Step | Hyp | Ref
| Expression |
1 | | isercoll.z |
. . . . . . . . . 10
⊢ 𝑍 =
(ℤ≥‘𝑀) |
2 | | uzssz 11707 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
3 | 1, 2 | eqsstri 3635 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℤ |
4 | | isercoll.g |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺:ℕ⟶𝑍) |
5 | 4 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ 𝑍) |
6 | 3, 5 | sseldi 3601 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℤ) |
7 | | nnz 11399 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℤ) |
8 | 7 | ad2antlr 763 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝑛 ∈ ℤ) |
9 | | fzfid 12772 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝑀...𝑚) ∈ Fin) |
10 | | ffun 6048 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺) |
11 | | funimacnv 5970 |
. . . . . . . . . . . . . . . 16
⊢ (Fun
𝐺 → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) = ((𝑀...𝑚) ∩ ran 𝐺)) |
12 | 4, 10, 11 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) = ((𝑀...𝑚) ∩ ran 𝐺)) |
13 | | inss1 3833 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀...𝑚) ∩ ran 𝐺) ⊆ (𝑀...𝑚) |
14 | 12, 13 | syl6eqss 3655 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚)) |
15 | 14 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚)) |
16 | | ssfi 8180 |
. . . . . . . . . . . . 13
⊢ (((𝑀...𝑚) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ⊆ (𝑀...𝑚)) → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ∈ Fin) |
17 | 9, 15, 16 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ∈ Fin) |
18 | | hashcl 13147 |
. . . . . . . . . . . 12
⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ∈ Fin → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ∈
ℕ0) |
19 | | nn0z 11400 |
. . . . . . . . . . . 12
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑚)))) ∈ ℕ0 →
(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ∈ ℤ) |
20 | 17, 18, 19 | 3syl 18 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ∈ ℤ) |
21 | | ssid 3624 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ℕ
⊆ ℕ |
22 | | isercoll.m |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ ℤ) |
23 | | isercoll.i |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
24 | 1, 22, 4, 23 | isercolllem1 14395 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ℕ ⊆ ℕ)
→ (𝐺 ↾ ℕ)
Isom < , < (ℕ, (𝐺 “ ℕ))) |
25 | 21, 24 | mpan2 707 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “
ℕ))) |
26 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ) |
27 | | fnresdm 6000 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐺 Fn ℕ → (𝐺 ↾ ℕ) = 𝐺) |
28 | | isoeq1 6567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐺 ↾ ℕ) = 𝐺 → ((𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “ ℕ)) ↔
𝐺 Isom < , <
(ℕ, (𝐺 “
ℕ)))) |
29 | 4, 26, 27, 28 | 4syl 19 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝐺 ↾ ℕ) Isom < , < (ℕ,
(𝐺 “ ℕ)) ↔
𝐺 Isom < , <
(ℕ, (𝐺 “
ℕ)))) |
30 | 25, 29 | mpbid 222 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐺 Isom < , < (ℕ, (𝐺 “
ℕ))) |
31 | | isof1o 6573 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺 Isom < , < (ℕ,
(𝐺 “ ℕ)) →
𝐺:ℕ–1-1-onto→(𝐺 “ ℕ)) |
32 | | f1ocnv 6149 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐺:ℕ–1-1-onto→(𝐺 “ ℕ) → ◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ) |
33 | | f1ofun 6139 |
. . . . . . . . . . . . . . . . . 18
⊢ (◡𝐺:(𝐺 “ ℕ)–1-1-onto→ℕ → Fun ◡𝐺) |
34 | 30, 31, 32, 33 | 4syl 19 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → Fun ◡𝐺) |
35 | | df-f1 5893 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:ℕ–1-1→𝑍 ↔ (𝐺:ℕ⟶𝑍 ∧ Fun ◡𝐺)) |
36 | 4, 34, 35 | sylanbrc 698 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐺:ℕ–1-1→𝑍) |
37 | 36 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝐺:ℕ–1-1→𝑍) |
38 | | elfznn 12370 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (1...𝑛) → 𝑦 ∈ ℕ) |
39 | 38 | ssriv 3607 |
. . . . . . . . . . . . . . 15
⊢
(1...𝑛) ⊆
ℕ |
40 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢
(1...𝑛) ∈
V |
41 | 40 | f1imaen 8018 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺:ℕ–1-1→𝑍 ∧ (1...𝑛) ⊆ ℕ) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) |
42 | 37, 39, 41 | sylancl 694 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) |
43 | | fzfid 12772 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (1...𝑛) ∈ Fin) |
44 | | enfii 8177 |
. . . . . . . . . . . . . . . 16
⊢
(((1...𝑛) ∈ Fin
∧ (𝐺 “ (1...𝑛)) ≈ (1...𝑛)) → (𝐺 “ (1...𝑛)) ∈ Fin) |
45 | 43, 42, 44 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (1...𝑛)) ∈ Fin) |
46 | | hashen 13135 |
. . . . . . . . . . . . . . 15
⊢ (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛))) |
47 | 45, 43, 46 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → ((#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛)) ↔ (𝐺 “ (1...𝑛)) ≈ (1...𝑛))) |
48 | 42, 47 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(𝐺 “ (1...𝑛))) = (#‘(1...𝑛))) |
49 | | nnnn0 11299 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
50 | 49 | ad2antlr 763 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝑛 ∈ ℕ0) |
51 | | hashfz1 13134 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
→ (#‘(1...𝑛)) =
𝑛) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(1...𝑛)) = 𝑛) |
53 | 48, 52 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(𝐺 “ (1...𝑛))) = 𝑛) |
54 | 38 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℕ) |
55 | | zssre 11384 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℤ
⊆ ℝ |
56 | 3, 55 | sstri 3612 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑍 ⊆
ℝ |
57 | 4 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝐺:ℕ⟶𝑍) |
58 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐺:ℕ⟶𝑍 ∧ 𝑦 ∈ ℕ) → (𝐺‘𝑦) ∈ 𝑍) |
59 | 57, 38, 58 | syl2an 494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ∈ 𝑍) |
60 | 56, 59 | sseldi 3601 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ∈ ℝ) |
61 | 5 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑛) ∈ 𝑍) |
62 | 56, 61 | sseldi 3601 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑛) ∈ ℝ) |
63 | | eluzelz 11697 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑚 ∈
(ℤ≥‘(𝐺‘𝑛)) → 𝑚 ∈ ℤ) |
64 | 63 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℤ) |
65 | 64 | zred 11482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑚 ∈ ℝ) |
66 | | elfzle2 12345 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 ∈ (1...𝑛) → 𝑦 ≤ 𝑛) |
67 | 66 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ≤ 𝑛) |
68 | 30 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Isom < , < (ℕ, (𝐺 “
ℕ))) |
69 | | simpllr 799 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℕ) |
70 | | isorel 6576 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐺 Isom < , < (ℕ,
(𝐺 “ ℕ)) ∧
(𝑛 ∈ ℕ ∧
𝑦 ∈ ℕ)) →
(𝑛 < 𝑦 ↔ (𝐺‘𝑛) < (𝐺‘𝑦))) |
71 | 68, 69, 54, 70 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑛 < 𝑦 ↔ (𝐺‘𝑛) < (𝐺‘𝑦))) |
72 | 71 | notbid 308 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (¬ 𝑛 < 𝑦 ↔ ¬ (𝐺‘𝑛) < (𝐺‘𝑦))) |
73 | 54 | nnred 11035 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ ℝ) |
74 | 69 | nnred 11035 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑛 ∈ ℝ) |
75 | 73, 74 | lenltd 10183 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦 ≤ 𝑛 ↔ ¬ 𝑛 < 𝑦)) |
76 | 60, 62 | lenltd 10183 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺‘𝑦) ≤ (𝐺‘𝑛) ↔ ¬ (𝐺‘𝑛) < (𝐺‘𝑦))) |
77 | 72, 75, 76 | 3bitr4d 300 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦 ≤ 𝑛 ↔ (𝐺‘𝑦) ≤ (𝐺‘𝑛))) |
78 | 67, 77 | mpbid 222 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ≤ (𝐺‘𝑛)) |
79 | | eluzle 11700 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑚 ∈
(ℤ≥‘(𝐺‘𝑛)) → (𝐺‘𝑛) ≤ 𝑚) |
80 | 79 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑛) ≤ 𝑚) |
81 | 60, 62, 65, 78, 80 | letrd 10194 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ≤ 𝑚) |
82 | 59, 1 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ∈ (ℤ≥‘𝑀)) |
83 | | elfz5 12334 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐺‘𝑦) ∈ (ℤ≥‘𝑀) ∧ 𝑚 ∈ ℤ) → ((𝐺‘𝑦) ∈ (𝑀...𝑚) ↔ (𝐺‘𝑦) ≤ 𝑚)) |
84 | 82, 64, 83 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → ((𝐺‘𝑦) ∈ (𝑀...𝑚) ↔ (𝐺‘𝑦) ≤ 𝑚)) |
85 | 81, 84 | mpbird 247 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝐺‘𝑦) ∈ (𝑀...𝑚)) |
86 | 57, 26 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝐺 Fn ℕ) |
87 | 86 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝐺 Fn ℕ) |
88 | | elpreima 6337 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺 Fn ℕ → (𝑦 ∈ (◡𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺‘𝑦) ∈ (𝑀...𝑚)))) |
89 | 87, 88 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → (𝑦 ∈ (◡𝐺 “ (𝑀...𝑚)) ↔ (𝑦 ∈ ℕ ∧ (𝐺‘𝑦) ∈ (𝑀...𝑚)))) |
90 | 54, 85, 89 | mpbir2and 957 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) ∧ 𝑦 ∈ (1...𝑛)) → 𝑦 ∈ (◡𝐺 “ (𝑀...𝑚))) |
91 | 90 | ex 450 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝑦 ∈ (1...𝑛) → 𝑦 ∈ (◡𝐺 “ (𝑀...𝑚)))) |
92 | 91 | ssrdv 3609 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (1...𝑛) ⊆ (◡𝐺 “ (𝑀...𝑚))) |
93 | | imass2 5501 |
. . . . . . . . . . . . . . 15
⊢
((1...𝑛) ⊆
(◡𝐺 “ (𝑀...𝑚)) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) |
94 | 92, 93 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) |
95 | | ssdomg 8001 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ∈ Fin → ((𝐺 “ (1...𝑛)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) |
96 | 17, 94, 95 | sylc 65 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) |
97 | | hashdom 13168 |
. . . . . . . . . . . . . 14
⊢ (((𝐺 “ (1...𝑛)) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) ∈ Fin) → ((#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) |
98 | 45, 17, 97 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → ((#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ↔ (𝐺 “ (1...𝑛)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) |
99 | 96, 98 | mpbird 247 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(𝐺 “ (1...𝑛))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) |
100 | 53, 99 | eqbrtrrd 4677 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → 𝑛 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) |
101 | | eluz2 11693 |
. . . . . . . . . . 11
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑚)))) ∈
(ℤ≥‘𝑛) ↔ (𝑛 ∈ ℤ ∧ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ∈ ℤ ∧ 𝑛 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))))) |
102 | 8, 20, 100, 101 | syl3anbrc 1246 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) ∈
(ℤ≥‘𝑛)) |
103 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))))) |
104 | 103 | eleq1d 2686 |
. . . . . . . . . . . 12
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ↔ (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ)) |
105 | 103 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → ((seq1( + , 𝐻)‘𝑘) − 𝐴) = ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) |
106 | 105 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴))) |
107 | 106 | breq1d 4663 |
. . . . . . . . . . . 12
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → ((abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)) |
108 | 104, 107 | anbi12d 747 |
. . . . . . . . . . 11
⊢ (𝑘 = (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) → (((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
109 | 108 | rspcv 3305 |
. . . . . . . . . 10
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑚)))) ∈
(ℤ≥‘𝑛) → (∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
110 | 102, 109 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))) → (∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
111 | 110 | ralrimdva 2969 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈
(ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∀𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
112 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑗 = (𝐺‘𝑛) → (ℤ≥‘𝑗) =
(ℤ≥‘(𝐺‘𝑛))) |
113 | 112 | raleqdv 3144 |
. . . . . . . . 9
⊢ (𝑗 = (𝐺‘𝑛) → (∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∀𝑚 ∈ (ℤ≥‘(𝐺‘𝑛))((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
114 | 113 | rspcev 3309 |
. . . . . . . 8
⊢ (((𝐺‘𝑛) ∈ ℤ ∧ ∀𝑚 ∈
(ℤ≥‘(𝐺‘𝑛))((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)) |
115 | 6, 111, 114 | syl6an 568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑘 ∈
(ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
116 | 115 | rexlimdva 3031 |
. . . . . 6
⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
117 | | 1nn 11031 |
. . . . . . . . 9
⊢ 1 ∈
ℕ |
118 | | ffvelrn 6357 |
. . . . . . . . 9
⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) →
(𝐺‘1) ∈ 𝑍) |
119 | 4, 117, 118 | sylancl 694 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘1) ∈ 𝑍) |
120 | 119, 1 | syl6eleq 2711 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘1) ∈
(ℤ≥‘𝑀)) |
121 | | eluzelz 11697 |
. . . . . . 7
⊢ ((𝐺‘1) ∈
(ℤ≥‘𝑀) → (𝐺‘1) ∈ ℤ) |
122 | | eqid 2622 |
. . . . . . . 8
⊢
(ℤ≥‘(𝐺‘1)) =
(ℤ≥‘(𝐺‘1)) |
123 | 122 | rexuz3 14088 |
. . . . . . 7
⊢ ((𝐺‘1) ∈ ℤ →
(∃𝑗 ∈
(ℤ≥‘(𝐺‘1))∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
124 | 120, 121,
123 | 3syl 18 |
. . . . . 6
⊢ (𝜑 → (∃𝑗 ∈ (ℤ≥‘(𝐺‘1))∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
125 | 116, 124 | sylibrd 249 |
. . . . 5
⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) → ∃𝑗 ∈ (ℤ≥‘(𝐺‘1))∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
126 | | fzfid 12772 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) → (𝑀...𝑗) ∈ Fin) |
127 | | funimacnv 5970 |
. . . . . . . . . . . 12
⊢ (Fun
𝐺 → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺)) |
128 | 4, 10, 127 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) = ((𝑀...𝑗) ∩ ran 𝐺)) |
129 | | inss1 3833 |
. . . . . . . . . . 11
⊢ ((𝑀...𝑗) ∩ ran 𝐺) ⊆ (𝑀...𝑗) |
130 | 128, 129 | syl6eqss 3655 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗)) |
131 | 130 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗)) |
132 | | ssfi 8180 |
. . . . . . . . 9
⊢ (((𝑀...𝑗) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ⊆ (𝑀...𝑗)) → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin) |
133 | 126, 131,
132 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin) |
134 | | hashcl 13147 |
. . . . . . . 8
⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈
ℕ0) |
135 | | nn0p1nn 11332 |
. . . . . . . 8
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 →
((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ) |
136 | 133, 134,
135 | 3syl 18 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) →
((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ) |
137 | | eluzle 11700 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1)) → ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘) |
138 | 137 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘) |
139 | 133 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin) |
140 | | nn0z 11400 |
. . . . . . . . . . . . . . . 16
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 →
(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℤ) |
141 | 139, 134,
140 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℤ) |
142 | | eluzelz 11697 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1)) → 𝑘 ∈ ℤ) |
143 | 142 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℤ) |
144 | | zltp1le 11427 |
. . . . . . . . . . . . . . 15
⊢
(((#‘(𝐺
“ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)) |
145 | 141, 143,
144 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) ≤ 𝑘)) |
146 | 138, 145 | mpbird 247 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) < 𝑘) |
147 | | nn0re 11301 |
. . . . . . . . . . . . . . . 16
⊢
((#‘(𝐺 “
(◡𝐺 “ (𝑀...𝑗)))) ∈ ℕ0 →
(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℝ) |
148 | 133, 134,
147 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) →
(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℝ) |
149 | 148 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ∈ ℝ) |
150 | | eluznn 11758 |
. . . . . . . . . . . . . . . 16
⊢
((((#‘(𝐺
“ (◡𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ) |
151 | 136, 150 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℕ) |
152 | 151 | nnred 11035 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑘 ∈ ℝ) |
153 | 149, 152 | ltnled 10184 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) < 𝑘 ↔ ¬ 𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))))) |
154 | 146, 153 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
155 | | fzss2 12381 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘(𝐺‘𝑘)) → (𝑀...(𝐺‘𝑘)) ⊆ (𝑀...𝑗)) |
156 | | imass2 5501 |
. . . . . . . . . . . . . 14
⊢ ((𝑀...(𝐺‘𝑘)) ⊆ (𝑀...𝑗) → (◡𝐺 “ (𝑀...(𝐺‘𝑘))) ⊆ (◡𝐺 “ (𝑀...𝑗))) |
157 | | imass2 5501 |
. . . . . . . . . . . . . 14
⊢ ((◡𝐺 “ (𝑀...(𝐺‘𝑘))) ⊆ (◡𝐺 “ (𝑀...𝑗)) → (𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) |
158 | 155, 156,
157 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘(𝐺‘𝑘)) → (𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) |
159 | | ssdomg 8001 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin → ((𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) → (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
160 | 139, 159 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) → (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
161 | 4 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ⟶𝑍) |
162 | 161 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ 𝑍) |
163 | 162, 1 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑥) ∈ (ℤ≥‘𝑀)) |
164 | 161, 151 | ffvelrnd 6360 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺‘𝑘) ∈ 𝑍) |
165 | 3, 164 | sseldi 3601 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺‘𝑘) ∈ ℤ) |
166 | 165 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺‘𝑘) ∈ ℤ) |
167 | | elfz5 12334 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐺‘𝑥) ∈ (ℤ≥‘𝑀) ∧ (𝐺‘𝑘) ∈ ℤ) → ((𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘)) ↔ (𝐺‘𝑥) ≤ (𝐺‘𝑘))) |
168 | 163, 166,
167 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘)) ↔ (𝐺‘𝑥) ≤ (𝐺‘𝑘))) |
169 | 30 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺 Isom < , < (ℕ, (𝐺 “
ℕ))) |
170 | | nnssre 11024 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℕ
⊆ ℝ |
171 | | ressxr 10083 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℝ
⊆ ℝ* |
172 | 170, 171 | sstri 3612 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ
⊆ ℝ* |
173 | 172 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ℕ ⊆
ℝ*) |
174 | | imassrn 5477 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐺 “ ℕ) ⊆ ran
𝐺 |
175 | 161 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝐺:ℕ⟶𝑍) |
176 | | frn 6053 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝐺:ℕ⟶𝑍 → ran 𝐺 ⊆ 𝑍) |
177 | 175, 176 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ 𝑍) |
178 | 177, 56 | syl6ss 3615 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ran 𝐺 ⊆ ℝ) |
179 | 174, 178 | syl5ss 3614 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆
ℝ) |
180 | 179, 171 | syl6ss 3615 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝐺 “ ℕ) ⊆
ℝ*) |
181 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℕ) |
182 | 151 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → 𝑘 ∈ ℕ) |
183 | | leisorel 13244 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐺 Isom < , < (ℕ,
(𝐺 “ ℕ)) ∧
(ℕ ⊆ ℝ* ∧ (𝐺 “ ℕ) ⊆
ℝ*) ∧ (𝑥 ∈ ℕ ∧ 𝑘 ∈ ℕ)) → (𝑥 ≤ 𝑘 ↔ (𝐺‘𝑥) ≤ (𝐺‘𝑘))) |
184 | 169, 173,
180, 181, 182, 183 | syl122anc 1335 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → (𝑥 ≤ 𝑘 ↔ (𝐺‘𝑥) ≤ (𝐺‘𝑘))) |
185 | 168, 184 | bitr4d 271 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) ∧ 𝑥 ∈ ℕ) → ((𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘)) ↔ 𝑥 ≤ 𝑘)) |
186 | 185 | pm5.32da 673 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘))) ↔ (𝑥 ∈ ℕ ∧ 𝑥 ≤ 𝑘))) |
187 | | elpreima 6337 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐺 Fn ℕ → (𝑥 ∈ (◡𝐺 “ (𝑀...(𝐺‘𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘))))) |
188 | 161, 26, 187 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (◡𝐺 “ (𝑀...(𝐺‘𝑘))) ↔ (𝑥 ∈ ℕ ∧ (𝐺‘𝑥) ∈ (𝑀...(𝐺‘𝑘))))) |
189 | | fznn 12408 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℤ → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥 ≤ 𝑘))) |
190 | 143, 189 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (1...𝑘) ↔ (𝑥 ∈ ℕ ∧ 𝑥 ≤ 𝑘))) |
191 | 186, 188,
190 | 3bitr4d 300 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑥 ∈ (◡𝐺 “ (𝑀...(𝐺‘𝑘))) ↔ 𝑥 ∈ (1...𝑘))) |
192 | 191 | eqrdv 2620 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (◡𝐺 “ (𝑀...(𝐺‘𝑘))) = (1...𝑘)) |
193 | 192 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) = (𝐺 “ (1...𝑘))) |
194 | 193 | sseq1d 3632 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ↔ (𝐺 “ (1...𝑘)) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
195 | 36 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝐺:ℕ–1-1→𝑍) |
196 | | elfznn 12370 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ (1...𝑘) → 𝑥 ∈ ℕ) |
197 | 196 | ssriv 3607 |
. . . . . . . . . . . . . . . . . . 19
⊢
(1...𝑘) ⊆
ℕ |
198 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(1...𝑘) ∈
V |
199 | 198 | f1imaen 8018 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺:ℕ–1-1→𝑍 ∧ (1...𝑘) ⊆ ℕ) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) |
200 | 195, 197,
199 | sylancl 694 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) |
201 | | fzfid 12772 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (1...𝑘) ∈ Fin) |
202 | | enfii 8177 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((1...𝑘) ∈ Fin
∧ (𝐺 “ (1...𝑘)) ≈ (1...𝑘)) → (𝐺 “ (1...𝑘)) ∈ Fin) |
203 | 201, 200,
202 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺 “ (1...𝑘)) ∈ Fin) |
204 | | hashen 13135 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (1...𝑘) ∈ Fin) → ((#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘))) |
205 | 203, 201,
204 | syl2anc 693 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘)) ↔ (𝐺 “ (1...𝑘)) ≈ (1...𝑘))) |
206 | 200, 205 | mpbird 247 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (1...𝑘))) = (#‘(1...𝑘))) |
207 | | nnnn0 11299 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
208 | | hashfz1 13134 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ (#‘(1...𝑘)) =
𝑘) |
209 | 151, 207,
208 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(1...𝑘)) = 𝑘) |
210 | 206, 209 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (1...𝑘))) = 𝑘) |
211 | 210 | breq1d 4663 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (1...𝑘))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ↔ 𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))))) |
212 | | hashdom 13168 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺 “ (1...𝑘)) ∈ Fin ∧ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) ∈ Fin) → ((#‘(𝐺 “ (1...𝑘))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
213 | 203, 139,
212 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((#‘(𝐺 “ (1...𝑘))) ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
214 | 211, 213 | bitr3d 270 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) ↔ (𝐺 “ (1...𝑘)) ≼ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))))) |
215 | 160, 194,
214 | 3imtr4d 283 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) ⊆ (𝐺 “ (◡𝐺 “ (𝑀...𝑗))) → 𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))))) |
216 | 158, 215 | syl5 34 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ≥‘(𝐺‘𝑘)) → 𝑘 ≤ (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))))) |
217 | 154, 216 | mtod 189 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ¬ 𝑗 ∈ (ℤ≥‘(𝐺‘𝑘))) |
218 | | eluzelz 11697 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘(𝐺‘1)) → 𝑗 ∈ ℤ) |
219 | 218 | ad2antlr 763 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → 𝑗 ∈ ℤ) |
220 | | uztric 11709 |
. . . . . . . . . . . . 13
⊢ (((𝐺‘𝑘) ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (ℤ≥‘(𝐺‘𝑘)) ∨ (𝐺‘𝑘) ∈ (ℤ≥‘𝑗))) |
221 | 165, 219,
220 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝑗 ∈ (ℤ≥‘(𝐺‘𝑘)) ∨ (𝐺‘𝑘) ∈ (ℤ≥‘𝑗))) |
222 | 221 | ord 392 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (¬ 𝑗 ∈ (ℤ≥‘(𝐺‘𝑘)) → (𝐺‘𝑘) ∈ (ℤ≥‘𝑗))) |
223 | 217, 222 | mpd 15 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (𝐺‘𝑘) ∈ (ℤ≥‘𝑗)) |
224 | | oveq2 6658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = (𝐺‘𝑘) → (𝑀...𝑚) = (𝑀...(𝐺‘𝑘))) |
225 | 224 | imaeq2d 5466 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝐺‘𝑘) → (◡𝐺 “ (𝑀...𝑚)) = (◡𝐺 “ (𝑀...(𝐺‘𝑘)))) |
226 | 225 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝐺‘𝑘) → (𝐺 “ (◡𝐺 “ (𝑀...𝑚))) = (𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘))))) |
227 | 226 | fveq2d 6195 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝐺‘𝑘) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))) = (#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) |
228 | 227 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝐺‘𝑘) → (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘))))))) |
229 | 228 | eleq1d 2686 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝐺‘𝑘) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ)) |
230 | 228 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝐺‘𝑘) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴) = ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) |
231 | 230 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝐺‘𝑘) → (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴))) |
232 | 231 | breq1d 4663 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝐺‘𝑘) → ((abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥)) |
233 | 229, 232 | anbi12d 747 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝐺‘𝑘) → (((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ ∧
(abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥))) |
234 | 233 | rspcv 3305 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑘) ∈ (ℤ≥‘𝑗) → (∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ ∧
(abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥))) |
235 | 223, 234 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ ∧
(abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥))) |
236 | 193 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘))))) = (#‘(𝐺 “ (1...𝑘)))) |
237 | 236, 210 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘))))) = 𝑘) |
238 | 237 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) = (seq1( + , 𝐻)‘𝑘)) |
239 | 238 | eleq1d 2686 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ ↔ (seq1( + , 𝐻)‘𝑘) ∈ ℂ)) |
240 | 238 | oveq1d 6665 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴) = ((seq1( + , 𝐻)‘𝑘) − 𝐴)) |
241 | 240 | fveq2d 6195 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (abs‘((seq1( + ,
𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) = (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴))) |
242 | 241 | breq1d 4663 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → ((abs‘((seq1( + ,
𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥 ↔ (abs‘((seq1( + , 𝐻)‘𝑘) − 𝐴)) < 𝑥)) |
243 | 239, 242 | anbi12d 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) ∈ ℂ ∧
(abs‘((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...(𝐺‘𝑘)))))) − 𝐴)) < 𝑥) ↔ ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
244 | 235, 243 | sylibd 229 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) → (∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
245 | 244 | ralrimdva 2969 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) →
(∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∀𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
246 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑛 = ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) →
(ℤ≥‘𝑛) =
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))) |
247 | 246 | raleqdv 3144 |
. . . . . . . 8
⊢ (𝑛 = ((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
248 | 247 | rspcev 3309 |
. . . . . . 7
⊢
((((#‘(𝐺
“ (◡𝐺 “ (𝑀...𝑗)))) + 1) ∈ ℕ ∧ ∀𝑘 ∈
(ℤ≥‘((#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑗)))) + 1))((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥)) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥)) |
249 | 136, 245,
248 | syl6an 568 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘(𝐺‘1))) →
(∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
250 | 249 | rexlimdva 3031 |
. . . . 5
⊢ (𝜑 → (∃𝑗 ∈ (ℤ≥‘(𝐺‘1))∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥))) |
251 | 125, 250 | impbid 202 |
. . . 4
⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ (ℤ≥‘(𝐺‘1))∀𝑚 ∈
(ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
252 | 251 | ralbidv 2986 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ ℝ+
∃𝑛 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘(𝐺‘1))∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥))) |
253 | 252 | anbi2d 740 |
. 2
⊢ (𝜑 → ((𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑛 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥)) ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘(𝐺‘1))∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))) |
254 | | nnuz 11723 |
. . 3
⊢ ℕ =
(ℤ≥‘1) |
255 | | 1zzd 11408 |
. . 3
⊢ (𝜑 → 1 ∈
ℤ) |
256 | | seqex 12803 |
. . . 4
⊢ seq1( + ,
𝐻) ∈
V |
257 | 256 | a1i 11 |
. . 3
⊢ (𝜑 → seq1( + , 𝐻) ∈ V) |
258 | | eqidd 2623 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (seq1( + , 𝐻)‘𝑘)) |
259 | 254, 255,
257, 258 | clim2 14235 |
. 2
⊢ (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑛 ∈ ℕ
∀𝑘 ∈
(ℤ≥‘𝑛)((seq1( + , 𝐻)‘𝑘) ∈ ℂ ∧ (abs‘((seq1( + ,
𝐻)‘𝑘) − 𝐴)) < 𝑥)))) |
260 | 120, 121 | syl 17 |
. . 3
⊢ (𝜑 → (𝐺‘1) ∈ ℤ) |
261 | | seqex 12803 |
. . . 4
⊢ seq𝑀( + , 𝐹) ∈ V |
262 | 261 | a1i 11 |
. . 3
⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ V) |
263 | | isercoll.0 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0) |
264 | | isercoll.f |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ) |
265 | | isercoll.h |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘))) |
266 | 1, 22, 4, 23, 263, 264, 265 | isercolllem3 14397 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑚) = (seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚)))))) |
267 | 122, 260,
262, 266 | clim2 14235 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐹) ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘(𝐺‘1))∀𝑚 ∈ (ℤ≥‘𝑗)((seq1( + , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) ∈ ℂ ∧ (abs‘((seq1(
+ , 𝐻)‘(#‘(𝐺 “ (◡𝐺 “ (𝑀...𝑚))))) − 𝐴)) < 𝑥)))) |
268 | 253, 259,
267 | 3bitr4d 300 |
1
⊢ (𝜑 → (seq1( + , 𝐻) ⇝ 𝐴 ↔ seq𝑀( + , 𝐹) ⇝ 𝐴)) |