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Theorem cotrclrcl 40436
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 40414 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dfrcl4 40370 . 2 r* = (𝑏 ∈ V ↦ 𝑗 ∈ {0, 1} (𝑏𝑟𝑗))
3 dfrtrcl3 40427 . 2 t* = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑐𝑟𝑘))
4 nnex 11635 . 2 ℕ ∈ V
5 prex 5301 . 2 {0, 1} ∈ V
6 df-n0 11890 . . 3 0 = (ℕ ∪ {0})
7 df-pr 4531 . . . . . 6 {0, 1} = ({0} ∪ {1})
87equncomi 4085 . . . . 5 {0, 1} = ({1} ∪ {0})
98uneq2i 4090 . . . 4 (ℕ ∪ {0, 1}) = (ℕ ∪ ({1} ∪ {0}))
10 unass 4096 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ ({1} ∪ {0}))
11 1nn 11640 . . . . . . 7 1 ∈ ℕ
12 snssi 4704 . . . . . . 7 (1 ∈ ℕ → {1} ⊆ ℕ)
1311, 12ax-mp 5 . . . . . 6 {1} ⊆ ℕ
14 ssequn2 4113 . . . . . 6 ({1} ⊆ ℕ ↔ (ℕ ∪ {1}) = ℕ)
1513, 14mpbi 233 . . . . 5 (ℕ ∪ {1}) = ℕ
1615uneq1i 4089 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ {0})
179, 10, 163eqtr2ri 2831 . . 3 (ℕ ∪ {0}) = (ℕ ∪ {0, 1})
186, 17eqtri 2824 . 2 0 = (ℕ ∪ {0, 1})
19 oveq2 7147 . . . 4 (𝑘 = 𝑖 → (𝑑𝑟𝑘) = (𝑑𝑟𝑖))
2019cbviunv 4930 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ ℕ (𝑑𝑟𝑖)
21 ss2iun 4902 . . . 4 (∀𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) → 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
22 1ex 10630 . . . . . . . 8 1 ∈ V
2322prid2 4662 . . . . . . 7 1 ∈ {0, 1}
24 oveq2 7147 . . . . . . . . 9 (𝑗 = 1 → (𝑑𝑟𝑗) = (𝑑𝑟1))
25 relexp1g 14381 . . . . . . . . . 10 (𝑑 ∈ V → (𝑑𝑟1) = 𝑑)
2625elv 3449 . . . . . . . . 9 (𝑑𝑟1) = 𝑑
2724, 26eqtrdi 2852 . . . . . . . 8 (𝑗 = 1 → (𝑑𝑟𝑗) = 𝑑)
2827ssiun2s 4938 . . . . . . 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 7172 . . . . . . 7 (𝑑𝑟𝑗) ∈ V
325, 31iunex 7655 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V
3332a1i 11 . . . . 5 (𝑖 ∈ ℕ → 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V)
34 nnnn0 11896 . . . . 5 (𝑖 ∈ ℕ → 𝑖 ∈ ℕ0)
3530, 33, 34relexpss1d 40399 . . . 4 (𝑖 ∈ ℕ → (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
3621, 35mprg 3123 . . 3 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
3720, 36eqsstri 3952 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
38 oveq2 7147 . . . . 5 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1))
39 relexp1g 14381 . . . . . . 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 7147 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
4241cbviunv 4930 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4340, 42eqtri 2824 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4438, 43eqtrdi 2852 . . . 4 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘))
4544ssiun2s 4938 . . 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 4935 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
48 iuneq1 4900 . . . . . . . 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 4982 . . . . . . 7 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗) = ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗))
51 c0ex 10628 . . . . . . . . 9 0 ∈ V
52 oveq2 7147 . . . . . . . . 9 (𝑗 = 0 → (𝑑𝑟𝑗) = (𝑑𝑟0))
5351, 52iunxsn 4979 . . . . . . . 8 𝑗 ∈ {0} (𝑑𝑟𝑗) = (𝑑𝑟0)
5422, 24iunxsn 4979 . . . . . . . 8 𝑗 ∈ {1} (𝑑𝑟𝑗) = (𝑑𝑟1)
5553, 54uneq12i 4091 . . . . . . 7 ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗)) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5649, 50, 553eqtri 2828 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5756oveq1i 7149 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖)
58 oveq2 7147 . . . . . . 7 (𝑥 = 1 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1))
5958sseq1d 3949 . . . . . 6 (𝑥 = 1 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
60 oveq2 7147 . . . . . . 7 (𝑥 = 𝑦 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦))
6160sseq1d 3949 . . . . . 6 (𝑥 = 𝑦 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
62 oveq2 7147 . . . . . . 7 (𝑥 = (𝑦 + 1) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)))
6362sseq1d 3949 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
64 oveq2 7147 . . . . . . 7 (𝑥 = 𝑖 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖))
6564sseq1d 3949 . . . . . 6 (𝑥 = 𝑖 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
66 ovex 7172 . . . . . . . . 9 (𝑑𝑟0) ∈ V
67 ovex 7172 . . . . . . . . 9 (𝑑𝑟1) ∈ V
6866, 67unex 7453 . . . . . . . 8 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V
69 relexp1g 14381 . . . . . . . 8 (((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1)))
7068, 69ax-mp 5 . . . . . . 7 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
71 0nn0 11904 . . . . . . . . 9 0 ∈ ℕ0
72 oveq2 7147 . . . . . . . . . 10 (𝑘 = 0 → (𝑑𝑟𝑘) = (𝑑𝑟0))
7372ssiun2s 4938 . . . . . . . . 9 (0 ∈ ℕ0 → (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7471, 73ax-mp 5 . . . . . . . 8 (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
75 1nn0 11905 . . . . . . . . 9 1 ∈ ℕ0
76 oveq2 7147 . . . . . . . . . 10 (𝑘 = 1 → (𝑑𝑟𝑘) = (𝑑𝑟1))
7776ssiun2s 4938 . . . . . . . . 9 (1 ∈ ℕ0 → (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7875, 77ax-mp 5 . . . . . . . 8 (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
7974, 78unssi 4115 . . . . . . 7 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
8070, 79eqsstri 3952 . . . . . 6 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
81 simpl 486 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
82 relexpsucnnr 14380 . . . . . . . . 9 ((((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V ∧ 𝑦 ∈ ℕ) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
8368, 81, 82sylancr 590 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
84 coss1 5694 . . . . . . . . . 10 ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
85 coundi 6071 . . . . . . . . . . 11 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) = (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)))
86 relexp0g 14377 . . . . . . . . . . . . . . . 16 (𝑑 ∈ V → (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
8786elv 3449 . . . . . . . . . . . . . . 15 (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑))
8887coeq2i 5699 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
89 coiun1 40346 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
90 coires1 6088 . . . . . . . . . . . . . . . 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 4905 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
9388, 89, 923eqtri 2828 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
94 ss2iun 4902 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
95 resss 5847 . . . . . . . . . . . . . . 15 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘)
9695a1i 11 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘))
9794, 96mprg 3123 . . . . . . . . . . . . 13 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
9893, 97eqsstri 3952 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
99 coiun1 40346 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1))
100 iunss2 4939 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ ℕ0𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖))
101 peano2nn0 11929 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
102 sbcel1v 3789 . . . . . . . . . . . . . . . . . . 19 ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0 ↔ (𝑘 + 1) ∈ ℕ0)
103101, 102sylibr 237 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0)
104 vex 3447 . . . . . . . . . . . . . . . . . . . . 21 𝑑 ∈ V
105 relexpaddss 40412 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ 1 ∈ ℕ0𝑑 ∈ V) → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
10675, 104, 105mp3an23 1450 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
107 ovex 7172 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 + 1) ∈ V
108 csbconstg 3850 . . . . . . . . . . . . . . . . . . . . 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 7185 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖(𝑑𝑟𝑖) = (𝑑𝑟(𝑘 + 1) / 𝑖𝑖))
111 csbvarg 4342 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖𝑖 = (𝑘 + 1))
112111oveq2d 7155 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑑𝑟(𝑘 + 1) / 𝑖𝑖) = (𝑑𝑟(𝑘 + 1)))
113110, 112eqtrd 2836 . . . . . . . . . . . . . . . . . . . . 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 3963 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0(𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑘 + 1) / 𝑖(𝑑𝑟𝑖))
116 sbcssg 4424 . . . . . . . . . . . . . . . . . . . 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 237 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
119 sbcan 3771 . . . . . . . . . . . . . . . . . 18 ([(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)) ↔ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
120103, 118, 119sylanbrc 586 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
121120spesbcd 3815 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 → ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
122 df-rex 3115 . . . . . . . . . . . . . . . 16 (∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) ↔ ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
123121, 122sylibr 237 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
124100, 123mprg 3123 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
12599, 124eqsstri 3952 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
126 oveq2 7147 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑟𝑖) = (𝑑𝑟𝑘))
127126cbviunv 4930 . . . . . . . . . . . . 13 𝑖 ∈ ℕ0 (𝑑𝑟𝑖) = 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
128125, 127sseqtri 3954 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
12998, 128unssi 4115 . . . . . . . . . . 11 (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13085, 129eqsstri 3952 . . . . . . . . . 10 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13184, 130sstrdi 3930 . . . . . . . . 9 ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
132131adantl 485 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13383, 132eqsstrd 3956 . . . . . . 7 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
134133ex 416 . . . . . 6 (𝑦 ∈ ℕ → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
13559, 61, 63, 65, 80, 134nnind 11647 . . . . 5 (𝑖 ∈ ℕ → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13657, 135eqsstrid 3966 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13747, 136mprgbir 3124 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
138 iuneq1 4900 . . . 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 3954 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑𝑟𝑘)
1411, 2, 3, 4, 5, 18, 37, 46, 140comptiunov2i 40400 1 (t+ ∘ r*) = t*
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wrex 3110  Vcvv 3444  [wsbc 3723  csb 3831  cun 3882  wss 3884  {csn 4528  {cpr 4530   ciun 4884   I cid 5427  dom cdm 5523  ran crn 5524  cres 5525  ccom 5527  (class class class)co 7139  0cc0 10530  1c1 10531   + caddc 10533  cn 11629  0cn0 11889  t+ctcl 14340  t*crtcl 14341  𝑟crelexp 14374  r*crcl 40366
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-seq 13369  df-trcl 14342  df-rtrcl 14343  df-relexp 14375  df-rcl 40367
This theorem is referenced by:  cortrclrcl  40437  cotrclrtrcl  40438  cortrclrtrcl  40439
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