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Theorem relexpaddss 41980
Description: The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where 𝑅 is a relation as shown by relexpaddd 14939 or when the sum of the powers isn't 1 as shown by relexpaddg 14938. (Contributed by RP, 3-Jun-2020.)
Assertion
Ref Expression
relexpaddss ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))

Proof of Theorem relexpaddss
StepHypRef Expression
1 elnn0 12415 . . 3 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2 elnn0 12415 . . . . . 6 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
32biimpi 215 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 relexpaddnn 14936 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
5 eqimss 4000 . . . . . . . 8 (((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
64, 5syl 17 . . . . . . 7 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
763exp 1119 . . . . . 6 (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(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))
1110biimpi 215 . . . . . . . . . . . . . 14 (Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
129, 11ax-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))
1514coeq2i 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))
1713, 15, 163eqtri 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)))
1917, 18ax-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)))
2119, 20ax-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)
2422, 23ax-mp 5 . . . . . . . . . . . . . 14 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
2521, 24sstri 3953 . . . . . . . . . . . . 13 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
2612, 25eqsstrri 3979 . . . . . . . . . . . 12 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
27 cnvcnvss 6146 . . . . . . . . . . . 12 𝑅𝑅
2826, 27sstri 3953 . . . . . . . . . . 11 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
2928a1i 11 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅)
30 simp1 1136 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑁 = 0)
3130oveq2d 7373 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
32 relexp0g 14907 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33323ad2ant3 1135 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3431, 33eqtrd 2776 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35 simp2 1137 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑀 = 1)
3635oveq2d 7373 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟1))
37 relexp1g 14911 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
38373ad2ant3 1135 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2776 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = 𝑅)
4034, 39coeq12d 5820 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
4130, 35oveq12d 7375 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 1))
42 1cnd 11150 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 1 ∈ ℂ)
4342addid2d 11356 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (0 + 1) = 1)
4441, 43eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
4544oveq2d 7373 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
4645, 38eqtrd 2776 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
4729, 40, 463sstr4d 3991 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
48473exp 1119 . . . . . . . 8 (𝑁 = 0 → (𝑀 = 1 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
49 coires1 6216 . . . . . . . . . . . . . 14 ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
50 simp2 1137 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ (ℤ‘2))
51 cnvexg 7861 . . . . . . . . . . . . . . . . 17 (𝑅𝑉𝑅 ∈ V)
52513ad2ant3 1135 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
53 relexpuzrel 14937 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
5450, 52, 53syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
55 eluz2nn 12809 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
5650, 55syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
57 relexpnndm 14926 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ ∧ 𝑅 ∈ V) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
5856, 52, 57syl2anc 584 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
59 df-rn 5644 . . . . . . . . . . . . . . . . 17 ran 𝑅 = dom 𝑅
60 ssun2 4133 . . . . . . . . . . . . . . . . 17 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6159, 60eqsstrri 3979 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6258, 61sstrdi 3956 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63 relssres 5978 . . . . . . . . . . . . . . 15 ((Rel (𝑅𝑟𝑀) ∧ dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
6454, 62, 63syl2anc 584 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
6549, 64eqtrid 2788 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
66 cnvco 5841 . . . . . . . . . . . . . 14 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67 simp3 1138 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
68 eluzge2nn0 12812 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ0)
6950, 68syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
7067, 69relexpcnvd 14921 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
7114a1i 11 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7270, 71coeq12d 5820 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7366, 72eqtrid 2788 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7465, 73, 703eqtr4d 2786 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
75 relco 6060 . . . . . . . . . . . . 13 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀))
76 relexpuzrel 14937 . . . . . . . . . . . . . 14 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
77763adant1 1130 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
78 cnveqb 6148 . . . . . . . . . . . . 13 ((Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) ∧ Rel (𝑅𝑟𝑀)) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀)))
7975, 77, 78sylancr 587 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀)))
8074, 79mpbird 256 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
81 simp1 1136 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
8281oveq2d 7373 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
83323ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8482, 83eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8584coeq1d 5817 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)))
8681oveq1d 7372 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
87 eluzelcn 12775 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
8850, 87syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
8988addid2d 11356 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
9086, 89eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
9190oveq2d 7373 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
9280, 85, 913eqtr4d 2786 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
9392, 5syl 17 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
94933exp 1119 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (ℤ‘2) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
9548, 94jaod 857 . . . . . . 7 (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
968, 95biimtrid 241 . . . . . 6 (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
977, 96jaoi 855 . . . . 5 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
983, 97syl 17 . . . 4 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
99 elnn1uz2 12850 . . . . . . . 8 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
10099biimpi 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
103101, 102eqsstri 3978 . . . . . . . . . . 11 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅
104103a1i 11 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅)
105 simp1 1136 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 1)
106105oveq2d 7373 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟1))
107373ad2ant3 1135 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
108106, 107eqtrd 2776 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = 𝑅)
109 simp2 1137 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
110109oveq2d 7373 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
111323ad2ant3 1135 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
112110, 111eqtrd 2776 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113108, 112coeq12d 5820 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
114105, 109oveq12d 7375 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (1 + 0))
115 1cnd 11150 . . . . . . . . . . . . . 14 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 1 ∈ ℂ)
116115addid1d 11355 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (1 + 0) = 1)
117114, 116eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
118117oveq2d 7373 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
119118, 107eqtrd 2776 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
120104, 113, 1193sstr4d 3991 . . . . . . . . 9 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1211203exp 1119 . . . . . . . 8 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
122 coires1 6216 . . . . . . . . . . . 12 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
123 relexpuzrel 14937 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
1241233adant2 1131 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
125 simp1 1136 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
126 eluz2nn 12809 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
127125, 126syl 17 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
128 simp3 1138 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
129 relexpnndm 14926 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
130127, 128, 129syl2anc 584 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
131 ssun1 4132 . . . . . . . . . . . . . 14 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
132130, 131sstrdi 3956 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
133 relssres 5978 . . . . . . . . . . . . 13 ((Rel (𝑅𝑟𝑁) ∧ dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
134124, 132, 133syl2anc 584 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
135122, 134eqtrid 2788 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
136 simp2 1137 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
137136oveq2d 7373 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
138323ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
139137, 138eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140139coeq2d 5818 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
141136oveq2d 7373 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
142 eluzelcn 12775 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
143125, 142syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
144143addid1d 11355 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
145141, 144eqtrd 2776 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
146145oveq2d 7373 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
147135, 140, 1463eqtr4d 2786 . . . . . . . . . 10 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
148147, 5syl 17 . . . . . . . . 9 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1491483exp 1119 . . . . . . . 8 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
150121, 149jaoi 855 . . . . . . 7 ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
151100, 150syl 17 . . . . . 6 (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(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)
155154reseq2i 5934 . . . . . . . . . 10 ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
156152, 153, 1553eqtri 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 ∧ 𝑅𝑉) → 𝑁 = 0)
158157oveq2d 7373 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
159323ad2ant3 1135 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
160158, 159eqtrd 2776 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161 simp2 1137 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
162161oveq2d 7373 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
163162, 159eqtrd 2776 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
164160, 163coeq12d 5820 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
165157, 161oveq12d 7375 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
166 00id 11330 . . . . . . . . . . . . 13 (0 + 0) = 0
167166a1i 11 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (0 + 0) = 0)
168165, 167eqtrd 2776 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
169168oveq2d 7373 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
170169, 159eqtrd 2776 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
171156, 164, 1703eqtr4a 2802 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
172171, 5syl 17 . . . . . . 7 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1731723exp 1119 . . . . . 6 (𝑁 = 0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
174151, 173jaoi 855 . . . . 5 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1753, 174syl 17 . . . 4 (𝑁 ∈ ℕ0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
17698, 175jaod 857 . . 3 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1771, 176biimtrid 241 . 2 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1781773imp 1111 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  cin 3909  wss 3910   I cid 5530  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  ccom 5637  Rel wrel 5638  cfv 6496  (class class class)co 7357  cc 11049  0cc0 11051  1c1 11052   + caddc 11054  cn 12153  2c2 12208  0cn0 12413  cuz 12763  𝑟crelexp 14904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907  df-relexp 14905
This theorem is referenced by:  iunrelexpuztr  41981  cotrclrcl  42004
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