Proof of Theorem relexpaddss
Step | Hyp | Ref
| Expression |
1 | | elnn0 11921 |
. . 3
⊢ (𝑀 ∈ ℕ0
↔ (𝑀 ∈ ℕ
∨ 𝑀 =
0)) |
2 | | elnn0 11921 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
3 | 2 | biimpi 219 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
4 | | relexpaddnn 14443 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
5 | | eqimss 3944 |
. . . . . . . 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 1117 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
8 | | elnn1uz2 12350 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈
(ℤ≥‘2))) |
9 | | relco 6067 |
. . . . . . . . . . . . . 14
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) |
10 | | dfrel2 6011 |
. . . . . . . . . . . . . . 15
⊢ (Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ↔ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
11 | 10 | biimpi 219 |
. . . . . . . . . . . . . 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 5718 |
. . . . . . . . . . . . . . . . 17
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
14 | | cnvresid 6407 |
. . . . . . . . . . . . . . . . . 18
⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
15 | 14 | coeq2i 5693 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
16 | | coires1 6087 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
17 | 13, 15, 16 | 3eqtri 2786 |
. . . . . . . . . . . . . . . 16
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
18 | | eqimss 3944 |
. . . . . . . . . . . . . . . 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 5705 |
. . . . . . . . . . . . . . 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 5841 |
. . . . . . . . . . . . . . 15
⊢ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅 |
23 | | cnvss 5705 |
. . . . . . . . . . . . . . 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 3897 |
. . . . . . . . . . . . 13
⊢ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
26 | 12, 25 | eqsstrri 3923 |
. . . . . . . . . . . 12
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
27 | | cnvcnvss 6016 |
. . . . . . . . . . . 12
⊢ ◡◡𝑅 ⊆ 𝑅 |
28 | 26, 27 | sstri 3897 |
. . . . . . . . . . 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 1134 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
31 | 30 | oveq2d 7159 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
32 | | relexp0g 14414 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
33 | 32 | 3ad2ant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
34 | 31, 33 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
35 | | simp2 1135 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 1) |
36 | 35 | oveq2d 7159 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟1)) |
37 | | relexp1g 14418 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) |
38 | 37 | 3ad2ant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
39 | 36, 38 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = 𝑅) |
40 | 34, 39 | coeq12d 5697 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
41 | 30, 35 | oveq12d 7161 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 1)) |
42 | | 1cnd 10659 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
43 | 42 | addid2d 10864 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (0 + 1) = 1) |
44 | 41, 43 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
45 | 44 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
46 | 45, 38 | eqtrd 2794 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
47 | 29, 40, 46 | 3sstr4d 3935 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
48 | 47 | 3exp 1117 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
49 | | coires1 6087 |
. . . . . . . . . . . . . 14
⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) |
50 | | simp2 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈
(ℤ≥‘2)) |
51 | | cnvexg 7627 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V) |
52 | 51 | 3ad2ant3 1133 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V) |
53 | | relexpuzrel 14444 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈
(ℤ≥‘2) ∧ ◡𝑅 ∈ V) → Rel (◡𝑅↑𝑟𝑀)) |
54 | 50, 52, 53 | syl2anc 588 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → Rel (◡𝑅↑𝑟𝑀)) |
55 | | eluz2nn 12309 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℕ) |
56 | 50, 55 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ) |
57 | | relexpnndm 14433 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℕ ∧ ◡𝑅 ∈ V) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅) |
58 | 56, 52, 57 | syl2anc 588 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ dom ◡𝑅) |
59 | | df-rn 5528 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 = dom ◡𝑅 |
60 | | ssun2 4074 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
61 | 59, 60 | eqsstrri 3923 |
. . . . . . . . . . . . . . . 16
⊢ dom ◡𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
62 | 58, 61 | sstrdi 3900 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
63 | | relssres 5857 |
. . . . . . . . . . . . . . 15
⊢ ((Rel
(◡𝑅↑𝑟𝑀) ∧ dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀)) |
64 | 54, 62, 63 | syl2anc 588 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (◡𝑅↑𝑟𝑀)) |
65 | 49, 64 | syl5eq 2806 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅↑𝑟𝑀)) |
66 | | cnvco 5718 |
. . . . . . . . . . . . . 14
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
67 | | eluzge2nn0 12312 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈
ℕ0) |
68 | 50, 67 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈
ℕ0) |
69 | | simp3 1136 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
70 | | relexpcnv 14427 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ0
∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀)) |
71 | 68, 69, 70 | syl2anc 588 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑀) = (◡𝑅↑𝑟𝑀)) |
72 | 14 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
73 | 71, 72 | coeq12d 5697 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
74 | 66, 73 | syl5eq 2806 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
75 | 65, 74, 71 | 3eqtr4d 2804 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀)) |
76 | | relco 6067 |
. . . . . . . . . . . . 13
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ (𝑅↑𝑟𝑀)) |
77 | | relexpuzrel 14444 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
78 | 77 | 3adant1 1128 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
79 | | cnveqb 6018 |
. . . . . . . . . . . . 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))) |
80 | 76, 78, 79 | sylancr 591 |
. . . . . . . . . . . 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))) |
81 | 75, 80 | mpbird 260 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀)) |
82 | | simp1 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
83 | 82 | oveq2d 7159 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
84 | 32 | 3ad2ant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
85 | 83, 84 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
86 | 85 | coeq1d 5694 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀))) |
87 | 82 | oveq1d 7158 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀)) |
88 | | eluzelcn 12279 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℂ) |
89 | 50, 88 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ) |
90 | 89 | addid2d 10864 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀) |
91 | 87, 90 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀) |
92 | 91 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑀)) |
93 | 81, 86, 92 | 3eqtr4d 2804 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
94 | 93, 5 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
95 | 94 | 3exp 1117 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2)
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
96 | 48, 95 | jaod 857 |
. . . . . . 7
⊢ (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2))
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
97 | 8, 96 | syl5bi 245 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
98 | 7, 97 | jaoi 855 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
99 | 3, 98 | syl 17 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑀 ∈ ℕ
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
100 | | elnn1uz2 12350 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
101 | 100 | biimpi 219 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
102 | | coires1 6087 |
. . . . . . . . . . . 12
⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
103 | | resss 5841 |
. . . . . . . . . . . 12
⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅 |
104 | 102, 103 | eqsstri 3922 |
. . . . . . . . . . 11
⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅 |
105 | 104 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅) |
106 | | simp1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 1) |
107 | 106 | oveq2d 7159 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1)) |
108 | 37 | 3ad2ant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
109 | 107, 108 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = 𝑅) |
110 | | simp2 1135 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
111 | 110 | oveq2d 7159 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
112 | 32 | 3ad2ant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
113 | 111, 112 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
114 | 109, 113 | coeq12d 5697 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
115 | 106, 110 | oveq12d 7161 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (1 + 0)) |
116 | | 1cnd 10659 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
117 | 116 | addid1d 10863 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (1 + 0) = 1) |
118 | 115, 117 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
119 | 118 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
120 | 119, 108 | eqtrd 2794 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
121 | 105, 114,
120 | 3sstr4d 3935 |
. . . . . . . . 9
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
122 | 121 | 3exp 1117 |
. . . . . . . 8
⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
123 | | coires1 6087 |
. . . . . . . . . . . 12
⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) |
124 | | relexpuzrel 14444 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
125 | 124 | 3adant2 1129 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
126 | | simp1 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈
(ℤ≥‘2)) |
127 | | eluz2nn 12309 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℕ) |
128 | 126, 127 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) |
129 | | simp3 1136 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
130 | | relexpnndm 14433 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
131 | 128, 129,
130 | syl2anc 588 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) |
132 | | ssun1 4073 |
. . . . . . . . . . . . . 14
⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
133 | 131, 132 | sstrdi 3900 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
134 | | relssres 5857 |
. . . . . . . . . . . . 13
⊢ ((Rel
(𝑅↑𝑟𝑁) ∧ dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁)) |
135 | 125, 133,
134 | syl2anc 588 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅↑𝑟𝑁)) |
136 | 123, 135 | syl5eq 2806 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁)) |
137 | | simp2 1135 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
138 | 137 | oveq2d 7159 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
139 | 32 | 3ad2ant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
140 | 138, 139 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
141 | 140 | coeq2d 5695 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
142 | 137 | oveq2d 7159 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0)) |
143 | | eluzelcn 12279 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℂ) |
144 | 126, 143 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ) |
145 | 144 | addid1d 10863 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁) |
146 | 142, 145 | eqtrd 2794 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁) |
147 | 146 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁)) |
148 | 136, 141,
147 | 3eqtr4d 2804 |
. . . . . . . . . 10
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
149 | 148, 5 | syl 17 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
150 | 149 | 3exp 1117 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
151 | 122, 150 | jaoi 855 |
. . . . . . 7
⊢ ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ≥‘2))
→ (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
152 | 101, 151 | syl 17 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
153 | | coires1 6087 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) |
154 | | resres 5829 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) |
155 | | inidm 4119 |
. . . . . . . . . . 11
⊢ ((dom
𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅) |
156 | 155 | reseq2i 5813 |
. . . . . . . . . 10
⊢ ( I
↾ ((dom 𝑅 ∪ ran
𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
157 | 153, 154,
156 | 3eqtri 2786 |
. . . . . . . . 9
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
158 | | simp1 1134 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
159 | 158 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
160 | 32 | 3ad2ant3 1133 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
161 | 159, 160 | eqtrd 2794 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
162 | | simp2 1135 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
163 | 162 | oveq2d 7159 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
164 | 163, 160 | eqtrd 2794 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
165 | 161, 164 | coeq12d 5697 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
166 | 158, 162 | oveq12d 7161 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0)) |
167 | | 00id 10838 |
. . . . . . . . . . . . 13
⊢ (0 + 0) =
0 |
168 | 167 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (0 + 0) = 0) |
169 | 166, 168 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0) |
170 | 169 | oveq2d 7159 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0)) |
171 | 170, 160 | eqtrd 2794 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
172 | 157, 165,
171 | 3eqtr4a 2820 |
. . . . . . . 8
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
173 | 172, 5 | syl 17 |
. . . . . . 7
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
174 | 173 | 3exp 1117 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
175 | 152, 174 | jaoi 855 |
. . . . 5
⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
176 | 3, 175 | syl 17 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
177 | 99, 176 | jaod 857 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ ((𝑀 ∈ ℕ
∨ 𝑀 = 0) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
178 | 1, 177 | syl5bi 245 |
. 2
⊢ (𝑁 ∈ ℕ0
→ (𝑀 ∈
ℕ0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
179 | 178 | 3imp 1109 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝑀 ∈
ℕ0 ∧ 𝑅
∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |