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Theorem relexpaddss 41326
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 14765 or when the sum of the powers isn't 1 as shown by relexpaddg 14764. (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 12235 . . 3 (𝑀 ∈ ℕ0 ↔ (𝑀 ∈ ℕ ∨ 𝑀 = 0))
2 elnn0 12235 . . . . . 6 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
32biimpi 215 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0))
4 relexpaddnn 14762 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
5 eqimss 3977 . . . . . . . 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 12665 . . . . . . 7 (𝑀 ∈ ℕ ↔ (𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)))
9 relco 6148 . . . . . . . . . . . . . 14 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅)
10 dfrel2 6092 . . . . . . . . . . . . . . 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 5794 . . . . . . . . . . . . . . . . 17 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅)))
14 cnvresid 6513 . . . . . . . . . . . . . . . . . 18 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
1514coeq2i 5769 . . . . . . . . . . . . . . . . 17 (𝑅( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
16 coires1 6168 . . . . . . . . . . . . . . . . 17 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
1713, 15, 163eqtri 2770 . . . . . . . . . . . . . . . 16 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
18 eqimss 3977 . . . . . . . . . . . . . . . 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 5781 . . . . . . . . . . . . . . 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 5916 . . . . . . . . . . . . . . 15 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
23 cnvss 5781 . . . . . . . . . . . . . . 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 3930 . . . . . . . . . . . . 13 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
2612, 25eqsstrri 3956 . . . . . . . . . . . 12 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅) ⊆ 𝑅
27 cnvcnvss 6097 . . . . . . . . . . . 12 𝑅𝑅
2826, 27sstri 3930 . . . . . . . . . . 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 7291 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
32 relexp0g 14733 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
33323ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
3431, 33eqtrd 2778 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
35 simp2 1136 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 𝑀 = 1)
3635oveq2d 7291 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟1))
37 relexp1g 14737 . . . . . . . . . . . . 13 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
38373ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
3936, 38eqtrd 2778 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = 𝑅)
4034, 39coeq12d 5773 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ 𝑅))
4130, 35oveq12d 7293 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 1))
42 1cnd 10970 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → 1 ∈ ℂ)
4342addid2d 11176 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (0 + 1) = 1)
4441, 43eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
4544oveq2d 7291 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
4645, 38eqtrd 2778 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 1 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
4729, 40, 463sstr4d 3968 . . . . . . . . 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 6168 . . . . . . . . . . . . . 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 7771 . . . . . . . . . . . . . . . . 17 (𝑅𝑉𝑅 ∈ V)
52513ad2ant3 1134 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅 ∈ V)
53 relexpuzrel 14763 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅 ∈ V) → Rel (𝑅𝑟𝑀))
5450, 52, 53syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
55 eluz2nn 12624 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ)
5650, 55syl 17 . . . . . . . . . . . . . . . . 17 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ)
57 relexpnndm 14752 . . . . . . . . . . . . . . . . 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 5600 . . . . . . . . . . . . . . . . 17 ran 𝑅 = dom 𝑅
60 ssun2 4107 . . . . . . . . . . . . . . . . 17 ran 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6159, 60eqsstrri 3956 . . . . . . . . . . . . . . . 16 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
6258, 61sstrdi 3933 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → dom (𝑅𝑟𝑀) ⊆ (dom 𝑅 ∪ ran 𝑅))
63 relssres 5932 . . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑀))
66 cnvco 5794 . . . . . . . . . . . . . 14 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
67 eluzge2nn0 12627 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℕ0)
6850, 67syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℕ0)
69 simp3 1137 . . . . . . . . . . . . . . . 16 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑅𝑉)
70 relexpcnv 14746 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
7168, 69, 70syl2anc 584 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟𝑀))
7214a1i 11 . . . . . . . . . . . . . . 15 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
7371, 72coeq12d 5773 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7466, 73eqtrid 2790 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑀) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
7565, 74, 713eqtr4d 2788 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
76 relco 6148 . . . . . . . . . . . . 13 Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀))
77 relexpuzrel 14763 . . . . . . . . . . . . . 14 ((𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
78773adant1 1129 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑀))
79 cnveqb 6099 . . . . . . . . . . . . 13 ((Rel (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) ∧ Rel (𝑅𝑟𝑀)) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀)))
8076, 78, 79sylancr 587 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀)))
8175, 80mpbird 256 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟𝑀))
82 simp1 1135 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑁 = 0)
8382oveq2d 7291 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
84323ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8583, 84eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
8685coeq1d 5770 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ (𝑅𝑟𝑀)))
8782oveq1d 7290 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 𝑀))
88 eluzelcn 12594 . . . . . . . . . . . . . . 15 (𝑀 ∈ (ℤ‘2) → 𝑀 ∈ ℂ)
8950, 88syl 17 . . . . . . . . . . . . . 14 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → 𝑀 ∈ ℂ)
9089addid2d 11176 . . . . . . . . . . . . 13 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (0 + 𝑀) = 𝑀)
9187, 90eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑀)
9291oveq2d 7291 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑀))
9381, 86, 923eqtr4d 2788 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
9493, 5syl 17 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 ∈ (ℤ‘2) ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
95943exp 1118 . . . . . . . 8 (𝑁 = 0 → (𝑀 ∈ (ℤ‘2) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
9648, 95jaod 856 . . . . . . 7 (𝑁 = 0 → ((𝑀 = 1 ∨ 𝑀 ∈ (ℤ‘2)) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
978, 96syl5bi 241 . . . . . 6 (𝑁 = 0 → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
987, 97jaoi 854 . . . . 5 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
993, 98syl 17 . . . 4 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
100 elnn1uz2 12665 . . . . . . . 8 (𝑁 ∈ ℕ ↔ (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
101100biimpi 215 . . . . . . 7 (𝑁 ∈ ℕ → (𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)))
102 coires1 6168 . . . . . . . . . . . 12 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅))
103 resss 5916 . . . . . . . . . . . 12 (𝑅 ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑅
104102, 103eqsstri 3955 . . . . . . . . . . 11 (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅
105104a1i 11 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) ⊆ 𝑅)
106 simp1 1135 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 1)
107106oveq2d 7291 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟1))
108373ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟1) = 𝑅)
109107, 108eqtrd 2778 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = 𝑅)
110 simp2 1136 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
111110oveq2d 7291 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
112323ad2ant3 1134 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
113111, 112eqtrd 2778 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
114109, 113coeq12d 5773 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
115106, 110oveq12d 7293 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (1 + 0))
116 1cnd 10970 . . . . . . . . . . . . . 14 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 1 ∈ ℂ)
117116addid1d 11175 . . . . . . . . . . . . 13 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (1 + 0) = 1)
118115, 117eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 1)
119118oveq2d 7291 . . . . . . . . . . 11 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟1))
120119, 108eqtrd 2778 . . . . . . . . . 10 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = 𝑅)
121105, 114, 1203sstr4d 3968 . . . . . . . . 9 ((𝑁 = 1 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1221213exp 1118 . . . . . . . 8 (𝑁 = 1 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
123 coires1 6168 . . . . . . . . . . . 12 ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅))
124 relexpuzrel 14763 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
1251243adant2 1130 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))
126 simp1 1135 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ (ℤ‘2))
127 eluz2nn 12624 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℕ)
128126, 127syl 17 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℕ)
129 simp3 1137 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑅𝑉)
130 relexpnndm 14752 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
131128, 129, 130syl2anc 584 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)
132 ssun1 4106 . . . . . . . . . . . . . 14 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
133131, 132sstrdi 3933 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
134 relssres 5932 . . . . . . . . . . . . 13 ((Rel (𝑅𝑟𝑁) ∧ dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
135125, 133, 134syl2anc 584 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ↾ (dom 𝑅 ∪ ran 𝑅)) = (𝑅𝑟𝑁))
136123, 135eqtrid 2790 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅𝑟𝑁))
137 simp2 1136 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
138137oveq2d 7291 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
139323ad2ant3 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
140138, 139eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
141140coeq2d 5771 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = ((𝑅𝑟𝑁) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
142137oveq2d 7291 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (𝑁 + 0))
143 eluzelcn 12594 . . . . . . . . . . . . . . 15 (𝑁 ∈ (ℤ‘2) → 𝑁 ∈ ℂ)
144126, 143syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 ∈ ℂ)
145144addid1d 11175 . . . . . . . . . . . . 13 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 0) = 𝑁)
146142, 145eqtrd 2778 . . . . . . . . . . . 12 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 𝑁)
147146oveq2d 7291 . . . . . . . . . . 11 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟𝑁))
148136, 141, 1473eqtr4d 2788 . . . . . . . . . 10 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
149148, 5syl 17 . . . . . . . . 9 ((𝑁 ∈ (ℤ‘2) ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1501493exp 1118 . . . . . . . 8 (𝑁 ∈ (ℤ‘2) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
151122, 150jaoi 854 . . . . . . 7 ((𝑁 = 1 ∨ 𝑁 ∈ (ℤ‘2)) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
152101, 151syl 17 . . . . . 6 (𝑁 ∈ ℕ → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
153 coires1 6168 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅))
154 resres 5904 . . . . . . . . . 10 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)))
155 inidm 4152 . . . . . . . . . . 11 ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
156155reseq2i 5888 . . . . . . . . . 10 ( I ↾ ((dom 𝑅 ∪ ran 𝑅) ∩ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
157153, 154, 1563eqtri 2770 . . . . . . . . 9 (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))
158 simp1 1135 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑁 = 0)
159158oveq2d 7291 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟0))
160323ad2ant3 1134 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
161159, 160eqtrd 2778 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
162 simp2 1136 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → 𝑀 = 0)
163162oveq2d 7291 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = (𝑅𝑟0))
164163, 160eqtrd 2778 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟𝑀) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
165161, 164coeq12d 5773 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
166158, 162oveq12d 7293 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = (0 + 0))
167 00id 11150 . . . . . . . . . . . . 13 (0 + 0) = 0
168167a1i 11 . . . . . . . . . . . 12 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (0 + 0) = 0)
169166, 168eqtrd 2778 . . . . . . . . . . 11 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑁 + 𝑀) = 0)
170169oveq2d 7291 . . . . . . . . . 10 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = (𝑅𝑟0))
171170, 160eqtrd 2778 . . . . . . . . 9 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → (𝑅𝑟(𝑁 + 𝑀)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
172157, 165, 1713eqtr4a 2804 . . . . . . . 8 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))
173172, 5syl 17 . . . . . . 7 ((𝑁 = 0 ∧ 𝑀 = 0 ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
1741733exp 1118 . . . . . 6 (𝑁 = 0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
175152, 174jaoi 854 . . . . 5 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1763, 175syl 17 . . . 4 (𝑁 ∈ ℕ0 → (𝑀 = 0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
17799, 176jaod 856 . . 3 (𝑁 ∈ ℕ0 → ((𝑀 ∈ ℕ ∨ 𝑀 = 0) → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1781, 177syl5bi 241 . 2 (𝑁 ∈ ℕ0 → (𝑀 ∈ ℕ0 → (𝑅𝑉 → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))))
1791783imp 1110 1 ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 844  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  cun 3885  cin 3886  wss 3887   I cid 5488  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  ccom 5593  Rel wrel 5594  cfv 6433  (class class class)co 7275  cc 10869  0cc0 10871  1c1 10872   + caddc 10874  cn 11973  2c2 12028  0cn0 12233  cuz 12582  𝑟crelexp 14730
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-seq 13722  df-relexp 14731
This theorem is referenced by:  iunrelexpuztr  41327  cotrclrcl  41350
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