Step | Hyp | Ref
| Expression |
1 | | incsequz 35643 |
. 2
⊢ ((𝐹:ℕ⟶ℕ ∧
∀𝑚 ∈ ℕ
(𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ≥‘𝐴)) |
2 | | nnssre 11834 |
. . . . . . . 8
⊢ ℕ
⊆ ℝ |
3 | | ltso 10913 |
. . . . . . . . 9
⊢ < Or
ℝ |
4 | | sopo 5487 |
. . . . . . . . 9
⊢ ( < Or
ℝ → < Po ℝ) |
5 | 3, 4 | ax-mp 5 |
. . . . . . . 8
⊢ < Po
ℝ |
6 | | poss 5470 |
. . . . . . . 8
⊢ (ℕ
⊆ ℝ → ( < Po ℝ → < Po
ℕ)) |
7 | 2, 5, 6 | mp2 9 |
. . . . . . 7
⊢ < Po
ℕ |
8 | | seqpo 35642 |
. . . . . . 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 690 |
. . . . . 6
⊢ (𝐹:ℕ⟶ℕ →
(∀𝑚 ∈ ℕ
(𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ↔ ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞))) |
10 | 9 | biimpd 232 |
. . . . 5
⊢ (𝐹:ℕ⟶ℕ →
(∀𝑚 ∈ ℕ
(𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) → ∀𝑝 ∈ ℕ ∀𝑞 ∈ (ℤ≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞))) |
11 | 10 | imdistani 572 |
. . . 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 12475 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑛) → (𝑘 = 𝑛 ∨ 𝑘 ∈ (ℤ≥‘(𝑛 + 1)))) |
13 | | fveq2 6717 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
14 | 13 | adantl 485 |
. . . . . . . . . . . 12
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 = 𝑛) → (𝐹‘𝑘) = (𝐹‘𝑛)) |
15 | | ffvelrn 6902 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) →
(𝐹‘𝑛) ∈ ℕ) |
16 | 15 | nnzd 12281 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) →
(𝐹‘𝑛) ∈ ℤ) |
17 | | uzid 12453 |
. . . . . . . . . . . . . 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 484 |
. . . . . . . . . . . 12
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 = 𝑛) → (𝐹‘𝑛) ∈ (ℤ≥‘(𝐹‘𝑛))) |
20 | 14, 19 | eqeltrd 2838 |
. . . . . . . . . . 11
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 = 𝑛) → (𝐹‘𝑘) ∈ (ℤ≥‘(𝐹‘𝑛))) |
21 | 20 | adantllr 719 |
. . . . . . . . . 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 7236 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑛 → (ℤ≥‘(𝑝 + 1)) =
(ℤ≥‘(𝑛 + 1))) |
23 | | fveq2 6717 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 = 𝑛 → (𝐹‘𝑝) = (𝐹‘𝑛)) |
24 | 23 | breq1d 5063 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 = 𝑛 → ((𝐹‘𝑝) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑞))) |
25 | 22, 24 | raleqbidv 3313 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = 𝑛 → (∀𝑞 ∈ (ℤ≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ↔ ∀𝑞 ∈ (ℤ≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞))) |
26 | 25 | rspccva 3536 |
. . . . . . . . . . . . 13
⊢
((∀𝑝 ∈
ℕ ∀𝑞 ∈
(ℤ≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ∧ 𝑛 ∈ ℕ) → ∀𝑞 ∈
(ℤ≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞)) |
27 | | fveq2 6717 |
. . . . . . . . . . . . . . 15
⊢ (𝑞 = 𝑘 → (𝐹‘𝑞) = (𝐹‘𝑘)) |
28 | 27 | breq2d 5065 |
. . . . . . . . . . . . . 14
⊢ (𝑞 = 𝑘 → ((𝐹‘𝑛) < (𝐹‘𝑞) ↔ (𝐹‘𝑛) < (𝐹‘𝑘))) |
29 | 28 | rspccva 3536 |
. . . . . . . . . . . . 13
⊢
((∀𝑞 ∈
(ℤ≥‘(𝑛 + 1))(𝐹‘𝑛) < (𝐹‘𝑞) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘)) |
30 | 26, 29 | sylan 583 |
. . . . . . . . . . . 12
⊢
(((∀𝑝 ∈
ℕ ∀𝑞 ∈
(ℤ≥‘(𝑝 + 1))(𝐹‘𝑝) < (𝐹‘𝑞) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → (𝐹‘𝑛) < (𝐹‘𝑘)) |
31 | 30 | adantlll 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)) |
32 | 16 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘(𝑛 + 1))) → (𝐹‘𝑛) ∈ ℤ) |
33 | | peano2nn 11842 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
34 | | elnnuz 12478 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 + 1) ∈ ℕ ↔
(𝑛 + 1) ∈
(ℤ≥‘1)) |
35 | 33, 34 | sylib 221 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
(ℤ≥‘1)) |
36 | | uztrn 12456 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈
(ℤ≥‘(𝑛 + 1)) ∧ (𝑛 + 1) ∈
(ℤ≥‘1)) → 𝑘 ∈
(ℤ≥‘1)) |
37 | 36 | ancoms 462 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 + 1) ∈
(ℤ≥‘1) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈
(ℤ≥‘1)) |
38 | | elnnuz 12478 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈
(ℤ≥‘1)) |
39 | 37, 38 | sylibr 237 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑛 + 1) ∈
(ℤ≥‘1) ∧ 𝑘 ∈ (ℤ≥‘(𝑛 + 1))) → 𝑘 ∈
ℕ) |
40 | 35, 39 | sylan 583 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘(𝑛 + 1))) → 𝑘 ∈ ℕ) |
41 | | ffvelrn 6902 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:ℕ⟶ℕ ∧
𝑘 ∈ ℕ) →
(𝐹‘𝑘) ∈ ℕ) |
42 | 41 | nnzd 12281 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:ℕ⟶ℕ ∧
𝑘 ∈ ℕ) →
(𝐹‘𝑘) ∈ ℤ) |
43 | 40, 42 | sylan2 596 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:ℕ⟶ℕ ∧
(𝑛 ∈ ℕ ∧
𝑘 ∈
(ℤ≥‘(𝑛 + 1)))) → (𝐹‘𝑘) ∈ ℤ) |
44 | 43 | anassrs 471 |
. . . . . . . . . . . . 13
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘(𝑛 + 1))) → (𝐹‘𝑘) ∈ ℤ) |
45 | | zre 12180 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹‘𝑛) ∈ ℤ → (𝐹‘𝑛) ∈ ℝ) |
46 | | zre 12180 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹‘𝑘) ∈ ℤ → (𝐹‘𝑘) ∈ ℝ) |
47 | | ltle 10921 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑛) ∈ ℝ ∧ (𝐹‘𝑘) ∈ ℝ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘))) |
48 | 45, 46, 47 | syl2an 599 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑛) ≤ (𝐹‘𝑘))) |
49 | | eluz 12452 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑘) ∈ (ℤ≥‘(𝐹‘𝑛)) ↔ (𝐹‘𝑛) ≤ (𝐹‘𝑘))) |
50 | 48, 49 | sylibrd 262 |
. . . . . . . . . . . . 13
⊢ (((𝐹‘𝑛) ∈ ℤ ∧ (𝐹‘𝑘) ∈ ℤ) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑘) ∈ (ℤ≥‘(𝐹‘𝑛)))) |
51 | 32, 44, 50 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (((𝐹:ℕ⟶ℕ ∧
𝑛 ∈ ℕ) ∧
𝑘 ∈
(ℤ≥‘(𝑛 + 1))) → ((𝐹‘𝑛) < (𝐹‘𝑘) → (𝐹‘𝑘) ∈ (ℤ≥‘(𝐹‘𝑛)))) |
52 | 51 | adantllr 719 |
. . . . . . . . . . 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 958 |
. . . . . . . . 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 596 |
. . . . . . . 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 12456 |
. . . . . . . . 9
⊢ (((𝐹‘𝑘) ∈ (ℤ≥‘(𝐹‘𝑛)) ∧ (𝐹‘𝑛) ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) ∈ (ℤ≥‘𝐴)) |
57 | 56 | ex 416 |
. . . . . . . 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 719 |
. . . . . 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 3110 |
. . . . 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 416 |
. . . 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 1784 |
. . 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 3191 |
. 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)) |