqeltrd 2306	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
31	23	oveq2d 6029	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32		elfznn 10279	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
33	32	adantl 277	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
34	33	nncnd 9147	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35		peano2cn 8304	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
36	34, 35	syl 14	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37		peano2nn 9145	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
38	33, 37	syl 14	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
39	33, 38	nnmulcld 9182	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
40	39	nncnd 9147	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
41	39	nnap0d 9179	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) # 0)
42	36, 34, 40, 41	divsubdirapd 9000	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43		ax-1cn 8115	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
44		pncan2 8376	. . . . . . . . . 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 6028	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
47	36	mulridd 8186	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
48	36, 34	mulcomd 8191	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
49	47, 48	oveq12d 6031	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50		1cnd 8185	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
51	33	nnap0d 9179	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 # 0)
52	38	nnap0d 9179	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) # 0)
53	50, 34, 36, 51, 52	divcanap5d 8987	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
54	49, 53	eqtr3d 2264	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
55	34	mulridd 8186	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
56	55	oveq1d 6028	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
57	50, 36, 34, 52, 51	divcanap5d 8987	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
58	56, 57	eqtr3d 2264	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
59	54, 58	oveq12d 6031	. . . . . . . 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 2270	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
61	60	sumeq2dv 11919	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62		oveq2 6021	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63		oveq2 6021	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64		oveq2 6021	. . . . . . . 8 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65		1div1e1 8874	. . . . . . . 8 ⊢ (1 / 1) = 1
66	64, 65	eqtrdi 2278	. . . . . . 7 ⊢ (𝑛 = 1 → (1 / 𝑛) = 1)
67		nnz 9488	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68	67	adantl 277	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
69	12, 1	eleqtrdi 2322	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ≥‘1))
70		elfznn 10279	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
71	70	adantl 277	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
72	71	nnrecred 9180	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
73	72	recnd 8198	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
74	62, 63, 66, 14, 68, 69, 73	telfsum 12019	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
75	61, 74	eqtrd 2262	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76		elnnuz 9783	. . . . . . . . 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 9755	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘1) → 𝑗 ∈ ℤ)
80	79	adantl 277	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℤ)
81	80	zcnd 9593	. . . . . . . . 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 8190	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
84	78	nnap0d 9179	. . . . . . . . 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 9179	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 + 1) # 0)
87	81, 82, 84, 86	mulap0d 8828	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) # 0)
88	83, 87	recclapd 8951	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
89		id 19	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → 𝑛 = 𝑗)
90		oveq1 6020	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
91	89, 90	oveq12d 6031	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
92	91	oveq2d 6029	. . . . . . . 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 5718	. . . . . . 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 2322	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
97	95, 96, 88	fsum3ser 11948	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
98	31, 75, 97	3eqtr2rd 2269	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
99	1, 2, 26, 3, 28, 30, 98	climsubc2 11884	. . 3 ⊢ (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
100	99	mptru 1404	. 2 ⊢ seq1( + , 𝐹) ⇝ (1 − 0)
101		1m0e1 9246	. 2 ⊢ (1 − 0) = 1
102	100, 101	breqtri 4111	1 ⊢ seq1( + , 𝐹) ⇝ 1

Syntax hints:   ∧ wa 104   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  Vcvv 2800   class class class wbr 4086   ↦ cmpt 4148  ‘cfv 5324  (class class class)co 6013  ℂcc 8020  ℝcr 8021  0cc0 8022  1c1 8023   + caddc 8025   · cmul 8027   − cmin 8340   / cdiv 8842  ℕcn 9133  ℤcz 9469  ℤ_≥cuz 9745  ...cfz 10233  seqcseq 10699   ⇝ cli 11829  Σcsu 11904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-shft 11366  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905

This theorem is referenced by:  trirecip  12052