incsequz2 - Mathbox for Jeff Madsen

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Theorem incsequz2 37157

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 37156	. 2 ⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ_≥‘𝐴))
2		nnssre 12238	. . . . . . . 8 ⊢ ℕ ⊆ ℝ
3		ltso 11316	. . . . . . . . 9 ⊢ < Or ℝ
4		sopo 5603	. . . . . . . . 9 ⊢ ( < Or ℝ → < Po ℝ)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ < Po ℝ
6		poss 5586	. . . . . . . 8 ⊢ (ℕ ⊆ ℝ → ( < Po ℝ → < Po ℕ))
7	2, 5, 6	mp2 9	. . . . . . 7 ⊢ < Po ℕ
8		seqpo 37155	. . . . . . 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 568	. . . 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 12885	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → (𝑘 = 𝑛 ∨ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))))
13		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
14	13	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑘) = (𝐹‘𝑛))
15		ffvelcdm 7085	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℕ)
16	15	nnzd 12607	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℤ)
17		uzid 12859	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (𝐹‘𝑛) ∈ (ℤ_≥‘(𝐹‘𝑛)))
20	14, 19	eqeltrd 2828	. . . . . . . . . . 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 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → (ℤ_≥‘(𝑝 + 1)) = (ℤ_≥‘(𝑛 + 1)))
23		fveq2 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = 𝑛 → (𝐹‘𝑝) = (𝐹‘𝑛))
24	23	breq1d 5152	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑛 → ((𝐹‘𝑝) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑞)))
25	22, 24	raleqbidv 3337	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑛 → (∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ↔ ∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞)))
26	25	rspccva 3606	. . . . . . . . . . . . 13 ⊢ ((∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ_≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ∧ 𝑛 ∈ ℕ) → ∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞))
27		fveq2 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 𝑘 → (𝐹‘𝑞) = (𝐹‘𝑘))
28	27	breq2d 5154	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 𝑘 → ((𝐹‘𝑛) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑘)))
29	28	rspccva 3606	. . . . . . . . . . . . 13 ⊢ ((∀𝑞 ∈ (ℤ_≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘))
30	26, 29	sylan 579	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑛) ∈ ℤ)
33		peano2nn 12246	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
34		elnnuz 12888	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 + 1) ∈ ℕ ↔ (𝑛 + 1) ∈ (ℤ_≥‘1))
35	33, 34	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ (ℤ_≥‘1))
36		uztrn 12862	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (ℤ_≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈ (ℤ_≥‘1)) → 𝑘 ∈ (ℤ_≥‘1))
37	36	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 + 1) ∈ (ℤ_≥‘1) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ (ℤ_≥‘1))
38		elnnuz 12888	. . . . . . . . . . . . . . . . 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 579	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ)
41		ffvelcdm 7085	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℕ)
42	41	nnzd 12607	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:ℕ⟶ℕ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℤ)
43	40, 42	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝐹:ℕ⟶ℕ ∧ (𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1)))) → (𝐹‘𝑘) ∈ ℤ)
44	43	anassrs 467	. . . . . . . . . . . . 13 ⊢ (((𝐹:ℕ⟶ℕ ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘(𝑛 + 1))) → (𝐹‘𝑘) ∈ ℤ)
45		zre 12584	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑛) ∈ ℤ → (𝐹‘𝑛) ∈ ℝ)
46		zre 12584	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) ∈ ℤ → (𝐹‘𝑘) ∈ ℝ)
47		ltle 11324	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘)))
48	45, 46, 47	syl2an 595	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘)))
49		eluz 12858	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . 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 956	. . . . . . . . 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 592	. . . . . . . 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 12862	. . . . . . . . 9 ⊢ (((𝐹‘𝑘) ∈ (ℤ_≥‘(𝐹‘𝑛)) ∧ (𝐹‘𝑛) ∈ (ℤ_≥‘𝐴)) → (𝐹‘𝑘) ∈ (ℤ_≥‘𝐴))
57	56	ex 412	. . . . . . . 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 3149	. . . . 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 412	. . . 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 1771	. . 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 3160	. 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 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ⊆ wss 3944 class class class wbr 5142 Po wpo 5582 Or wor 5583 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 1c1 11131 + caddc 11133 < clt 11270 ≤ cle 11271 ℕcn 12234 ℤcz 12580 ℤ_≥cuz 12844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845

This theorem is referenced by: (None)

W3C validator