ostth2lem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ostth2lem1		Structured version Visualization version GIF version

Theorem ostth2lem1 25107

Description: Lemma for ostth2 25126, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 25126. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, 𝑛 ∈ 𝑜(𝐴↑𝑛) for any 1 < 𝐴. (Contributed by Mario Carneiro, 10-Sep-2014.)

**Hypotheses**
Ref	Expression
ostth2lem1.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ostth2lem1.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ostth2lem1.3	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ≤ (𝑛 · 𝐵))

**Assertion**
Ref	Expression
ostth2lem1	⊢ (𝜑 → 𝐴 ≤ 1)

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝜑,𝑛

Proof of Theorem ostth2lem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2re 10967	. . . . . 6 ⊢ 2 ∈ ℝ
2		ostth2lem1.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 1 < 𝐴) → 𝐵 ∈ ℝ)
4		remulcl 9900	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (2 · 𝐵) ∈ ℝ)
5	1, 3, 4	sylancr 694	. . . . 5 ⊢ ((𝜑 ∧ 1 < 𝐴) → (2 · 𝐵) ∈ ℝ)
6		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ 1 < 𝐴) → 1 < 𝐴)
7		1re 9918	. . . . . . 7 ⊢ 1 ∈ ℝ
8		ostth2lem1.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < 𝐴) → 𝐴 ∈ ℝ)
10		difrp 11744	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (1 < 𝐴 ↔ (𝐴 − 1) ∈ ℝ⁺))
11	7, 9, 10	sylancr 694	. . . . . 6 ⊢ ((𝜑 ∧ 1 < 𝐴) → (1 < 𝐴 ↔ (𝐴 − 1) ∈ ℝ⁺))
12	6, 11	mpbid 221	. . . . 5 ⊢ ((𝜑 ∧ 1 < 𝐴) → (𝐴 − 1) ∈ ℝ⁺)
13	5, 12	rerpdivcld 11779	. . . 4 ⊢ ((𝜑 ∧ 1 < 𝐴) → ((2 · 𝐵) / (𝐴 − 1)) ∈ ℝ)
14		expnbnd 12855	. . . 4 ⊢ ((((2 · 𝐵) / (𝐴 − 1)) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 < 𝐴) → ∃𝑘 ∈ ℕ ((2 · 𝐵) / (𝐴 − 1)) < (𝐴↑𝑘))
15	13, 9, 6, 14	syl3anc 1318	. . 3 ⊢ ((𝜑 ∧ 1 < 𝐴) → ∃𝑘 ∈ ℕ ((2 · 𝐵) / (𝐴 − 1)) < (𝐴↑𝑘))
16		nnnn0 11176	. . . . . 6 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
17		reexpcl 12739	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → (𝐴↑𝑘) ∈ ℝ)
18	9, 16, 17	syl2an 493	. . . . 5 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) ∈ ℝ)
19	13	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((2 · 𝐵) / (𝐴 − 1)) ∈ ℝ)
20	12	rpred 11748	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < 𝐴) → (𝐴 − 1) ∈ ℝ)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴 − 1) ∈ ℝ)
22		nnre 10904	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
23	22	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
24	21, 23	remulcld 9949	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐴 − 1) · 𝑘) ∈ ℝ)
25	24, 18	remulcld 9949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) · (𝐴↑𝑘)) ∈ ℝ)
26	8	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
27		2nn 11062	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ
28		simpr 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
29		nnmulcl 10920	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ)
30	27, 28, 29	sylancr 694	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ)
31		nnnn0 11176	. . . . . . . . . . 11 ⊢ ((2 · 𝑘) ∈ ℕ → (2 · 𝑘) ∈ ℕ₀)
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℕ₀)
33	26, 32	reexpcld 12887	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑(2 · 𝑘)) ∈ ℝ)
34	30	nnred 10912	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) ∈ ℝ)
35	2	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℝ)
36	34, 35	remulcld 9949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((2 · 𝑘) · 𝐵) ∈ ℝ)
37		0red 9920	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < 𝐴) → 0 ∈ ℝ)
38	7	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < 𝐴) → 1 ∈ ℝ)
39		0lt1 10429	. . . . . . . . . . . . . . 15 ⊢ 0 < 1
40	39	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 1 < 𝐴) → 0 < 1)
41	37, 38, 9, 40, 6	lttrd 10077	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 1 < 𝐴) → 0 < 𝐴)
42	9, 41	elrpd 11745	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < 𝐴) → 𝐴 ∈ ℝ⁺)
43		nnz 11276	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
44		rpexpcl 12741	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → (𝐴↑𝑘) ∈ ℝ⁺)
45	42, 43, 44	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) ∈ ℝ⁺)
46		peano2re 10088	. . . . . . . . . . . . 13 ⊢ (((𝐴 − 1) · 𝑘) ∈ ℝ → (((𝐴 − 1) · 𝑘) + 1) ∈ ℝ)
47	24, 46	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) + 1) ∈ ℝ)
48	24	ltp1d 10833	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐴 − 1) · 𝑘) < (((𝐴 − 1) · 𝑘) + 1))
49	16	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ₀)
50	42	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
51	50	rpge0d 11752	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 0 ≤ 𝐴)
52		bernneq2 12853	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑘) + 1) ≤ (𝐴↑𝑘))
53	26, 49, 51, 52	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) + 1) ≤ (𝐴↑𝑘))
54	24, 47, 18, 48, 53	ltletrd 10076	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐴 − 1) · 𝑘) < (𝐴↑𝑘))
55	24, 18, 45, 54	ltmul1dd 11803	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) · (𝐴↑𝑘)) < ((𝐴↑𝑘) · (𝐴↑𝑘)))
56	23	recnd 9947	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
57	56	2timesd 11152	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (2 · 𝑘) = (𝑘 + 𝑘))
58	57	oveq2d 6565	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑(2 · 𝑘)) = (𝐴↑(𝑘 + 𝑘)))
59	26	recnd 9947	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℂ)
60	59, 49, 49	expaddd 12872	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑(𝑘 + 𝑘)) = ((𝐴↑𝑘) · (𝐴↑𝑘)))
61	58, 60	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑(2 · 𝑘)) = ((𝐴↑𝑘) · (𝐴↑𝑘)))
62	55, 61	breqtrrd 4611	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) · (𝐴↑𝑘)) < (𝐴↑(2 · 𝑘)))
63		ostth2lem1.3	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴↑𝑛) ≤ (𝑛 · 𝐵))
64	63	ralrimiva 2949	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴↑𝑛) ≤ (𝑛 · 𝐵))
65	64	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ ℕ (𝐴↑𝑛) ≤ (𝑛 · 𝐵))
66		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑛 = (2 · 𝑘) → (𝐴↑𝑛) = (𝐴↑(2 · 𝑘)))
67		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑛 = (2 · 𝑘) → (𝑛 · 𝐵) = ((2 · 𝑘) · 𝐵))
68	66, 67	breq12d 4596	. . . . . . . . . . 11 ⊢ (𝑛 = (2 · 𝑘) → ((𝐴↑𝑛) ≤ (𝑛 · 𝐵) ↔ (𝐴↑(2 · 𝑘)) ≤ ((2 · 𝑘) · 𝐵)))
69	68	rspcv 3278	. . . . . . . . . 10 ⊢ ((2 · 𝑘) ∈ ℕ → (∀𝑛 ∈ ℕ (𝐴↑𝑛) ≤ (𝑛 · 𝐵) → (𝐴↑(2 · 𝑘)) ≤ ((2 · 𝑘) · 𝐵)))
70	30, 65, 69	sylc 63	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑(2 · 𝑘)) ≤ ((2 · 𝑘) · 𝐵))
71	25, 33, 36, 62, 70	ltletrd 10076	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · 𝑘) · (𝐴↑𝑘)) < ((2 · 𝑘) · 𝐵))
72	21	recnd 9947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴 − 1) ∈ ℂ)
73	18	recnd 9947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) ∈ ℂ)
74	72, 73, 56	mul32d 10125	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · (𝐴↑𝑘)) · 𝑘) = (((𝐴 − 1) · 𝑘) · (𝐴↑𝑘)))
75		2cnd 10970	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 2 ∈ ℂ)
76	35	recnd 9947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ)
77	75, 76, 56	mul32d 10125	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((2 · 𝐵) · 𝑘) = ((2 · 𝑘) · 𝐵))
78	71, 74, 77	3brtr4d 4615	. . . . . . 7 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · (𝐴↑𝑘)) · 𝑘) < ((2 · 𝐵) · 𝑘))
79	21, 18	remulcld 9949	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐴 − 1) · (𝐴↑𝑘)) ∈ ℝ)
80	5	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (2 · 𝐵) ∈ ℝ)
81		nngt0 10926	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → 0 < 𝑘)
82	81	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 0 < 𝑘)
83		ltmul1 10752	. . . . . . . 8 ⊢ ((((𝐴 − 1) · (𝐴↑𝑘)) ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → (((𝐴 − 1) · (𝐴↑𝑘)) < (2 · 𝐵) ↔ (((𝐴 − 1) · (𝐴↑𝑘)) · 𝑘) < ((2 · 𝐵) · 𝑘)))
84	79, 80, 23, 82, 83	syl112anc 1322	. . . . . . 7 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · (𝐴↑𝑘)) < (2 · 𝐵) ↔ (((𝐴 − 1) · (𝐴↑𝑘)) · 𝑘) < ((2 · 𝐵) · 𝑘)))
85	78, 84	mpbird 246	. . . . . 6 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ((𝐴 − 1) · (𝐴↑𝑘)) < (2 · 𝐵))
86	12	rpgt0d 11751	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < 𝐴) → 0 < (𝐴 − 1))
87	86	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → 0 < (𝐴 − 1))
88		ltmuldiv2 10776	. . . . . . 7 ⊢ (((𝐴↑𝑘) ∈ ℝ ∧ (2 · 𝐵) ∈ ℝ ∧ ((𝐴 − 1) ∈ ℝ ∧ 0 < (𝐴 − 1))) → (((𝐴 − 1) · (𝐴↑𝑘)) < (2 · 𝐵) ↔ (𝐴↑𝑘) < ((2 · 𝐵) / (𝐴 − 1))))
89	18, 80, 21, 87, 88	syl112anc 1322	. . . . . 6 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (((𝐴 − 1) · (𝐴↑𝑘)) < (2 · 𝐵) ↔ (𝐴↑𝑘) < ((2 · 𝐵) / (𝐴 − 1))))
90	85, 89	mpbid 221	. . . . 5 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → (𝐴↑𝑘) < ((2 · 𝐵) / (𝐴 − 1)))
91	18, 19, 90	ltnsymd 10065	. . . 4 ⊢ (((𝜑 ∧ 1 < 𝐴) ∧ 𝑘 ∈ ℕ) → ¬ ((2 · 𝐵) / (𝐴 − 1)) < (𝐴↑𝑘))
92	91	nrexdv 2984	. . 3 ⊢ ((𝜑 ∧ 1 < 𝐴) → ¬ ∃𝑘 ∈ ℕ ((2 · 𝐵) / (𝐴 − 1)) < (𝐴↑𝑘))
93	15, 92	pm2.65da 598	. 2 ⊢ (𝜑 → ¬ 1 < 𝐴)
94		lenlt 9995	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ≤ 1 ↔ ¬ 1 < 𝐴))
95	8, 7, 94	sylancl 693	. 2 ⊢ (𝜑 → (𝐴 ≤ 1 ↔ ¬ 1 < 𝐴))
96	93, 95	mpbird 246	1 ⊢ (𝜑 → 𝐴 ≤ 1)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℝ⁺crp 11708 ↑cexp 12722

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723

This theorem is referenced by: ostth2lem4 25125

W3C validator