Theorem trireciplem 11450

Description: Lemma for trirecip 11451. 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 9509	. . . 4 ⊢ ℕ = (ℤ≥‘1)
2		1zzd 9226	. . . 4 ⊢ (⊤ → 1 ∈ ℤ)
3		1cnd 7923	. . . . . 6 ⊢ (⊤ → 1 ∈ ℂ)
4		divcnv 11447	. . . . . 6 ⊢ (1 ∈ ℂ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
5	3, 4	syl 14	. . . . 5 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0)
6		nnex 8871	. . . . . . . 8 ⊢ ℕ ∈ V
7	6	mptex 5719	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V
8	7	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ∈ V)
9	6	mptex 5719	. . . . . . 7 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V
10	9	a1i 9	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ∈ V)
11		peano2nn 8877	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
12	11	adantl 275	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
13	12	nnrecred 8912	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℝ)
14		oveq2 5858	. . . . . . . . 9 ⊢ (𝑛 = (𝑘 + 1) → (1 / 𝑛) = (1 / (𝑘 + 1)))
15		eqid 2170	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) = (𝑛 ∈ ℕ ↦ (1 / 𝑛))
16	14, 15	fvmptg 5570	. . . . . . . 8 ⊢ (((𝑘 + 1) ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
17	12, 13, 16	syl2anc 409	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = (1 / (𝑘 + 1)))
18		simpr 109	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
19		oveq1 5857	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1))
20	19	oveq2d 5866	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (1 / (𝑛 + 1)) = (1 / (𝑘 + 1)))
21		eqid 2170	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) = (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))
22	20, 21	fvmptg 5570	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / (𝑘 + 1)) ∈ ℝ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
23	18, 13, 22	syl2anc 409	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) = (1 / (𝑘 + 1)))
24	17, 23	eqtr4d 2206	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / 𝑛))‘(𝑘 + 1)) = ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘))
25	1, 2, 2, 8, 10, 24	climshft2 11256	. . . . 5 ⊢ (⊤ → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0 ↔ (𝑛 ∈ ℕ ↦ (1 / 𝑛)) ⇝ 0))
26	5, 25	mpbird 166	. . . 4 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1))) ⇝ 0)
27		seqex 10390	. . . . 5 ⊢ seq1( + , 𝐹) ∈ V
28	27	a1i 9	. . . 4 ⊢ (⊤ → seq1( + , 𝐹) ∈ V)
29	13	recnd 7935	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 / (𝑘 + 1)) ∈ ℂ)
30	23, 29	eqeltrd 2247	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘) ∈ ℂ)
31	23	oveq2d 5866	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)) = (1 − (1 / (𝑘 + 1))))
32		elfznn 9997	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑘) → 𝑗 ∈ ℕ)
33	32	adantl 275	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℕ)
34	33	nncnd 8879	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 ∈ ℂ)
35		peano2cn 8041	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℂ → (𝑗 + 1) ∈ ℂ)
36	34, 35	syl 14	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℂ)
37		peano2nn 8877	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
38	33, 37	syl 14	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) ∈ ℕ)
39	33, 38	nnmulcld 8914	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℕ)
40	39	nncnd 8879	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
41	39	nnap0d 8911	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · (𝑗 + 1)) # 0)
42	36, 34, 40, 41	divsubdirapd 8734	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (((𝑗 + 1) / (𝑗 · (𝑗 + 1))) − (𝑗 / (𝑗 · (𝑗 + 1)))))
43		ax-1cn 7854	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
44		pncan2 8113	. . . . . . . . . 10 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 𝑗) = 1)
45	34, 43, 44	sylancl 411	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) − 𝑗) = 1)
46	45	oveq1d 5865	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) − 𝑗) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 · (𝑗 + 1))))
47	36	mulid1d 7924	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 1) = (𝑗 + 1))
48	36, 34	mulcomd 7928	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) · 𝑗) = (𝑗 · (𝑗 + 1)))
49	47, 48	oveq12d 5868	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = ((𝑗 + 1) / (𝑗 · (𝑗 + 1))))
50		1cnd 7923	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 1 ∈ ℂ)
51	33	nnap0d 8911	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → 𝑗 # 0)
52	38	nnap0d 8911	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 + 1) # 0)
53	50, 34, 36, 51, 52	divcanap5d 8721	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (((𝑗 + 1) · 1) / ((𝑗 + 1) · 𝑗)) = (1 / 𝑗))
54	49, 53	eqtr3d 2205	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 + 1) / (𝑗 · (𝑗 + 1))) = (1 / 𝑗))
55	34	mulid1d 7924	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 · 1) = 𝑗)
56	55	oveq1d 5865	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (𝑗 / (𝑗 · (𝑗 + 1))))
57	50, 36, 34, 52, 51	divcanap5d 8721	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → ((𝑗 · 1) / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
58	56, 57	eqtr3d 2205	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (𝑗 / (𝑗 · (𝑗 + 1))) = (1 / (𝑗 + 1)))
59	54, 58	oveq12d 5868	. . . . . . . 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 2211	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (1...𝑘)) → (1 / (𝑗 · (𝑗 + 1))) = ((1 / 𝑗) − (1 / (𝑗 + 1))))
61	60	sumeq2dv 11318	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))))
62		oveq2 5858	. . . . . . 7 ⊢ (𝑛 = 𝑗 → (1 / 𝑛) = (1 / 𝑗))
63		oveq2 5858	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (1 / 𝑛) = (1 / (𝑗 + 1)))
64		oveq2 5858	. . . . . . . 8 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
65		1div1e1 8608	. . . . . . . 8 ⊢ (1 / 1) = 1
66	64, 65	eqtrdi 2219	. . . . . . 7 ⊢ (𝑛 = 1 → (1 / 𝑛) = 1)
67		nnz 9218	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
68	67	adantl 275	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
69	12, 1	eleqtrdi 2263	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ≥‘1))
70		elfznn 9997	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(𝑘 + 1)) → 𝑛 ∈ ℕ)
71	70	adantl 275	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → 𝑛 ∈ ℕ)
72	71	nnrecred 8912	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℝ)
73	72	recnd 7935	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...(𝑘 + 1))) → (1 / 𝑛) ∈ ℂ)
74	62, 63, 66, 14, 68, 69, 73	telfsum 11418	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)((1 / 𝑗) − (1 / (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
75	61, 74	eqtrd 2203	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (1 − (1 / (𝑘 + 1))))
76		elnnuz 9510	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ ↔ 𝑗 ∈ (ℤ≥‘1))
77	76	biimpri 132	. . . . . . . 8 ⊢ (𝑗 ∈ (ℤ≥‘1) → 𝑗 ∈ ℕ)
78	77	adantl 275	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℕ)
79		eluzelz 9483	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ≥‘1) → 𝑗 ∈ ℤ)
80	79	adantl 275	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → 𝑗 ∈ ℤ)
81	80	zcnd 9322	. . . . . . . . 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 7927	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) ∈ ℂ)
84	78	nnap0d 8911	. . . . . . . . 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 8911	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 + 1) # 0)
87	81, 82, 84, 86	mulap0d 8563	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝑗 · (𝑗 + 1)) # 0)
88	83, 87	recclapd 8685	. . . . . . 7 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ)
89		id 19	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → 𝑛 = 𝑗)
90		oveq1 5857	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (𝑛 + 1) = (𝑗 + 1))
91	89, 90	oveq12d 5868	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (𝑛 · (𝑛 + 1)) = (𝑗 · (𝑗 + 1)))
92	91	oveq2d 5866	. . . . . . . 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 5570	. . . . . . 7 ⊢ ((𝑗 ∈ ℕ ∧ (1 / (𝑗 · (𝑗 + 1))) ∈ ℂ) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
95	78, 88, 94	syl2anc 409	. . . . . 6 ⊢ (((⊤ ∧ 𝑘 ∈ ℕ) ∧ 𝑗 ∈ (ℤ≥‘1)) → (𝐹‘𝑗) = (1 / (𝑗 · (𝑗 + 1))))
96	18, 1	eleqtrdi 2263	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1))
97	95, 96, 88	fsum3ser 11347	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → Σ𝑗 ∈ (1...𝑘)(1 / (𝑗 · (𝑗 + 1))) = (seq1( + , 𝐹)‘𝑘))
98	31, 75, 97	3eqtr2rd 2210	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐹)‘𝑘) = (1 − ((𝑛 ∈ ℕ ↦ (1 / (𝑛 + 1)))‘𝑘)))
99	1, 2, 26, 3, 28, 30, 98	climsubc2 11283	. . 3 ⊢ (⊤ → seq1( + , 𝐹) ⇝ (1 − 0))
100	99	mptru 1357	. 2 ⊢ seq1( + , 𝐹) ⇝ (1 − 0)
101		1m0e1 8978	. 2 ⊢ (1 − 0) = 1
102	100, 101	breqtri 4012	1 ⊢ seq1( + , 𝐹) ⇝ 1

Syntax hints:   ∧ wa 103   = wceq 1348  ⊤wtru 1349   ∈ wcel 2141  Vcvv 2730   class class class wbr 3987   ↦ cmpt 4048  ‘cfv 5196  (class class class)co 5850  ℂcc 7759  ℝcr 7760  0cc0 7761  1c1 7762   + caddc 7764   · cmul 7766   − cmin 8077   / cdiv 8576  ℕcn 8865  ℤcz 9199  ℤ_≥cuz 9474  ...cfz 9952  seqcseq 10388   ⇝ cli 11228  Σcsu 11303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-frec 6367  df-1o 6392  df-oadd 6396  df-er 6509  df-en 6715  df-dom 6716  df-fin 6717  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-seqfrec 10389  df-exp 10463  df-ihash 10697  df-shft 10766  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-clim 11229  df-sumdc 11304

This theorem is referenced by:  trirecip  11451