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Theorem relexpaddss 43707
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 15089 or when the sum of the powers isn't 1 as shown by relexpaddg 15088. (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 12525 . . 3 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2 elnn0 12525 . . . . . 6 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
32biimpi 216 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 relexpaddnn 15086 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
5 eqimss 4053 . . . . . . . 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 1118 . . . . . 6 (𝑁 ∈ ℕ → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
8 elnn1uz2 12964 . . . . . . 7 (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)))
9 relco 6128 . . . . . . . . . . . . . 14 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
10 dfrel2 6210 . . . . . . . . . . . . . . 15 (Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
1110biimpi 216 . . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . . . 17 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14 cnvresid 6646 . . . . . . . . . . . . . . . . . 18 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
1514coeq2i 5873 . . . . . . . . . . . . . . . . 17 (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
16 coires1 6285 . . . . . . . . . . . . . . . . 17 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
1713, 15, 163eqtri 2766 . . . . . . . . . . . . . . . 16 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
18 eqimss 4053 . . . . . . . . . . . . . . . 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 5885 . . . . . . . . . . . . . . 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 6021 . . . . . . . . . . . . . . 15 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
23 cnvss 5885 . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . 13 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
2612, 25eqsstrri 4030 . . . . . . . . . . . 12 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
27 cnvcnvss 6215 . . . . . . . . . . . 12 𝑅𝑅
2826, 27sstri 4004 . . . . . . . . . . 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 1135 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑁 = 0)
3130oveq2d 7446 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
32 relexp0g 15057 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33323ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3431, 33eqtrd 2774 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35 simp2 1136 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑀 = 1)
3635oveq2d 7446 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟1))
37 relexp1g 15061 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
38373ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2774 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = 𝑅)
4034, 39coeq12d 5877 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
4130, 35oveq12d 7448 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 1))
42 1cnd 11253 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 1 ∈ ℂ)
4342addlidd 11459 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (0 + 1) = 1)
4441, 43eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
4544oveq2d 7446 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
4645, 38eqtrd 2774 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
4729, 40, 463sstr4d 4042 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
48473exp 1118 . . . . . . . 8 (𝑁 = 0 → (𝑀 = 1 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
49 coires1 6285 . . . . . . . . . . . . . 14 ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ↾ (dom 𝑅 ∪ ran 𝑅))
50 simp2 1136 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ (ℤ‘2))
51 cnvexg 7946 . . . . . . . . . . . . . . . . 17 (𝑅𝑉𝑅 ∈ V)
52513ad2ant3 1134 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
53 relexpuzrel 15087 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
5450, 52, 53syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
55 eluz2nn 12921 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
5650, 55syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
57 relexpnndm 15076 . . . . . . . . . . . . . . . . 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 5699 . . . . . . . . . . . . . . . . 17 ran 𝑅 = dom 𝑅
60 ssun2 4188 . . . . . . . . . . . . . . . . 17 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6159, 60eqsstrri 4030 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6258, 61sstrdi 4007 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63 relssres 6041 . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
66 cnvco 5898 . . . . . . . . . . . . . 14 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67 simp3 1137 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
68 eluzge2nn0 12926 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ0)
6950, 68syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
7067, 69relexpcnvd 15071 . . . . . . . . . . . . . . 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 5877 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7366, 72eqtrid 2786 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7465, 73, 703eqtr4d 2784 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
75 relco 6128 . . . . . . . . . . . . 13 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀))
76 relexpuzrel 15087 . . . . . . . . . . . . . 14 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
77763adant1 1129 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
78 cnveqb 6217 . . . . . . . . . . . . 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 257 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
81 simp1 1135 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
8281oveq2d 7446 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
83323ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8482, 83eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8584coeq1d 5874 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)))
8681oveq1d 7445 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
87 eluzelcn 12887 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
8850, 87syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
8988addlidd 11459 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
9086, 89eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
9190oveq2d 7446 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
9280, 85, 913eqtr4d 2784 . . . . . . . . . 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 1118 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (ℤ‘2) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
9548, 94jaod 859 . . . . . . 7 (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
968, 95biimtrid 242 . . . . . 6 (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
977, 96jaoi 857 . . . . 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 12964 . . . . . . . 8 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
10099biimpi 216 . . . . . . 7 (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
101 coires1 6285 . . . . . . . . . . . 12 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
102 resss 6021 . . . . . . . . . . . 12 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
103101, 102eqsstri 4029 . . . . . . . . . . 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 1135 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 1)
106105oveq2d 7446 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟1))
107373ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
108106, 107eqtrd 2774 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = 𝑅)
109 simp2 1136 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
110109oveq2d 7446 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
111323ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
112110, 111eqtrd 2774 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113108, 112coeq12d 5877 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
114105, 109oveq12d 7448 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (1 + 0))
115 1cnd 11253 . . . . . . . . . . . . . 14 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 1 ∈ ℂ)
116115addridd 11458 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (1 + 0) = 1)
117114, 116eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
118117oveq2d 7446 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
119118, 107eqtrd 2774 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
120104, 113, 1193sstr4d 4042 . . . . . . . . 9 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1211203exp 1118 . . . . . . . 8 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
122 coires1 6285 . . . . . . . . . . . 12 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
123 relexpuzrel 15087 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
1241233adant2 1130 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
125 simp1 1135 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
126 eluz2nn 12921 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
127125, 126syl 17 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
128 simp3 1137 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
129 relexpnndm 15076 . . . . . . . . . . . . . . 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 4187 . . . . . . . . . . . . . 14 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
132130, 131sstrdi 4007 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
133 relssres 6041 . . . . . . . . . . . . 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 2786 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
136 simp2 1136 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
137136oveq2d 7446 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
138323ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
139137, 138eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140139coeq2d 5875 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
141136oveq2d 7446 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
142 eluzelcn 12887 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
143125, 142syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
144143addridd 11458 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
145141, 144eqtrd 2774 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
146145oveq2d 7446 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
147135, 140, 1463eqtr4d 2784 . . . . . . . . . 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 1118 . . . . . . . 8 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
150121, 149jaoi 857 . . . . . . 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 6285 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅))
153 resres 6012 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)))
154 inidm 4234 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
155154reseq2i 5996 . . . . . . . . . 10 ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
156152, 153, 1553eqtri 2766 . . . . . . . . 9 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
157 simp1 1135 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
158157oveq2d 7446 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
159323ad2ant3 1134 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
160158, 159eqtrd 2774 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161 simp2 1136 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
162161oveq2d 7446 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
163162, 159eqtrd 2774 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
164160, 163coeq12d 5877 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
165157, 161oveq12d 7448 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
166 00id 11433 . . . . . . . . . . . . 13 (0 + 0) = 0
167166a1i 11 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (0 + 0) = 0)
168165, 167eqtrd 2774 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
169168oveq2d 7446 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
170169, 159eqtrd 2774 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
171156, 164, 1703eqtr4a 2800 . . . . . . . 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 1118 . . . . . 6 (𝑁 = 0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
174151, 173jaoi 857 . . . . 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 859 . . 3 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1771, 176biimtrid 242 . 2 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1781773imp 1110 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847  w3a 1086   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  cin 3961  wss 3962   I cid 5581  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  ccom 5692  Rel wrel 5693  cfv 6562  (class class class)co 7430  cc 11150  0cc0 11152  1c1 11153   + caddc 11155  cn 12263  2c2 12318  0cn0 12523  cuz 12875  𝑟crelexp 15054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-seq 14039  df-relexp 15055
This theorem is referenced by:  iunrelexpuztr  43708  cotrclrcl  43731
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