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Theorem cotrclrcl 43760
Description: The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
Assertion
Ref Expression
cotrclrcl (t+ ∘ r*) = t*

Proof of Theorem cotrclrcl
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑖 𝑗 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftrcl3 43738 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dfrcl4 43694 . 2 r* = (𝑏 ∈ V ↦ 𝑗 ∈ {0, 1} (𝑏𝑟𝑗))
3 dfrtrcl3 43751 . 2 t* = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑐𝑟𝑘))
4 nnex 12273 . 2 ℕ ∈ V
5 prex 5436 . 2 {0, 1} ∈ V
6 df-n0 12529 . . 3 0 = (ℕ ∪ {0})
7 df-pr 4628 . . . . . 6 {0, 1} = ({0} ∪ {1})
87equncomi 4159 . . . . 5 {0, 1} = ({1} ∪ {0})
98uneq2i 4164 . . . 4 (ℕ ∪ {0, 1}) = (ℕ ∪ ({1} ∪ {0}))
10 unass 4171 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ ({1} ∪ {0}))
11 1nn 12278 . . . . . . 7 1 ∈ ℕ
12 snssi 4807 . . . . . . 7 (1 ∈ ℕ → {1} ⊆ ℕ)
1311, 12ax-mp 5 . . . . . 6 {1} ⊆ ℕ
14 ssequn2 4188 . . . . . 6 ({1} ⊆ ℕ ↔ (ℕ ∪ {1}) = ℕ)
1513, 14mpbi 230 . . . . 5 (ℕ ∪ {1}) = ℕ
1615uneq1i 4163 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ {0})
179, 10, 163eqtr2ri 2771 . . 3 (ℕ ∪ {0}) = (ℕ ∪ {0, 1})
186, 17eqtri 2764 . 2 0 = (ℕ ∪ {0, 1})
19 oveq2 7440 . . . 4 (𝑘 = 𝑖 → (𝑑𝑟𝑘) = (𝑑𝑟𝑖))
2019cbviunv 5039 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ ℕ (𝑑𝑟𝑖)
21 ss2iun 5009 . . . 4 (∀𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) → 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
22 1ex 11258 . . . . . . . 8 1 ∈ V
2322prid2 4762 . . . . . . 7 1 ∈ {0, 1}
24 oveq2 7440 . . . . . . . . 9 (𝑗 = 1 → (𝑑𝑟𝑗) = (𝑑𝑟1))
25 relexp1g 15066 . . . . . . . . . 10 (𝑑 ∈ V → (𝑑𝑟1) = 𝑑)
2625elv 3484 . . . . . . . . 9 (𝑑𝑟1) = 𝑑
2724, 26eqtrdi 2792 . . . . . . . 8 (𝑗 = 1 → (𝑑𝑟𝑗) = 𝑑)
2827ssiun2s 5047 . . . . . . 7 (1 ∈ {0, 1} → 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
2923, 28ax-mp 5 . . . . . 6 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)
3029a1i 11 . . . . 5 (𝑖 ∈ ℕ → 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
31 ovex 7465 . . . . . . 7 (𝑑𝑟𝑗) ∈ V
325, 31iunex 7994 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V
3332a1i 11 . . . . 5 (𝑖 ∈ ℕ → 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V)
34 nnnn0 12535 . . . . 5 (𝑖 ∈ ℕ → 𝑖 ∈ ℕ0)
3530, 33, 34relexpss1d 43723 . . . 4 (𝑖 ∈ ℕ → (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
3621, 35mprg 3066 . . 3 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
3720, 36eqsstri 4029 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
38 oveq2 7440 . . . . 5 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1))
39 relexp1g 15066 . . . . . . 7 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
4032, 39ax-mp 5 . . . . . 6 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)
41 oveq2 7440 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
4241cbviunv 5039 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4340, 42eqtri 2764 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4438, 43eqtrdi 2792 . . . 4 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘))
4544ssiun2s 5047 . . 3 (1 ∈ ℕ → 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
4611, 45ax-mp 5 . 2 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
47 iunss 5044 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
48 iuneq1 5007 . . . . . . . 8 ({0, 1} = ({0} ∪ {1}) → 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗))
497, 48ax-mp 5 . . . . . . 7 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗)
50 iunxun 5093 . . . . . . 7 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗) = ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗))
51 c0ex 11256 . . . . . . . . 9 0 ∈ V
52 oveq2 7440 . . . . . . . . 9 (𝑗 = 0 → (𝑑𝑟𝑗) = (𝑑𝑟0))
5351, 52iunxsn 5090 . . . . . . . 8 𝑗 ∈ {0} (𝑑𝑟𝑗) = (𝑑𝑟0)
5422, 24iunxsn 5090 . . . . . . . 8 𝑗 ∈ {1} (𝑑𝑟𝑗) = (𝑑𝑟1)
5553, 54uneq12i 4165 . . . . . . 7 ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗)) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5649, 50, 553eqtri 2768 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5756oveq1i 7442 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖)
58 oveq2 7440 . . . . . . 7 (𝑥 = 1 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1))
5958sseq1d 4014 . . . . . 6 (𝑥 = 1 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
60 oveq2 7440 . . . . . . 7 (𝑥 = 𝑦 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦))
6160sseq1d 4014 . . . . . 6 (𝑥 = 𝑦 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
62 oveq2 7440 . . . . . . 7 (𝑥 = (𝑦 + 1) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)))
6362sseq1d 4014 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
64 oveq2 7440 . . . . . . 7 (𝑥 = 𝑖 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖))
6564sseq1d 4014 . . . . . 6 (𝑥 = 𝑖 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
66 ovex 7465 . . . . . . . . 9 (𝑑𝑟0) ∈ V
67 ovex 7465 . . . . . . . . 9 (𝑑𝑟1) ∈ V
6866, 67unex 7765 . . . . . . . 8 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V
69 relexp1g 15066 . . . . . . . 8 (((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1)))
7068, 69ax-mp 5 . . . . . . 7 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
71 0nn0 12543 . . . . . . . . 9 0 ∈ ℕ0
72 oveq2 7440 . . . . . . . . . 10 (𝑘 = 0 → (𝑑𝑟𝑘) = (𝑑𝑟0))
7372ssiun2s 5047 . . . . . . . . 9 (0 ∈ ℕ0 → (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7471, 73ax-mp 5 . . . . . . . 8 (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
75 1nn0 12544 . . . . . . . . 9 1 ∈ ℕ0
76 oveq2 7440 . . . . . . . . . 10 (𝑘 = 1 → (𝑑𝑟𝑘) = (𝑑𝑟1))
7776ssiun2s 5047 . . . . . . . . 9 (1 ∈ ℕ0 → (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7875, 77ax-mp 5 . . . . . . . 8 (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
7974, 78unssi 4190 . . . . . . 7 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
8070, 79eqsstri 4029 . . . . . 6 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
81 simpl 482 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
82 relexpsucnnr 15065 . . . . . . . . 9 ((((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V ∧ 𝑦 ∈ ℕ) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
8368, 81, 82sylancr 587 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
84 coss1 5865 . . . . . . . . . 10 ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
85 coundi 6266 . . . . . . . . . . 11 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) = (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)))
86 relexp0g 15062 . . . . . . . . . . . . . . . 16 (𝑑 ∈ V → (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
8786elv 3484 . . . . . . . . . . . . . . 15 (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑))
8887coeq2i 5870 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
89 coiun1 43670 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
90 coires1 6283 . . . . . . . . . . . . . . . 16 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
9190a1i 11 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)))
9291iuneq2i 5012 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
9388, 89, 923eqtri 2768 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
94 ss2iun 5009 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
95 resss 6018 . . . . . . . . . . . . . . 15 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘)
9695a1i 11 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘))
9794, 96mprg 3066 . . . . . . . . . . . . 13 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
9893, 97eqsstri 4029 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
99 coiun1 43670 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1))
100 iunss2 5048 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ ℕ0𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖))
101 peano2nn0 12568 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
102 sbcel1v 3855 . . . . . . . . . . . . . . . . . . 19 ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0 ↔ (𝑘 + 1) ∈ ℕ0)
103101, 102sylibr 234 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0)
104 vex 3483 . . . . . . . . . . . . . . . . . . . . 21 𝑑 ∈ V
105 relexpaddss 43736 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ 1 ∈ ℕ0𝑑 ∈ V) → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
10675, 104, 105mp3an23 1454 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
107 ovex 7465 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 + 1) ∈ V
108 csbconstg 3917 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) = ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)))
109107, 108ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) = ((𝑑𝑟𝑘) ∘ (𝑑𝑟1))
110 csbov2g 7480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖(𝑑𝑟𝑖) = (𝑑𝑟(𝑘 + 1) / 𝑖𝑖))
111 csbvarg 4433 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖𝑖 = (𝑘 + 1))
112111oveq2d 7448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑑𝑟(𝑘 + 1) / 𝑖𝑖) = (𝑑𝑟(𝑘 + 1)))
113110, 112eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖(𝑑𝑟𝑖) = (𝑑𝑟(𝑘 + 1)))
114107, 113ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑘 + 1) / 𝑖(𝑑𝑟𝑖) = (𝑑𝑟(𝑘 + 1))
115106, 109, 1143sstr4g 4036 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0(𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑘 + 1) / 𝑖(𝑑𝑟𝑖))
116 sbcssg 4519 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 + 1) ∈ V → ([(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) ↔ (𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑘 + 1) / 𝑖(𝑑𝑟𝑖)))
117107, 116ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ([(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) ↔ (𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑘 + 1) / 𝑖(𝑑𝑟𝑖))
118115, 117sylibr 234 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
119 sbcan 3837 . . . . . . . . . . . . . . . . . 18 ([(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)) ↔ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
120103, 118, 119sylanbrc 583 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
121120spesbcd 3882 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 → ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
122 df-rex 3070 . . . . . . . . . . . . . . . 16 (∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) ↔ ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
123121, 122sylibr 234 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
124100, 123mprg 3066 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
12599, 124eqsstri 4029 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
126 oveq2 7440 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑟𝑖) = (𝑑𝑟𝑘))
127126cbviunv 5039 . . . . . . . . . . . . 13 𝑖 ∈ ℕ0 (𝑑𝑟𝑖) = 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
128125, 127sseqtri 4031 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
12998, 128unssi 4190 . . . . . . . . . . 11 (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13085, 129eqsstri 4029 . . . . . . . . . 10 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13184, 130sstrdi 3995 . . . . . . . . 9 ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
132131adantl 481 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13383, 132eqsstrd 4017 . . . . . . 7 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
134133ex 412 . . . . . 6 (𝑦 ∈ ℕ → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
13559, 61, 63, 65, 80, 134nnind 12285 . . . . 5 (𝑖 ∈ ℕ → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13657, 135eqsstrid 4021 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13747, 136mprgbir 3067 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
138 iuneq1 5007 . . . 4 (ℕ0 = (ℕ ∪ {0, 1}) → 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑𝑟𝑘))
13918, 138ax-mp 5 . . 3 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑𝑟𝑘)
140137, 139sseqtri 4031 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑𝑟𝑘)
1411, 2, 3, 4, 5, 18, 37, 46, 140comptiunov2i 43724 1 (t+ ∘ r*) = t*
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  wrex 3069  Vcvv 3479  [wsbc 3787  csb 3898  cun 3948  wss 3950  {csn 4625  {cpr 4627   ciun 4990   I cid 5576  dom cdm 5684  ran crn 5685  cres 5686  ccom 5688  (class class class)co 7432  0cc0 11156  1c1 11157   + caddc 11159  cn 12267  0cn0 12528  t+ctcl 15025  t*crtcl 15026  𝑟crelexp 15059  r*crcl 43690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-seq 14044  df-trcl 15027  df-rtrcl 15028  df-relexp 15060  df-rcl 43691
This theorem is referenced by:  cortrclrcl  43761  cotrclrtrcl  43762  cortrclrtrcl  43763
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