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Theorem relexpaddss 42469
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 15001 or when the sum of the powers isn't 1 as shown by relexpaddg 15000. (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 12474 . . 3 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2 elnn0 12474 . . . . . 6 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
32biimpi 215 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 relexpaddnn 14998 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
5 eqimss 4041 . . . . . . . 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 1120 . . . . . 6 (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
8 elnn1uz2 12909 . . . . . . 7 (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)))
9 relco 6108 . . . . . . . . . . . . . 14 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
10 dfrel2 6189 . . . . . . . . . . . . . . 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 5886 . . . . . . . . . . . . . . . . 17 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14 cnvresid 6628 . . . . . . . . . . . . . . . . . 18 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
1514coeq2i 5861 . . . . . . . . . . . . . . . . 17 (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
16 coires1 6264 . . . . . . . . . . . . . . . . 17 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
1713, 15, 163eqtri 2765 . . . . . . . . . . . . . . . 16 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
18 eqimss 4041 . . . . . . . . . . . . . . . 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 5873 . . . . . . . . . . . . . . 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 6007 . . . . . . . . . . . . . . 15 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
23 cnvss 5873 . . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . 13 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
2612, 25eqsstrri 4018 . . . . . . . . . . . 12 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
27 cnvcnvss 6194 . . . . . . . . . . . 12 𝑅𝑅
2826, 27sstri 3992 . . . . . . . . . . 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 1137 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑁 = 0)
3130oveq2d 7425 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
32 relexp0g 14969 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33323ad2ant3 1136 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3431, 33eqtrd 2773 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35 simp2 1138 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑀 = 1)
3635oveq2d 7425 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟1))
37 relexp1g 14973 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
38373ad2ant3 1136 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2773 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = 𝑅)
4034, 39coeq12d 5865 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
4130, 35oveq12d 7427 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 1))
42 1cnd 11209 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 1 ∈ ℂ)
4342addlidd 11415 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (0 + 1) = 1)
4441, 43eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
4544oveq2d 7425 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
4645, 38eqtrd 2773 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
4729, 40, 463sstr4d 4030 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
48473exp 1120 . . . . . . . 8 (𝑁 = 0 → (𝑀 = 1 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
49 coires1 6264 . . . . . . . . . . . . . 14 ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
50 simp2 1138 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ (ℤ‘2))
51 cnvexg 7915 . . . . . . . . . . . . . . . . 17 (𝑅𝑉𝑅 ∈ V)
52513ad2ant3 1136 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
53 relexpuzrel 14999 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
5450, 52, 53syl2anc 585 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
55 eluz2nn 12868 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
5650, 55syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
57 relexpnndm 14988 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ ∧ 𝑅 ∈ V) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
5856, 52, 57syl2anc 585 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ dom 𝑅)
59 df-rn 5688 . . . . . . . . . . . . . . . . 17 ran 𝑅 = dom 𝑅
60 ssun2 4174 . . . . . . . . . . . . . . . . 17 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6159, 60eqsstrri 4018 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6258, 61sstrdi 3995 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63 relssres 6023 . . . . . . . . . . . . . . 15 ((Rel (𝑅𝑟𝑀) ∧ dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
6454, 62, 63syl2anc 585 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑀))
6549, 64eqtrid 2785 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
66 cnvco 5886 . . . . . . . . . . . . . 14 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67 simp3 1139 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
68 eluzge2nn0 12871 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ0)
6950, 68syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
7067, 69relexpcnvd 14983 . . . . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7366, 72eqtrid 2785 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7465, 73, 703eqtr4d 2783 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
75 relco 6108 . . . . . . . . . . . . 13 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀))
76 relexpuzrel 14999 . . . . . . . . . . . . . 14 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
77763adant1 1131 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
78 cnveqb 6196 . . . . . . . . . . . . 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 588 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀)))
8074, 79mpbird 257 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
81 simp1 1137 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
8281oveq2d 7425 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
83323ad2ant3 1136 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8482, 83eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8584coeq1d 5862 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)))
8681oveq1d 7424 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
87 eluzelcn 12834 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
8850, 87syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
8988addlidd 11415 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
9086, 89eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
9190oveq2d 7425 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
9280, 85, 913eqtr4d 2783 . . . . . . . . . 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 1120 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (ℤ‘2) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
9548, 94jaod 858 . . . . . . 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 856 . . . . 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 12909 . . . . . . . 8 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
10099biimpi 215 . . . . . . 7 (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
101 coires1 6264 . . . . . . . . . . . 12 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
102 resss 6007 . . . . . . . . . . . 12 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
103101, 102eqsstri 4017 . . . . . . . . . . 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 1137 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 1)
106105oveq2d 7425 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟1))
107373ad2ant3 1136 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
108106, 107eqtrd 2773 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = 𝑅)
109 simp2 1138 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
110109oveq2d 7425 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
111323ad2ant3 1136 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
112110, 111eqtrd 2773 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113108, 112coeq12d 5865 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
114105, 109oveq12d 7427 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (1 + 0))
115 1cnd 11209 . . . . . . . . . . . . . 14 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 1 ∈ ℂ)
116115addridd 11414 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (1 + 0) = 1)
117114, 116eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
118117oveq2d 7425 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
119118, 107eqtrd 2773 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
120104, 113, 1193sstr4d 4030 . . . . . . . . 9 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1211203exp 1120 . . . . . . . 8 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
122 coires1 6264 . . . . . . . . . . . 12 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
123 relexpuzrel 14999 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
1241233adant2 1132 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
125 simp1 1137 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
126 eluz2nn 12868 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
127125, 126syl 17 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
128 simp3 1139 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
129 relexpnndm 14988 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
130127, 128, 129syl2anc 585 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
131 ssun1 4173 . . . . . . . . . . . . . 14 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
132130, 131sstrdi 3995 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
133 relssres 6023 . . . . . . . . . . . . 13 ((Rel (𝑅𝑟𝑁) ∧ dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
134124, 132, 133syl2anc 585 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
135122, 134eqtrid 2785 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
136 simp2 1138 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
137136oveq2d 7425 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
138323ad2ant3 1136 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
139137, 138eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140139coeq2d 5863 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
141136oveq2d 7425 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
142 eluzelcn 12834 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
143125, 142syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
144143addridd 11414 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
145141, 144eqtrd 2773 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
146145oveq2d 7425 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
147135, 140, 1463eqtr4d 2783 . . . . . . . . . 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 1120 . . . . . . . 8 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
150121, 149jaoi 856 . . . . . . 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 6264 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅))
153 resres 5995 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)))
154 inidm 4219 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
155154reseq2i 5979 . . . . . . . . . 10 ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
156152, 153, 1553eqtri 2765 . . . . . . . . 9 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
157 simp1 1137 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
158157oveq2d 7425 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
159323ad2ant3 1136 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
160158, 159eqtrd 2773 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161 simp2 1138 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
162161oveq2d 7425 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
163162, 159eqtrd 2773 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
164160, 163coeq12d 5865 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
165157, 161oveq12d 7427 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
166 00id 11389 . . . . . . . . . . . . 13 (0 + 0) = 0
167166a1i 11 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (0 + 0) = 0)
168165, 167eqtrd 2773 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
169168oveq2d 7425 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
170169, 159eqtrd 2773 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
171156, 164, 1703eqtr4a 2799 . . . . . . . 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 1120 . . . . . 6 (𝑁 = 0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
174151, 173jaoi 856 . . . . 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 858 . . 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 1112 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 846  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cun 3947  cin 3948  wss 3949   I cid 5574  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  ccom 5681  Rel wrel 5682  cfv 6544  (class class class)co 7409  cc 11108  0cc0 11110  1c1 11111   + caddc 11113  cn 12212  2c2 12267  0cn0 12472  cuz 12822  𝑟crelexp 14966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-relexp 14967
This theorem is referenced by:  iunrelexpuztr  42470  cotrclrcl  42493
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