| Step | Hyp | Ref
| Expression |
| 1 | | isercoll.z |
. . . . . . . . . . 11
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | uzssz 12899 |
. . . . . . . . . . 11
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
| 3 | 1, 2 | eqsstri 4030 |
. . . . . . . . . 10
⊢ 𝑍 ⊆
ℤ |
| 4 | | zssre 12620 |
. . . . . . . . . 10
⊢ ℤ
⊆ ℝ |
| 5 | 3, 4 | sstri 3993 |
. . . . . . . . 9
⊢ 𝑍 ⊆
ℝ |
| 6 | | isercoll.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:ℕ⟶𝑍) |
| 7 | 6 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝐺:ℕ⟶𝑍) |
| 8 | | simplrl 777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℕ) |
| 9 | 7, 8 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) ∈ 𝑍) |
| 10 | 5, 9 | sselid 3981 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) ∈ ℝ) |
| 11 | | simplrr 778 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℕ) |
| 12 | 11 | nnred 12281 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ) |
| 13 | 10, 12 | resubcld 11691 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) ∈ ℝ) |
| 14 | 8 | nnred 12281 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ) |
| 15 | 10, 14 | resubcld 11691 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑥) ∈ ℝ) |
| 16 | 7, 11 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑦) ∈ 𝑍) |
| 17 | 5, 16 | sselid 3981 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑦) ∈ ℝ) |
| 18 | 17, 12 | resubcld 11691 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑦) − 𝑦) ∈ ℝ) |
| 19 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 < 𝑦) |
| 20 | 14, 12, 10, 19 | ltsub2dd 11876 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑥) − 𝑥)) |
| 21 | 8 | nnzd 12640 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℤ) |
| 22 | 11 | nnzd 12640 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℤ) |
| 23 | 14, 12, 19 | ltled 11409 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑥 ≤ 𝑦) |
| 24 | | eluz2 12884 |
. . . . . . . . . 10
⊢ (𝑦 ∈
(ℤ≥‘𝑥) ↔ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑥 ≤ 𝑦)) |
| 25 | 21, 22, 23, 24 | syl3anbrc 1344 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → 𝑦 ∈ (ℤ≥‘𝑥)) |
| 26 | | elfzuz 13560 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑥...𝑦) → 𝑘 ∈ (ℤ≥‘𝑥)) |
| 27 | | eluznn 12960 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝑥)) → 𝑘 ∈ ℕ) |
| 28 | 8, 27 | sylan 580 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → 𝑘 ∈ ℕ) |
| 29 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) |
| 30 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘) |
| 31 | 29, 30 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑘) − 𝑘)) |
| 32 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛)) = (𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛)) |
| 33 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘𝑘) − 𝑘) ∈ V |
| 34 | 31, 32, 33 | fvmpt 7016 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) = ((𝐺‘𝑘) − 𝑘)) |
| 35 | 34 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) = ((𝐺‘𝑘) − 𝑘)) |
| 36 | 7 | ffvelcdmda 7104 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑍) |
| 37 | 5, 36 | sselid 3981 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
| 38 | | nnre 12273 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℝ) |
| 39 | 38 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ) |
| 40 | 37, 39 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ∈ ℝ) |
| 41 | 35, 40 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
| 42 | 28, 41 | syldan 591 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
| 43 | 26, 42 | sylan2 593 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (𝑥...𝑦)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ∈ ℝ) |
| 44 | | elfzuz 13560 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑥...(𝑦 − 1)) → 𝑘 ∈ (ℤ≥‘𝑥)) |
| 45 | | peano2nn 12278 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
| 46 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺:ℕ⟶𝑍 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑍) |
| 47 | 7, 45, 46 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑍) |
| 48 | 5, 47 | sselid 3981 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℝ) |
| 49 | | peano2rem 11576 |
. . . . . . . . . . . . . . 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 | | isercoll.i |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
| 52 | 51 | ad4ant14 752 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1))) |
| 53 | 3, 36 | sselid 3981 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℤ) |
| 54 | 3, 47 | sselid 3981 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℤ) |
| 55 | | zltlem1 12670 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐺‘𝑘) ∈ ℤ ∧ (𝐺‘(𝑘 + 1)) ∈ ℤ) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1))) |
| 56 | 53, 54, 55 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1))) |
| 57 | 52, 56 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ ((𝐺‘(𝑘 + 1)) − 1)) |
| 58 | 37, 50, 39, 57 | lesub1dd 11879 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ≤ (((𝐺‘(𝑘 + 1)) − 1) − 𝑘)) |
| 59 | 48 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ ℂ) |
| 60 | | 1cnd 11256 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 1 ∈
ℂ) |
| 61 | 39 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ) |
| 62 | 59, 60, 61 | sub32d 11652 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 1) − 𝑘) = (((𝐺‘(𝑘 + 1)) − 𝑘) − 1)) |
| 63 | 59, 61, 60 | subsub4d 11651 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 𝑘) − 1) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 64 | 62, 63 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1)) − 1) − 𝑘) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 65 | 58, 64 | breqtrd 5169 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) − 𝑘) ≤ ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 66 | 45 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
| 67 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑘 + 1))) |
| 68 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → 𝑛 = (𝑘 + 1)) |
| 69 | 67, 68 | oveq12d 7449 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 70 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ ((𝐺‘(𝑘 + 1)) − (𝑘 + 1)) ∈ V |
| 71 | 69, 32, 70 | fvmpt 7016 |
. . . . . . . . . . . . 13
⊢ ((𝑘 + 1) ∈ ℕ →
((𝑛 ∈ ℕ ↦
((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1)) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 72 | 66, 71 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1)) = ((𝐺‘(𝑘 + 1)) − (𝑘 + 1))) |
| 73 | 65, 35, 72 | 3brtr4d 5175 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1))) |
| 74 | 28, 73 | syldan 591 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) ∧ 𝑘 ∈ (ℤ≥‘𝑥)) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑘) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘(𝑘 + 1))) |
| 75 | 44, 74 | sylan2 593 |
. . . . . . . . 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))) |
| 76 | 25, 43, 75 | monoord 14073 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) ≤ ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦)) |
| 77 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑥 → (𝐺‘𝑛) = (𝐺‘𝑥)) |
| 78 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑥 → 𝑛 = 𝑥) |
| 79 | 77, 78 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑥 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑥) − 𝑥)) |
| 80 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑥) − 𝑥) ∈ V |
| 81 | 79, 32, 80 | fvmpt 7016 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) = ((𝐺‘𝑥) − 𝑥)) |
| 82 | 8, 81 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑥) = ((𝐺‘𝑥) − 𝑥)) |
| 83 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑦 → (𝐺‘𝑛) = (𝐺‘𝑦)) |
| 84 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑦 → 𝑛 = 𝑦) |
| 85 | 83, 84 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑦 → ((𝐺‘𝑛) − 𝑛) = ((𝐺‘𝑦) − 𝑦)) |
| 86 | | ovex 7464 |
. . . . . . . . . 10
⊢ ((𝐺‘𝑦) − 𝑦) ∈ V |
| 87 | 85, 32, 86 | fvmpt 7016 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦) = ((𝐺‘𝑦) − 𝑦)) |
| 88 | 11, 87 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝑛 ∈ ℕ ↦ ((𝐺‘𝑛) − 𝑛))‘𝑦) = ((𝐺‘𝑦) − 𝑦)) |
| 89 | 76, 82, 88 | 3brtr3d 5174 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑥) ≤ ((𝐺‘𝑦) − 𝑦)) |
| 90 | 13, 15, 18, 20, 89 | ltletrd 11421 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑦) − 𝑦)) |
| 91 | 10, 17, 12 | ltsub1d 11872 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → ((𝐺‘𝑥) < (𝐺‘𝑦) ↔ ((𝐺‘𝑥) − 𝑦) < ((𝐺‘𝑦) − 𝑦))) |
| 92 | 90, 91 | mpbird 257 |
. . . . 5
⊢ (((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) ∧ 𝑥 < 𝑦) → (𝐺‘𝑥) < (𝐺‘𝑦)) |
| 93 | 92 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ)) → (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
| 94 | 93 | ralrimivva 3202 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ ℕ ∀𝑦 ∈ ℕ (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
| 95 | | ss2ralv 4054 |
. . 3
⊢ (𝑆 ⊆ ℕ →
(∀𝑥 ∈ ℕ
∀𝑦 ∈ ℕ
(𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
| 96 | 94, 95 | mpan9 506 |
. 2
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦))) |
| 97 | | nnssre 12270 |
. . . . 5
⊢ ℕ
⊆ ℝ |
| 98 | | ltso 11341 |
. . . . 5
⊢ < Or
ℝ |
| 99 | | soss 5612 |
. . . . 5
⊢ (ℕ
⊆ ℝ → ( < Or ℝ → < Or
ℕ)) |
| 100 | 97, 98, 99 | mp2 9 |
. . . 4
⊢ < Or
ℕ |
| 101 | 100 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → < Or
ℕ) |
| 102 | | soss 5612 |
. . . . 5
⊢ (𝑍 ⊆ ℝ → ( <
Or ℝ → < Or 𝑍)) |
| 103 | 5, 98, 102 | mp2 9 |
. . . 4
⊢ < Or
𝑍 |
| 104 | 103 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → < Or 𝑍) |
| 105 | 6 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → 𝐺:ℕ⟶𝑍) |
| 106 | | simpr 484 |
. . 3
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → 𝑆 ⊆ ℕ) |
| 107 | | soisores 7347 |
. . 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)))) |
| 108 | 101, 104,
105, 106, 107 | syl22anc 839 |
. 2
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → ((𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 < 𝑦 → (𝐺‘𝑥) < (𝐺‘𝑦)))) |
| 109 | 96, 108 | mpbird 257 |
1
⊢ ((𝜑 ∧ 𝑆 ⊆ ℕ) → (𝐺 ↾ 𝑆) Isom < , < (𝑆, (𝐺 “ 𝑆))) |