incsequz2 - Mathbox for Jeff Madsen

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Theorem incsequz2 36617

Description: An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Assertion**
Ref	Expression
incsequz2	⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))

Distinct variable groups: 𝑘,𝐹,𝑚,𝑛 𝐴,𝑘,𝑚,𝑛

Proof of Theorem incsequz2 Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		incsequz 36616	. 2 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ_≥‘𝐴))
2		nnssre 12216	. . . . . . . 8 ⊢ ℕ ⊆ ℝ
3		ltso 11294	. . . . . . . . 9 ⊢ < Or ℝ
4		sopo 5608	. . . . . . . . 9 ⊢ ( < Or ℝ → < Po ℝ)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ < Po ℝ
6		poss 5591	. . . . . . . 8 ⊢ (ℕ ⊆ ℝ → ( < Po ℝ → < Po ℕ))
7	2, 5, 6	mp2 9	. . . . . . 7 ⊢ < Po ℕ
8		seqpo 36615	. . . . . . 7 ⊢ (( < Po ℕ ∧ 𝐹:ℕ⟶ℕ) → (∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ↔ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)))
9	7, 8	mpan 689	. . . . . 6 ⊢ (𝐹:ℕ⟶ℕ → (∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ↔ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)))
10	9	biimpd 228	. . . . 5 ⊢ (𝐹:ℕ⟶ℕ → (∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) → ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)))
11	10	imdistani 570	. . . 4 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1))) → (𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)))
12		uzp1 12863	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → (𝑘 = 𝑛 ∨ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))))
13		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
14	13	adantl 483	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑘) = (𝐹‘𝑛))
15		ffvelcdm 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℕ)
16	15	nnzd 12585	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℤ)
17		uzid 12837	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑛) ∈ ℤ → (𝐹‘𝑛) ∈ (ℤ_≥‘(𝐹‘𝑛)))
18	16, 17	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℤ_≥‘(𝐹‘𝑛)))
19	18	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑛) ∈ (ℤ_≥‘(𝐹‘𝑛)))
20	14, 19	eqeltrd 2834	. . . . . . . . . . 11 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)))
21	20	adantllr 718	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)))
22		fvoveq1 7432	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → (ℤ_≥‘(𝑝 + 1)) = (ℤ_≥‘(𝑛 + 1)))
23		fveq2 6892	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑛 → (𝐹‘𝑝) = (𝐹‘𝑛))
24	23	breq1d 5159	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → ((𝐹‘𝑝) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑞)))
25	22, 24	raleqbidv 3343	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → (∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ↔ ∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞)))
26	25	rspccva 3612	. . . . . . . . . . . . 13 ⊢ ((∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ∧ 𝑛 ∈ ℕ) → ∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞))
27		fveq2 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑘 → (𝐹‘𝑞) = (𝐹‘𝑘))
28	27	breq2d 5161	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑘 → ((𝐹‘𝑛) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑘)))
29	28	rspccva 3612	. . . . . . . . . . . . 13 ⊢ ((∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘))
30	26, 29	sylan 581	. . . . . . . . . . . 12 ⊢ (((∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘))
31	30	adantlll 717	. . . . . . . . . . 11 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘))
32	16	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) ∈ ℤ)
33		peano2nn 12224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
34		elnnuz 12866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 + 1) ∈ ℕ ↔ (𝑛 + 1) ∈ (ℤ_≥‘1))
35	33, 34	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ_≥‘1))
36		uztrn 12840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ_≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈ (ℤ_≥‘1)) → 𝑘 ∈ (ℤ_≥‘1))
37	36	ancoms 460	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘1) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ (ℤ_≥‘1))
38		elnnuz 12866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
39	37, 38	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘1) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ)
40	35, 39	sylan 581	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ)
41		ffvelcdm 7084	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℕ)
42	41	nnzd 12585	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
43	40, 42	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1)))) → (𝐹‘𝑘) ∈ ℤ)
44	43	anassrs 469	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑘) ∈ ℤ)
45		zre 12562	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ ℤ → (𝐹‘𝑛) ∈ ℝ)
46		zre 12562	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) ∈ ℤ → (𝐹‘𝑘) ∈ ℝ)
47		ltle 11302	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘)))
48	45, 46, 47	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘)))
49		eluz 12836	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)) ↔ (𝐹‘𝑛) ≤ (𝐹‘𝑘)))
50	48, 49	sylibrd 259	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛))))
51	32, 44, 50	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛))))
52	51	adantllr 718	. . . . . . . . . . 11 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛))))
53	31, 52	mpd 15	. . . . . . . . . 10 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)))
54	21, 53	jaodan 957	. . . . . . . . 9 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ (𝑘 = 𝑛 ∨ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1)))) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)))
55	12, 54	sylan2 594	. . . . . . . 8 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)))
56		uztrn 12840	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)) ∧ (𝐹‘𝑛) ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))
57	56	ex 414	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)) → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → (𝐹‘𝑘) ∈ (ℤ_≥‘𝐴)))
58	55, 57	syl 17	. . . . . . 7 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → (𝐹‘𝑘) ∈ (ℤ_≥‘𝐴)))
59	58	adantllr 718	. . . . . 6 ⊢ (((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝐴 ∈ ℕ) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → (𝐹‘𝑘) ∈ (ℤ_≥‘𝐴)))
60	59	ralrimdva 3155	. . . . 5 ⊢ ((((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝐴 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴)))
61	60	ex 414	. . . 4 ⊢ (((𝐹:ℕ⟶ℕ ∧ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞)) ∧ 𝐴 ∈ ℕ) → (𝑛 ∈ ℕ → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))))
62	11, 61	stoic3 1779	. . 3 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → (𝑛 ∈ ℕ → ((𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))))
63	62	reximdvai 3166	. 2 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → (∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ_≥‘𝐴) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴)))
64	1, 63	mpd 15	1 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ⊆ wss 3949 class class class wbr 5149 Po wpo 5587 Or wor 5588 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 ℕcn 12212 ℤcz 12558 ℤ_≥cuz 12822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823

This theorem is referenced by: (None)

W3C validator