Step | Hyp | Ref
| Expression |
1 | | isercoll.z |
. . . . . . . . . . 11
⊢ 𝑍 =
(ℤ≥‘𝑀) |
2 | | uzssz 11707 |
. . . . . . . . . . 11
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
3 | 1, 2 | eqsstri 3635 |
. . . . . . . . . 10
⊢ 𝑍 ⊆
ℤ |
4 | | zssre 11384 |
. . . . . . . . . 10
⊢ ℤ
⊆ ℝ |
5 | 3, 4 | sstri 3612 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℝ |
6 | | isercoll.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:ℕ⟶𝑍) |
7 | 6 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝐺:ℕ⟶𝑍) |
8 | | simplrl 800 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ) |
9 | 7, 8 | ffvelrnd 6360 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) ∈ 𝑍) |
10 | 5, 9 | sseldi 3601 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) ∈ ℝ) |
11 | | simplrr 801 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ) |
12 | 11 | nnred 11035 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ) |
13 | 10, 12 | resubcld 10458 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) ∈ ℝ) |
14 | 8 | nnred 11035 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ) |
15 | 10, 14 | resubcld 10458 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑥) ∈ ℝ) |
16 | 7, 11 | ffvelrnd 6360 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑦) ∈ 𝑍) |
17 | 5, 16 | sseldi 3601 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑦) ∈ ℝ) |
18 | 17, 12 | resubcld 10458 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑦) − 𝑦) ∈ ℝ) |
19 | | simpr 477 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦) |
20 | 14, 12, 10, 19 | ltsub2dd 10640 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑥) − 𝑥)) |
21 | 8 | nnzd 11481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℤ) |
22 | 11 | nnzd 11481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ) |
23 | 14, 12, 19 | ltled 10185 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦) |
24 | | eluz2 11693 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑥) ↔ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑥 ≤ 𝑦)) |
25 | 21, 22, 23, 24 | syl3anbrc 1246 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (ℤ≥‘𝑥)) |
26 | | elfzuz 12338 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑥...𝑦) → 𝑘 ∈ (ℤ≥‘𝑥)) |
27 | | eluznn 11758 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝑥)) → 𝑘 ∈ ℕ) |
28 | 8, 27 | sylan 488 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → 𝑘 ∈ ℕ) |
29 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) |
30 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) |
31 | 29, 30 | oveq12d 6668 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑘) − 𝑘)) |
32 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛)) = (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛)) |
33 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘𝑘) − 𝑘) ∈ V |
34 | 31, 32, 33 | fvmpt 6282 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) = ((𝐺‘𝑘) − 𝑘)) |
35 | 34 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) = ((𝐺‘𝑘) − 𝑘)) |
36 | 7 | ffvelrnda 6359 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑍) |
37 | 5, 36 | sseldi 3601 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
38 | | nnre 11027 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
39 | 38 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ) |
40 | 37, 39 | resubcld 10458 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ∈ ℝ) |
41 | 35, 40 | eqeltrd 2701 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
42 | 28, 41 | syldan 487 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
43 | 26, 42 | sylan2 491 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (𝑥...𝑦)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
44 | | elfzuz 12338 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑥...(𝑦 − 1)) → 𝑘 ∈ (ℤ≥‘𝑥)) |
45 | | peano2nn 11032 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
46 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺:ℕ⟶𝑍 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑍) |
47 | 7, 45, 46 | syl2an 494 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑍) |
48 | 5, 47 | sseldi 3601 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℝ) |
49 | | peano2rem 10348 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘(𝑘 + 1)) ∈ ℝ → ((𝐺‘(𝑘 + 1)) − 1) ∈
ℝ) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘(𝑘 + 1)) − 1) ∈
ℝ) |
51 | | simpll 790 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝜑) |
52 | | isercoll.i |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
53 | 51, 52 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
54 | 3, 36 | sseldi 3601 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℤ) |
55 | 3, 47 | sseldi 3601 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℤ) |
56 | | zltlem1 11430 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺‘𝑘) ∈ ℤ ∧ (𝐺‘(𝑘 + 1)) ∈ ℤ) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1))) |
57 | 54, 55, 56 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1))) |
58 | 53, 57 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1)) |
59 | 37, 50, 39, 58 | lesub1dd 10643 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ≤ (((𝐺‘(𝑘 + 1)) − 1) − 𝑘)) |
60 | 48 | recnd 10068 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℂ) |
61 | | 1cnd 10056 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 1 ∈
ℂ) |
62 | 39 | recnd 10068 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ) |
63 | 60, 61, 62 | sub32d 10424 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 1) − 𝑘) = (((𝐺‘(𝑘 + 1)) − 𝑘) − 1)) |
64 | 60, 62, 61 | subsub4d 10423 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 𝑘) − 1) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
65 | 63, 64 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 1) − 𝑘) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
66 | 59, 65 | breqtrd 4679 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ≤ ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
67 | 45 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
68 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑘 + 1))) |
69 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1)) |
70 | 68, 69 | oveq12d 6668 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
71 | | ovex 6678 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘(𝑘 + 1)) − (𝑘 + 1)) ∈ V |
72 | 70, 32, 71 | fvmpt 6282 |
. . . . . . . . . . . . 13
⊢ ((𝑘 + 1) ∈ ℕ →
((𝑛 ∈ ℕ ↦
((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1)) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
73 | 67, 72 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1)) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
74 | 66, 35, 73 | 3brtr4d 4685 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1))) |
75 | 28, 74 | syldan 487 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1))) |
76 | 44, 75 | sylan2 491 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (𝑥...(𝑦 − 1))) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1))) |
77 | 25, 43, 76 | monoord 12831 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦)) |
78 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑥 → (𝐺‘𝑛) = (𝐺‘𝑥)) |
79 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑥 → 𝑛 = 𝑥) |
80 | 78, 79 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑥 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑥) − 𝑥)) |
81 | | ovex 6678 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑥) − 𝑥) ∈ V |
82 | 80, 32, 81 | fvmpt 6282 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) = ((𝐺‘𝑥) − 𝑥)) |
83 | 8, 82 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) = ((𝐺‘𝑥) − 𝑥)) |
84 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑦 → (𝐺‘𝑛) = (𝐺‘𝑦)) |
85 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑦 → 𝑛 = 𝑦) |
86 | 84, 85 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑦 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑦) − 𝑦)) |
87 | | ovex 6678 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑦) − 𝑦) ∈ V |
88 | 86, 32, 87 | fvmpt 6282 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦) = ((𝐺‘𝑦) − 𝑦)) |
89 | 11, 88 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦) = ((𝐺‘𝑦) − 𝑦)) |
90 | 77, 83, 89 | 3brtr3d 4684 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑥) ≤ ((𝐺‘𝑦) − 𝑦)) |
91 | 13, 15, 18, 20, 90 | ltletrd 10197 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑦) − 𝑦)) |
92 | 10, 17, 12 | ltsub1d 10636 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) < (𝐺‘𝑦) ↔ ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑦) − 𝑦))) |
93 | 91, 92 | mpbird 247 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) < (𝐺‘𝑦)) |
94 | 93 | ex 450 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
95 | 94 | ralrimivva 2971 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
96 | | ssralv 3666 |
. . . . 5
⊢ (𝑆 ⊆ ℕ →
(∀𝑦 ∈ ℕ
(𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)) → ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
97 | 96 | ralimdv 2963 |
. . . 4
⊢ (𝑆 ⊆ ℕ →
(∀𝑥 ∈ ℕ
∀𝑦 ∈ ℕ
(𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)) → ∀𝑥 ∈ ℕ ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
98 | | ssralv 3666 |
. . . 4
⊢ (𝑆 ⊆ ℕ →
(∀𝑥 ∈ ℕ
∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
99 | 97, 98 | syld 47 |
. . 3
⊢ (𝑆 ⊆ ℕ →
(∀𝑥 ∈ ℕ
∀𝑦 ∈ ℕ
(𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
100 | 95, 99 | mpan9 486 |
. 2
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
101 | | nnssre 11024 |
. . . . 5
⊢ ℕ
⊆ ℝ |
102 | | ltso 10118 |
. . . . 5
⊢ < Or
ℝ |
103 | | soss 5053 |
. . . . 5
⊢ (ℕ
⊆ ℝ → ( < Or ℝ → < Or
ℕ)) |
104 | 101, 102,
103 | mp2 9 |
. . . 4
⊢ < Or
ℕ |
105 | 104 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → < Or
ℕ) |
106 | | soss 5053 |
. . . . 5
⊢ (𝑍 ⊆ ℝ → ( <
Or ℝ → < Or 𝑍)) |
107 | 5, 102, 106 | mp2 9 |
. . . 4
⊢ < Or
𝑍 |
108 | 107 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → < Or 𝑍) |
109 | 6 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → 𝐺:ℕ⟶𝑍) |
110 | | simpr 477 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → 𝑆 ⊆ ℕ) |
111 | | soisores 6577 |
. . 3
⊢ ((( <
Or ℕ ∧ < Or 𝑍)
∧ (𝐺:ℕ⟶𝑍 ∧ 𝑆 ⊆ ℕ)) → ((𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
112 | 105, 108,
109, 110, 111 | syl22anc 1327 |
. 2
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → ((𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
113 | 100, 112 | mpbird 247 |
1
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → (𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆))) |