Proof of Theorem relexpaddss
| Step | Hyp | Ref
| Expression |
| 1 | | elnn0 12528 |
. . 3
⊢ (𝑀 ∈ ℕ0
↔ (𝑀 ∈ ℕ
∨ 𝑀 =
0)) |
| 2 | | elnn0 12528 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
↔ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
| 3 | 2 | biimpi 216 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝑁 ∈ ℕ
∨ 𝑁 =
0)) |
| 4 | | relexpaddnn 15090 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
| 5 | | eqimss 4042 |
. . . . . . . 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 1120 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 8 | | elnn1uz2 12967 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈
(ℤ≥‘2))) |
| 9 | | relco 6126 |
. . . . . . . . . . . . . 14
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) |
| 10 | | dfrel2 6209 |
. . . . . . . . . . . . . . 15
⊢ (Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ↔ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
| 11 | 10 | biimpi 216 |
. . . . . . . . . . . . . 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 5896 |
. . . . . . . . . . . . . . . . 17
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 14 | | cnvresid 6645 |
. . . . . . . . . . . . . . . . . 18
⊢ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 15 | 14 | coeq2i 5871 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 16 | | coires1 6284 |
. . . . . . . . . . . . . . . . 17
⊢ (◡𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 17 | 13, 15, 16 | 3eqtri 2769 |
. . . . . . . . . . . . . . . 16
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 18 | | eqimss 4042 |
. . . . . . . . . . . . . . . 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 5883 |
. . . . . . . . . . . . . . 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 6019 |
. . . . . . . . . . . . . . 15
⊢ (◡𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ ◡𝑅 |
| 23 | | cnvss 5883 |
. . . . . . . . . . . . . . 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 3993 |
. . . . . . . . . . . . 13
⊢ ◡◡((
I ↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
| 26 | 12, 25 | eqsstrri 4031 |
. . . . . . . . . . . 12
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ 𝑅) ⊆ ◡◡𝑅 |
| 27 | | cnvcnvss 6214 |
. . . . . . . . . . . 12
⊢ ◡◡𝑅 ⊆ 𝑅 |
| 28 | 26, 27 | sstri 3993 |
. . . . . . . . . . 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 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
| 31 | 30 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
| 32 | | relexp0g 15061 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 33 | 32 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 34 | 31, 33 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 35 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 1) |
| 36 | 35 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟1)) |
| 37 | | relexp1g 15065 |
. . . . . . . . . . . . 13
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) |
| 38 | 37 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
| 39 | 36, 38 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = 𝑅) |
| 40 | 34, 39 | coeq12d 5875 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)) |
| 41 | 30, 35 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 1)) |
| 42 | | 1cnd 11256 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
| 43 | 42 | addlidd 11462 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (0 + 1) = 1) |
| 44 | 41, 43 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
| 45 | 44 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
| 46 | 45, 38 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
| 47 | 29, 40, 46 | 3sstr4d 4039 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
| 48 | 47 | 3exp 1120 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 = 1 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 49 | | coires1 6284 |
. . . . . . . . . . . . . 14
⊢ ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((◡𝑅↑𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 50 | | simp2 1138 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈
(ℤ≥‘2)) |
| 51 | | cnvexg 7946 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑅 ∈ 𝑉 → ◡𝑅 ∈ V) |
| 52 | 51 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡𝑅 ∈ V) |
| 53 | | relexpuzrel 15091 |
. . . . . . . . . . . . . . . 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 12924 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℕ) |
| 56 | 50, 55 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℕ) |
| 57 | | relexpnndm 15080 |
. . . . . . . . . . . . . . . . 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 5696 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 = dom ◡𝑅 |
| 60 | | ssun2 4179 |
. . . . . . . . . . . . . . . . 17
⊢ ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
| 61 | 59, 60 | eqsstrri 4031 |
. . . . . . . . . . . . . . . 16
⊢ dom ◡𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
| 62 | 58, 61 | sstrdi 3996 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → dom (◡𝑅↑𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
| 63 | | relssres 6040 |
. . . . . . . . . . . . . . 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 2789 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((◡𝑅↑𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (◡𝑅↑𝑟𝑀)) |
| 66 | | cnvco 5896 |
. . . . . . . . . . . . . 14
⊢ ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (◡(𝑅↑𝑟𝑀) ∘ ◡( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 67 | | simp3 1139 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
| 68 | | eluzge2nn0 12929 |
. . . . . . . . . . . . . . . . 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 15075 |
. . . . . . . . . . . . . . 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 5875 |
. . . . . . . . . . . . . 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 2789 |
. . . . . . . . . . . . 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 2787 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ◡(( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = ◡(𝑅↑𝑟𝑀)) |
| 75 | | relco 6126 |
. . . . . . . . . . . . 13
⊢ Rel (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ (𝑅↑𝑟𝑀)) |
| 76 | | relexpuzrel 15091 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
| 77 | 76 | 3adant1 1131 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑀)) |
| 78 | | cnveqb 6216 |
. . . . . . . . . . . . 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 257 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟𝑀)) |
| 81 | | simp1 1137 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
| 82 | 81 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
| 83 | 32 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 84 | 82, 83 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 85 | 84 | coeq1d 5872 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅↑𝑟𝑀))) |
| 86 | 81 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 𝑀)) |
| 87 | | eluzelcn 12890 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘2) → 𝑀 ∈ ℂ) |
| 88 | 50, 87 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → 𝑀 ∈ ℂ) |
| 89 | 88 | addlidd 11462 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (0 + 𝑀) = 𝑀) |
| 90 | 86, 89 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑀) |
| 91 | 90 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ≥‘2)
∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑀)) |
| 92 | 80, 85, 91 | 3eqtr4d 2787 |
. . . . . . . . . 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 1120 |
. . . . . . . 8
⊢ (𝑁 = 0 → (𝑀 ∈ (ℤ≥‘2)
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 95 | 48, 94 | jaod 860 |
. . . . . . 7
⊢ (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ≥‘2))
→ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 96 | 8, 95 | biimtrid 242 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 97 | 7, 96 | jaoi 858 |
. . . . 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 12967 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
| 100 | 99 | biimpi 216 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈
(ℤ≥‘2))) |
| 101 | | coires1 6284 |
. . . . . . . . . . . 12
⊢ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 102 | | resss 6019 |
. . . . . . . . . . . 12
⊢ (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅 |
| 103 | 101, 102 | eqsstri 4030 |
. . . . . . . . . . 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 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 1) |
| 106 | 105 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟1)) |
| 107 | 37 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
| 108 | 106, 107 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = 𝑅) |
| 109 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
| 110 | 109 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
| 111 | 32 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 112 | 110, 111 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 113 | 108, 112 | coeq12d 5875 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
| 114 | 105, 109 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (1 + 0)) |
| 115 | | 1cnd 11256 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 1 ∈ ℂ) |
| 116 | 115 | addridd 11461 |
. . . . . . . . . . . . 13
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (1 + 0) = 1) |
| 117 | 114, 116 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 1) |
| 118 | 117 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟1)) |
| 119 | 118, 107 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = 𝑅) |
| 120 | 104, 113,
119 | 3sstr4d 4039 |
. . . . . . . . 9
⊢ ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) |
| 121 | 120 | 3exp 1120 |
. . . . . . . 8
⊢ (𝑁 = 1 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 122 | | coires1 6284 |
. . . . . . . . . . . 12
⊢ ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅↑𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 123 | | relexpuzrel 15091 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
| 124 | 123 | 3adant2 1132 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) |
| 125 | | simp1 1137 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈
(ℤ≥‘2)) |
| 126 | | eluz2nn 12924 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℕ) |
| 127 | 125, 126 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) |
| 128 | | simp3 1139 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) |
| 129 | | relexpnndm 15080 |
. . . . . . . . . . . . . . 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 4178 |
. . . . . . . . . . . . . 14
⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) |
| 132 | 130, 131 | sstrdi 3996 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) |
| 133 | | relssres 6040 |
. . . . . . . . . . . . 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 2789 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅↑𝑟𝑁)) |
| 136 | | simp2 1138 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
| 137 | 136 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
| 138 | 32 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 139 | 137, 138 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 140 | 139 | coeq2d 5873 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = ((𝑅↑𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
| 141 | 136 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (𝑁 + 0)) |
| 142 | | eluzelcn 12890 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘2) → 𝑁 ∈ ℂ) |
| 143 | 125, 142 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℂ) |
| 144 | 143 | addridd 11461 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 0) = 𝑁) |
| 145 | 141, 144 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 𝑁) |
| 146 | 145 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈
(ℤ≥‘2) ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟𝑁)) |
| 147 | 135, 140,
146 | 3eqtr4d 2787 |
. . . . . . . . . 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 1120 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘2) → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 150 | 121, 149 | jaoi 858 |
. . . . . . 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 6284 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 153 | | resres 6010 |
. . . . . . . . . 10
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) |
| 154 | | inidm 4227 |
. . . . . . . . . . 11
⊢ ((dom
𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅) |
| 155 | 154 | reseq2i 5994 |
. . . . . . . . . 10
⊢ ( I
↾ ((dom 𝑅 ∪ ran
𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 156 | 152, 153,
155 | 3eqtri 2769 |
. . . . . . . . 9
⊢ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ∘ ( I ↾
(dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)) |
| 157 | | simp1 1137 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) |
| 158 | 157 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
| 159 | 32 | 3ad2ant3 1136 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾
(dom 𝑅 ∪ ran 𝑅))) |
| 160 | 158, 159 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 161 | | simp2 1138 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑀 = 0) |
| 162 | 161 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = (𝑅↑𝑟0)) |
| 163 | 162, 159 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 164 | 160, 163 | coeq12d 5875 |
. . . . . . . . 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 7449 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = (0 + 0)) |
| 166 | | 00id 11436 |
. . . . . . . . . . . . 13
⊢ (0 + 0) =
0 |
| 167 | 166 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (0 + 0) = 0) |
| 168 | 165, 167 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 𝑀) = 0) |
| 169 | 168 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = (𝑅↑𝑟0)) |
| 170 | 169, 159 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
| 171 | 156, 164,
170 | 3eqtr4a 2803 |
. . . . . . . 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 1120 |
. . . . . 6
⊢ (𝑁 = 0 → (𝑀 = 0 → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 174 | 151, 173 | jaoi 858 |
. . . . 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 860 |
. . 3
⊢ (𝑁 ∈ ℕ0
→ ((𝑀 ∈ ℕ
∨ 𝑀 = 0) → (𝑅 ∈ 𝑉 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))))) |
| 177 | 1, 176 | biimtrid 242 |
. 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))) |