Proof of Theorem relexpaddss
Step | Hyp | Ref
| Expression |
1 | | elnn0 12415 |
. . 3
⊢ (𝑀 ∈ ℕ0
↔ (𝑀 ∈ ℕ
∨ 𝑀 =
0)) |
2 | | elnn0 12415 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
3 | 2 | biimpi 215 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
4 | | relexpaddnn 14936 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
5 | | eqimss 4000 |
. . . . . . . 8
⊢ (((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀)) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
6 | 4, 5 | syl 17 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
7 | 6 | 3exp 1119 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
8 | | elnn1uz2 12850 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈
(ℤ≥‘2))) |
9 | | relco 6060 |
. . . . . . . . . . . . . 14
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) |
10 | | dfrel2 6141 |
. . . . . . . . . . . . . . 15
⊢ (Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ↔ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
11 | 10 | biimpi 215 |
. . . . . . . . . . . . . 14
⊢ (Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) → ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
12 | 9, 11 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) |
13 | | cnvco 5841 |
. . . . . . . . . . . . . . . . 17
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
14 | | cnvresid 6580 |
. . . . . . . . . . . . . . . . . 18
⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
15 | 14 | coeq2i 5816 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
16 | | coires1 6216 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
17 | 13, 15, 16 | 3eqtri 2768 |
. . . . . . . . . . . . . . . 16
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
18 | | eqimss 4000 |
. . . . . . . . . . . . . . . 16
⊢ (◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))) |
19 | 17, 18 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
20 | | cnvss 5828 |
. . . . . . . . . . . . . . 15
⊢ (◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) → ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))) |
21 | 19, 20 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
22 | | resss 5962 |
. . . . . . . . . . . . . . 15
⊢ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅 |
23 | | cnvss 5828 |
. . . . . . . . . . . . . . 15
⊢ ((◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅 → ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡◡𝑅) |
24 | 22, 23 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ◡(◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡◡𝑅 |
25 | 21, 24 | sstri 3953 |
. . . . . . . . . . . . 13
⊢ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
26 | 12, 25 | eqsstrri 3979 |
. . . . . . . . . . . 12
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
27 | | cnvcnvss 6146 |
. . . . . . . . . . . 12
⊢ ◡◡𝑅 ⊆ 𝑅 |
28 | 26, 27 | sstri 3953 |
. . . . . . . . . . 11
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ 𝑅 |
29 | 28 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅) |
30 | | simp1 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
31 | 30 | oveq2d 7373 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
32 | | relexp0g 14907 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
33 | 32 | 3ad2ant3 1135 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
34 | 31, 33 | eqtrd 2776 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
35 | | simp2 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 1) |
36 | 35 | oveq2d 7373 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟1)) |
37 | | relexp1g 14911 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) |
38 | 37 | 3ad2ant3 1135 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
39 | 36, 38 | eqtrd 2776 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = 𝑅) |
40 | 34, 39 | coeq12d 5820 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
41 | 30, 35 | oveq12d 7375 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 1)) |
42 | | 1cnd 11150 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
43 | 42 | addid2d 11356 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (0 + 1) = 1) |
44 | 41, 43 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
45 | 44 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
46 | 45, 38 | eqtrd 2776 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
47 | 29, 40, 46 | 3sstr4d 3991 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
48 | 47 | 3exp 1119 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
49 | | coires1 6216 |
. . . . . . . . . . . . . 14
⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) |
50 | | simp2 1137 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈
(ℤ≥‘2)) |
51 | | cnvexg 7861 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V) |
52 | 51 | 3ad2ant3 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V) |
53 | | relexpuzrel 14937 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈
(ℤ≥‘2) ∧ ◡𝑅 ∈ V) → Rel (◡𝑅↑𝑟𝑀)) |
54 | 50, 52, 53 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → Rel (◡𝑅↑𝑟𝑀)) |
55 | | eluz2nn 12809 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℕ) |
56 | 50, 55 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ) |
57 | | relexpnndm 14926 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ ∧ ◡𝑅 ∈ V) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅) |
58 | 56, 52, 57 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅) |
59 | | df-rn 5644 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 = dom ◡𝑅 |
60 | | ssun2 4133 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
61 | 59, 60 | eqsstrri 3979 |
. . . . . . . . . . . . . . . 16
⊢ dom ◡𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
62 | 58, 61 | sstrdi 3956 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
63 | | relssres 5978 |
. . . . . . . . . . . . . . 15
⊢ ((Rel
(◡𝑅↑𝑟𝑀) ∧ dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀)) |
64 | 54, 62, 63 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀)) |
65 | 49, 64 | eqtrid 2788 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅↑𝑟𝑀)) |
66 | | cnvco 5841 |
. . . . . . . . . . . . . 14
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
67 | | simp3 1138 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
68 | | eluzge2nn0 12812 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈
ℕ0) |
69 | 50, 68 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈
ℕ0) |
70 | 67, 69 | relexpcnvd 14921 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀)) |
71 | 14 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
72 | 70, 71 | coeq12d 5820 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
73 | 66, 72 | eqtrid 2788 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
74 | 65, 73, 70 | 3eqtr4d 2786 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀)) |
75 | | relco 6060 |
. . . . . . . . . . . . 13
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ (𝑅↑𝑟𝑀)) |
76 | | relexpuzrel 14937 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
77 | 76 | 3adant1 1130 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
78 | | cnveqb 6148 |
. . . . . . . . . . . . 13
⊢ ((Rel ((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ (𝑅↑𝑟𝑀)) ∧ Rel (𝑅↑𝑟𝑀)) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀) ↔ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀))) |
79 | 75, 77, 78 | sylancr 587 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((( I ↾ (dom
𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀) ↔ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀))) |
80 | 74, 79 | mpbird 256 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀)) |
81 | | simp1 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
82 | 81 | oveq2d 7373 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
83 | 32 | 3ad2ant3 1135 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
84 | 82, 83 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
85 | 84 | coeq1d 5817 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀))) |
86 | 81 | oveq1d 7372 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀)) |
87 | | eluzelcn 12775 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℂ) |
88 | 50, 87 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ) |
89 | 88 | addid2d 11356 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀) |
90 | 86, 89 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀) |
91 | 90 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑀)) |
92 | 80, 85, 91 | 3eqtr4d 2786 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
93 | 92, 5 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
94 | 93 | 3exp 1119 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2)
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
95 | 48, 94 | jaod 857 |
. . . . . . 7
⊢ (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2))
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
96 | 8, 95 | biimtrid 241 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
97 | 7, 96 | jaoi 855 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
98 | 3, 97 | syl 17 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑀 ∈ ℕ
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
99 | | elnn1uz2 12850 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
100 | 99 | biimpi 215 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
101 | | coires1 6216 |
. . . . . . . . . . . 12
⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
102 | | resss 5962 |
. . . . . . . . . . . 12
⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅 |
103 | 101, 102 | eqsstri 3978 |
. . . . . . . . . . 11
⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅 |
104 | 103 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅) |
105 | | simp1 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 1) |
106 | 105 | oveq2d 7373 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1)) |
107 | 37 | 3ad2ant3 1135 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
108 | 106, 107 | eqtrd 2776 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = 𝑅) |
109 | | simp2 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
110 | 109 | oveq2d 7373 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
111 | 32 | 3ad2ant3 1135 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
112 | 110, 111 | eqtrd 2776 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
113 | 108, 112 | coeq12d 5820 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
114 | 105, 109 | oveq12d 7375 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (1 + 0)) |
115 | | 1cnd 11150 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
116 | 115 | addid1d 11355 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (1 + 0) = 1) |
117 | 114, 116 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
118 | 117 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
119 | 118, 107 | eqtrd 2776 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
120 | 104, 113,
119 | 3sstr4d 3991 |
. . . . . . . . 9
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
121 | 120 | 3exp 1119 |
. . . . . . . 8
⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
122 | | coires1 6216 |
. . . . . . . . . . . 12
⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) |
123 | | relexpuzrel 14937 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
124 | 123 | 3adant2 1131 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
125 | | simp1 1136 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈
(ℤ≥‘2)) |
126 | | eluz2nn 12809 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℕ) |
127 | 125, 126 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) |
128 | | simp3 1138 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
129 | | relexpnndm 14926 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
130 | 127, 128,
129 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
131 | | ssun1 4132 |
. . . . . . . . . . . . . 14
⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
132 | 130, 131 | sstrdi 3956 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
133 | | relssres 5978 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁)) |
134 | 124, 132,
133 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁)) |
135 | 122, 134 | eqtrid 2788 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁)) |
136 | | simp2 1137 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
137 | 136 | oveq2d 7373 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
138 | 32 | 3ad2ant3 1135 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
139 | 137, 138 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
140 | 139 | coeq2d 5818 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
141 | 136 | oveq2d 7373 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0)) |
142 | | eluzelcn 12775 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℂ) |
143 | 125, 142 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ) |
144 | 143 | addid1d 11355 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁) |
145 | 141, 144 | eqtrd 2776 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁) |
146 | 145 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁)) |
147 | 135, 140,
146 | 3eqtr4d 2786 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
148 | 147, 5 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
149 | 148 | 3exp 1119 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
150 | 121, 149 | jaoi 855 |
. . . . . . 7
⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))
→ (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
151 | 100, 150 | syl 17 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
152 | | coires1 6216 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) |
153 | | resres 5950 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) |
154 | | inidm 4178 |
. . . . . . . . . . 11
⊢ ((dom
𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅) |
155 | 154 | reseq2i 5934 |
. . . . . . . . . 10
⊢ ( I
↾ ((dom 𝑅 ∪ ran
𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
156 | 152, 153,
155 | 3eqtri 2768 |
. . . . . . . . 9
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
157 | | simp1 1136 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
158 | 157 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
159 | 32 | 3ad2ant3 1135 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
160 | 158, 159 | eqtrd 2776 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
161 | | simp2 1137 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
162 | 161 | oveq2d 7373 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
163 | 162, 159 | eqtrd 2776 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
164 | 160, 163 | coeq12d 5820 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
165 | 157, 161 | oveq12d 7375 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0)) |
166 | | 00id 11330 |
. . . . . . . . . . . . 13
⊢ (0 + 0) =
0 |
167 | 166 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (0 + 0) = 0) |
168 | 165, 167 | eqtrd 2776 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0) |
169 | 168 | oveq2d 7373 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0)) |
170 | 169, 159 | eqtrd 2776 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
171 | 156, 164,
170 | 3eqtr4a 2802 |
. . . . . . . 8
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
172 | 171, 5 | syl 17 |
. . . . . . 7
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
173 | 172 | 3exp 1119 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
174 | 151, 173 | jaoi 855 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
175 | 3, 174 | syl 17 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
176 | 98, 175 | jaod 857 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ ((𝑀 ∈ ℕ
∨ 𝑀 = 0) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
177 | 1, 176 | biimtrid 241 |
. 2
⊢ (𝑁 ∈ ℕ0
→ (𝑀 ∈
ℕ0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
178 | 177 | 3imp 1111 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
ℕ0 ∧ 𝑅
∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |