Step | Hyp | Ref
| Expression |
1 | | dftrcl3 38012 |
. 2
⊢ t+ =
(𝑎 ∈ V ↦
∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖)) |
2 | | dfrcl4 37968 |
. 2
⊢ r* =
(𝑏 ∈ V ↦
∪ 𝑗 ∈ {0, 1} (𝑏↑𝑟𝑗)) |
3 | | dfrtrcl3 38025 |
. 2
⊢ t* =
(𝑐 ∈ V ↦
∪ 𝑘 ∈ ℕ0 (𝑐↑𝑟𝑘)) |
4 | | nnex 11026 |
. 2
⊢ ℕ
∈ V |
5 | | prex 4909 |
. 2
⊢ {0, 1}
∈ V |
6 | | df-n0 11293 |
. . 3
⊢
ℕ0 = (ℕ ∪ {0}) |
7 | | df-pr 4180 |
. . . . . 6
⊢ {0, 1} =
({0} ∪ {1}) |
8 | 7 | equncomi 3759 |
. . . . 5
⊢ {0, 1} =
({1} ∪ {0}) |
9 | 8 | uneq2i 3764 |
. . . 4
⊢ (ℕ
∪ {0, 1}) = (ℕ ∪ ({1} ∪ {0})) |
10 | | unass 3770 |
. . . 4
⊢ ((ℕ
∪ {1}) ∪ {0}) = (ℕ ∪ ({1} ∪ {0})) |
11 | | 1nn 11031 |
. . . . . . 7
⊢ 1 ∈
ℕ |
12 | | snssi 4339 |
. . . . . . 7
⊢ (1 ∈
ℕ → {1} ⊆ ℕ) |
13 | 11, 12 | ax-mp 5 |
. . . . . 6
⊢ {1}
⊆ ℕ |
14 | | ssequn2 3786 |
. . . . . 6
⊢ ({1}
⊆ ℕ ↔ (ℕ ∪ {1}) = ℕ) |
15 | 13, 14 | mpbi 220 |
. . . . 5
⊢ (ℕ
∪ {1}) = ℕ |
16 | 15 | uneq1i 3763 |
. . . 4
⊢ ((ℕ
∪ {1}) ∪ {0}) = (ℕ ∪ {0}) |
17 | 9, 10, 16 | 3eqtr2ri 2651 |
. . 3
⊢ (ℕ
∪ {0}) = (ℕ ∪ {0, 1}) |
18 | 6, 17 | eqtri 2644 |
. 2
⊢
ℕ0 = (ℕ ∪ {0, 1}) |
19 | | oveq2 6658 |
. . . 4
⊢ (𝑘 = 𝑖 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑖)) |
20 | 19 | cbviunv 4559 |
. . 3
⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ ℕ (𝑑↑𝑟𝑖) |
21 | | ss2iun 4536 |
. . . 4
⊢
(∀𝑖 ∈
ℕ (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) → ∪
𝑖 ∈ ℕ (𝑑↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖)) |
22 | | 1ex 10035 |
. . . . . . . 8
⊢ 1 ∈
V |
23 | 22 | prid2 4298 |
. . . . . . 7
⊢ 1 ∈
{0, 1} |
24 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑗 = 1 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟1)) |
25 | | vex 3203 |
. . . . . . . . . 10
⊢ 𝑑 ∈ V |
26 | | relexp1g 13766 |
. . . . . . . . . 10
⊢ (𝑑 ∈ V → (𝑑↑𝑟1) =
𝑑) |
27 | 25, 26 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝑑↑𝑟1) =
𝑑 |
28 | 24, 27 | syl6eq 2672 |
. . . . . . . 8
⊢ (𝑗 = 1 → (𝑑↑𝑟𝑗) = 𝑑) |
29 | 28 | ssiun2s 4564 |
. . . . . . 7
⊢ (1 ∈
{0, 1} → 𝑑 ⊆
∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)) |
30 | 23, 29 | ax-mp 5 |
. . . . . 6
⊢ 𝑑 ⊆ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) |
31 | 30 | a1i 11 |
. . . . 5
⊢ (𝑖 ∈ ℕ → 𝑑 ⊆ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)) |
32 | | ovex 6678 |
. . . . . . 7
⊢ (𝑑↑𝑟𝑗) ∈ V |
33 | 5, 32 | iunex 7147 |
. . . . . 6
⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) ∈ V |
34 | 33 | a1i 11 |
. . . . 5
⊢ (𝑖 ∈ ℕ → ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) ∈ V) |
35 | | nnnn0 11299 |
. . . . 5
⊢ (𝑖 ∈ ℕ → 𝑖 ∈
ℕ0) |
36 | 31, 34, 35 | relexpss1d 37997 |
. . . 4
⊢ (𝑖 ∈ ℕ → (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖)) |
37 | 21, 36 | mprg 2926 |
. . 3
⊢ ∪ 𝑖 ∈ ℕ (𝑑↑𝑟𝑖) ⊆ ∪
𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) |
38 | 20, 37 | eqsstri 3635 |
. 2
⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪
𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) |
39 | | oveq2 6658 |
. . . . 5
⊢ (𝑖 = 1 → (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪
𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟1)) |
40 | | relexp1g 13766 |
. . . . . . 7
⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) ∈ V → (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)) |
41 | 33, 40 | ax-mp 5 |
. . . . . 6
⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) |
42 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘)) |
43 | 42 | cbviunv 4559 |
. . . . . 6
⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) = ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) |
44 | 41, 43 | eqtri 2644 |
. . . . 5
⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) |
45 | 39, 44 | syl6eq 2672 |
. . . 4
⊢ (𝑖 = 1 → (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘)) |
46 | 45 | ssiun2s 4564 |
. . 3
⊢ (1 ∈
ℕ → ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ⊆ ∪
𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖)) |
47 | 11, 46 | ax-mp 5 |
. 2
⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ⊆ ∪
𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) |
48 | | iunss 4561 |
. . . 4
⊢ (∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪
𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪
𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘)) |
49 | | iuneq1 4534 |
. . . . . . . 8
⊢ ({0, 1} =
({0} ∪ {1}) → ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) = ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑𝑟𝑗)) |
50 | 7, 49 | ax-mp 5 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) = ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑𝑟𝑗) |
51 | | iunxun 4605 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑𝑟𝑗) = (∪
𝑗 ∈ {0} (𝑑↑𝑟𝑗) ∪ ∪ 𝑗 ∈ {1} (𝑑↑𝑟𝑗)) |
52 | | c0ex 10034 |
. . . . . . . . 9
⊢ 0 ∈
V |
53 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑗 = 0 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟0)) |
54 | 52, 53 | iunxsn 4603 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ {0} (𝑑↑𝑟𝑗) = (𝑑↑𝑟0) |
55 | 22, 24 | iunxsn 4603 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ {1} (𝑑↑𝑟𝑗) = (𝑑↑𝑟1) |
56 | 54, 55 | uneq12i 3765 |
. . . . . . 7
⊢ (∪ 𝑗 ∈ {0} (𝑑↑𝑟𝑗) ∪ ∪
𝑗 ∈ {1} (𝑑↑𝑟𝑗)) = ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)) |
57 | 50, 51, 56 | 3eqtri 2648 |
. . . . . 6
⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗) = ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)) |
58 | 57 | oveq1i 6660 |
. . . . 5
⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) = (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑖) |
59 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = 1 → (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑥) = (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟1)) |
60 | 59 | sseq1d 3632 |
. . . . . 6
⊢ (𝑥 = 1 → ((((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑥) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ↔ (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟1)
⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘))) |
61 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) = (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦)) |
62 | 61 | sseq1d 3632 |
. . . . . 6
⊢ (𝑥 = 𝑦 → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ↔ (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘))) |
63 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = (𝑦 + 1) → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) = (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1))) |
64 | 63 | sseq1d 3632 |
. . . . . 6
⊢ (𝑥 = (𝑦 + 1) → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ↔ (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘))) |
65 | | oveq2 6658 |
. . . . . . 7
⊢ (𝑥 = 𝑖 → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) = (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑖)) |
66 | 65 | sseq1d 3632 |
. . . . . 6
⊢ (𝑥 = 𝑖 → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑥) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ↔ (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑖) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘))) |
67 | | ovex 6678 |
. . . . . . . . 9
⊢ (𝑑↑𝑟0)
∈ V |
68 | | ovex 6678 |
. . . . . . . . 9
⊢ (𝑑↑𝑟1)
∈ V |
69 | 67, 68 | unex 6956 |
. . . . . . . 8
⊢ ((𝑑↑𝑟0)
∪ (𝑑↑𝑟1)) ∈
V |
70 | | relexp1g 13766 |
. . . . . . . 8
⊢ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1)) ∈ V →
(((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟1)
= ((𝑑↑𝑟0) ∪
(𝑑↑𝑟1))) |
71 | 69, 70 | ax-mp 5 |
. . . . . . 7
⊢ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟1)
= ((𝑑↑𝑟0) ∪
(𝑑↑𝑟1)) |
72 | | 0nn0 11307 |
. . . . . . . . 9
⊢ 0 ∈
ℕ0 |
73 | | oveq2 6658 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟0)) |
74 | 73 | ssiun2s 4564 |
. . . . . . . . 9
⊢ (0 ∈
ℕ0 → (𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) |
75 | 72, 74 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑑↑𝑟0)
⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
76 | | 1nn0 11308 |
. . . . . . . . 9
⊢ 1 ∈
ℕ0 |
77 | | oveq2 6658 |
. . . . . . . . . 10
⊢ (𝑘 = 1 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟1)) |
78 | 77 | ssiun2s 4564 |
. . . . . . . . 9
⊢ (1 ∈
ℕ0 → (𝑑↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) |
79 | 76, 78 | ax-mp 5 |
. . . . . . . 8
⊢ (𝑑↑𝑟1)
⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
80 | 75, 79 | unssi 3788 |
. . . . . . 7
⊢ ((𝑑↑𝑟0)
∪ (𝑑↑𝑟1)) ⊆
∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
81 | 71, 80 | eqsstri 3635 |
. . . . . 6
⊢ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟1)
⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
82 | | simpl 473 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) → 𝑦 ∈ ℕ) |
83 | | relexpsucnnr 13765 |
. . . . . . . . 9
⊢ ((((𝑑↑𝑟0)
∪ (𝑑↑𝑟1)) ∈ V ∧
𝑦 ∈ ℕ) →
(((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)))) |
84 | 69, 82, 83 | sylancr 695 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)))) |
85 | | coss1 5277 |
. . . . . . . . . 10
⊢ ((((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))) ⊆ (∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)))) |
86 | | coundi 5636 |
. . . . . . . . . . 11
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))) =
((∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟0)) ∪ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1))) |
87 | | relexp0g 13762 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ V → (𝑑↑𝑟0) = (
I ↾ (dom 𝑑 ∪ ran
𝑑))) |
88 | 25, 87 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ (𝑑↑𝑟0) = (
I ↾ (dom 𝑑 ∪ ran
𝑑)) |
89 | 88 | coeq2i 5282 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟0)) = (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) |
90 | | coiun1 37944 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ∪
𝑘 ∈
ℕ0 ((𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) |
91 | | coires1 5653 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) |
92 | 91 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ ((𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))) |
93 | 92 | iuneq2i 4539 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ∪
𝑘 ∈
ℕ0 ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) |
94 | 89, 90, 93 | 3eqtri 2648 |
. . . . . . . . . . . . 13
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟0)) = ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) |
95 | | ss2iun 4536 |
. . . . . . . . . . . . . 14
⊢
(∀𝑘 ∈
ℕ0 ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑𝑟𝑘) → ∪
𝑘 ∈
ℕ0 ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) |
96 | | resss 5422 |
. . . . . . . . . . . . . . 15
⊢ ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑𝑟𝑘) |
97 | 96 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑𝑟𝑘)) |
98 | 95, 97 | mprg 2926 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
99 | 94, 98 | eqsstri 3635 |
. . . . . . . . . . . 12
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟0)) ⊆
∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
100 | | coiun1 37944 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) = ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) |
101 | | iunss2 4565 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑘 ∈
ℕ0 ∃𝑖 ∈ ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖) → ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
∪ 𝑖 ∈ ℕ0 (𝑑↑𝑟𝑖)) |
102 | | peano2nn0 11333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ (𝑘 + 1) ∈
ℕ0) |
103 | | sbcel1v 3495 |
. . . . . . . . . . . . . . . . . . 19
⊢
([(𝑘 + 1) /
𝑖]𝑖 ∈ ℕ0 ↔ (𝑘 + 1) ∈
ℕ0) |
104 | 102, 103 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ [(𝑘 + 1) /
𝑖]𝑖 ∈ ℕ0) |
105 | | relexpaddss 38010 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ ℕ0
∧ 1 ∈ ℕ0 ∧ 𝑑 ∈ V) → ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟(𝑘 + 1))) |
106 | 76, 25, 105 | mp3an23 1416 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ ℕ0
→ ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟(𝑘 + 1))) |
107 | | ovex 6678 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 + 1) ∈ V |
108 | | csbconstg 3546 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 + 1) ∈ V →
⦋(𝑘 + 1) /
𝑖⦌((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) = ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1))) |
109 | 107, 108 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢
⦋(𝑘 +
1) / 𝑖⦌((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) = ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) |
110 | | csbov2g 6691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 + 1) ∈ V →
⦋(𝑘 + 1) /
𝑖⦌(𝑑↑𝑟𝑖) = (𝑑↑𝑟⦋(𝑘 + 1) / 𝑖⦌𝑖)) |
111 | | csbvarg 4003 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑘 + 1) ∈ V →
⦋(𝑘 + 1) /
𝑖⦌𝑖 = (𝑘 + 1)) |
112 | 111 | oveq2d 6666 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 + 1) ∈ V → (𝑑↑𝑟⦋(𝑘 + 1) / 𝑖⦌𝑖) = (𝑑↑𝑟(𝑘 + 1))) |
113 | 110, 112 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 + 1) ∈ V →
⦋(𝑘 + 1) /
𝑖⦌(𝑑↑𝑟𝑖) = (𝑑↑𝑟(𝑘 + 1))) |
114 | 107, 113 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢
⦋(𝑘 +
1) / 𝑖⦌(𝑑↑𝑟𝑖) = (𝑑↑𝑟(𝑘 + 1)) |
115 | 106, 109,
114 | 3sstr4g 3646 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ ℕ0
→ ⦋(𝑘 +
1) / 𝑖⦌((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
⦋(𝑘 + 1) /
𝑖⦌(𝑑↑𝑟𝑖)) |
116 | | sbcssg 4085 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 + 1) ∈ V →
([(𝑘 + 1) / 𝑖]((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖) ↔ ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
⦋(𝑘 + 1) /
𝑖⦌(𝑑↑𝑟𝑖))) |
117 | 107, 116 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
([(𝑘 + 1) /
𝑖]((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖) ↔ ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
⦋(𝑘 + 1) /
𝑖⦌(𝑑↑𝑟𝑖)) |
118 | 115, 117 | sylibr 224 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ0
→ [(𝑘 + 1) /
𝑖]((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖)) |
119 | | sbcan 3478 |
. . . . . . . . . . . . . . . . . 18
⊢
([(𝑘 + 1) /
𝑖](𝑖 ∈ ℕ0
∧ ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖)) ↔ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0 ∧
[(𝑘 + 1) / 𝑖]((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖))) |
120 | 104, 118,
119 | sylanbrc 698 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ0
→ [(𝑘 + 1) /
𝑖](𝑖 ∈ ℕ0
∧ ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖))) |
121 | 120 | spesbcd 3522 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ℕ0
→ ∃𝑖(𝑖 ∈ ℕ0
∧ ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖))) |
122 | | df-rex 2918 |
. . . . . . . . . . . . . . . 16
⊢
(∃𝑖 ∈
ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖) ↔ ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖))) |
123 | 121, 122 | sylibr 224 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ ∃𝑖 ∈
ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆ (𝑑↑𝑟𝑖)) |
124 | 101, 123 | mprg 2926 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑘 ∈ ℕ0 ((𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
∪ 𝑖 ∈ ℕ0 (𝑑↑𝑟𝑖) |
125 | 100, 124 | eqsstri 3635 |
. . . . . . . . . . . . 13
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
∪ 𝑖 ∈ ℕ0 (𝑑↑𝑟𝑖) |
126 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑘 → (𝑑↑𝑟𝑖) = (𝑑↑𝑟𝑘)) |
127 | 126 | cbviunv 4559 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑖 ∈ ℕ0 (𝑑↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
128 | 125, 127 | sseqtri 3637 |
. . . . . . . . . . . 12
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1)) ⊆
∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
129 | 99, 128 | unssi 3788 |
. . . . . . . . . . 11
⊢
((∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟0)) ∪ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ (𝑑↑𝑟1))) ⊆
∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
130 | 86, 129 | eqsstri 3635 |
. . . . . . . . . 10
⊢ (∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1)))
⊆ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) |
131 | 85, 130 | syl6ss 3615 |
. . . . . . . . 9
⊢ ((((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))) ⊆ ∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘)) |
132 | 131 | adantl 482 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) → ((((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟𝑦) ∘ ((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))) ⊆ ∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘)) |
133 | 84, 132 | eqsstrd 3639 |
. . . . . . 7
⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘)) |
134 | 133 | ex 450 |
. . . . . 6
⊢ (𝑦 ∈ ℕ → ((((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑦) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) → (((𝑑↑𝑟0) ∪ (𝑑↑𝑟1))↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘))) |
135 | 60, 62, 64, 66, 81, 134 | nnind 11038 |
. . . . 5
⊢ (𝑖 ∈ ℕ → (((𝑑↑𝑟0)
∪ (𝑑↑𝑟1))↑𝑟𝑖) ⊆ ∪
𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘)) |
136 | 58, 135 | syl5eqss 3649 |
. . . 4
⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪
𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘)) |
137 | 48, 136 | mprgbir 2927 |
. . 3
⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪
𝑘 ∈
ℕ0 (𝑑↑𝑟𝑘) |
138 | | iuneq1 4534 |
. . . 4
⊢
(ℕ0 = (ℕ ∪ {0, 1}) → ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑↑𝑟𝑘)) |
139 | 18, 138 | ax-mp 5 |
. . 3
⊢ ∪ 𝑘 ∈ ℕ0 (𝑑↑𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑↑𝑟𝑘) |
140 | 137, 139 | sseqtri 3637 |
. 2
⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪
𝑘 ∈ (ℕ ∪ {0,
1})(𝑑↑𝑟𝑘) |
141 | 1, 2, 3, 4, 5, 18,
38, 47, 140 | comptiunov2i 37998 |
1
⊢ (t+
∘ r*) = t* |