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Theorem cotrclrcl 37053
 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 37031 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dfrcl4 36987 . 2 r* = (𝑏 ∈ V ↦ 𝑗 ∈ {0, 1} (𝑏𝑟𝑗))
3 dfrtrcl3 37044 . 2 t* = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑐𝑟𝑘))
4 nnex 10903 . 2 ℕ ∈ V
5 prex 4836 . 2 {0, 1} ∈ V
6 df-n0 11170 . . 3 0 = (ℕ ∪ {0})
7 df-pr 4128 . . . . . 6 {0, 1} = ({0} ∪ {1})
87equncomi 3721 . . . . 5 {0, 1} = ({1} ∪ {0})
98uneq2i 3726 . . . 4 (ℕ ∪ {0, 1}) = (ℕ ∪ ({1} ∪ {0}))
10 unass 3732 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ ({1} ∪ {0}))
11 1nn 10908 . . . . . . 7 1 ∈ ℕ
12 snssi 4280 . . . . . . 7 (1 ∈ ℕ → {1} ⊆ ℕ)
1311, 12ax-mp 5 . . . . . 6 {1} ⊆ ℕ
14 ssequn2 3748 . . . . . 6 ({1} ⊆ ℕ ↔ (ℕ ∪ {1}) = ℕ)
1513, 14mpbi 219 . . . . 5 (ℕ ∪ {1}) = ℕ
1615uneq1i 3725 . . . 4 ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ {0})
179, 10, 163eqtr2ri 2639 . . 3 (ℕ ∪ {0}) = (ℕ ∪ {0, 1})
186, 17eqtri 2632 . 2 0 = (ℕ ∪ {0, 1})
19 oveq2 6557 . . . 4 (𝑘 = 𝑖 → (𝑑𝑟𝑘) = (𝑑𝑟𝑖))
2019cbviunv 4495 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ ℕ (𝑑𝑟𝑖)
21 ss2iun 4472 . . . 4 (∀𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) → 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
22 1ex 9914 . . . . . . . 8 1 ∈ V
2322prid2 4242 . . . . . . 7 1 ∈ {0, 1}
24 oveq2 6557 . . . . . . . . 9 (𝑗 = 1 → (𝑑𝑟𝑗) = (𝑑𝑟1))
25 vex 3176 . . . . . . . . . 10 𝑑 ∈ V
26 relexp1g 13614 . . . . . . . . . 10 (𝑑 ∈ V → (𝑑𝑟1) = 𝑑)
2725, 26ax-mp 5 . . . . . . . . 9 (𝑑𝑟1) = 𝑑
2824, 27syl6eq 2660 . . . . . . . 8 (𝑗 = 1 → (𝑑𝑟𝑗) = 𝑑)
2928ssiun2s 4500 . . . . . . 7 (1 ∈ {0, 1} → 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
3023, 29ax-mp 5 . . . . . 6 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)
3130a1i 11 . . . . 5 (𝑖 ∈ ℕ → 𝑑 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
32 ovex 6577 . . . . . . 7 (𝑑𝑟𝑗) ∈ V
335, 32iunex 7039 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V
3433a1i 11 . . . . 5 (𝑖 ∈ ℕ → 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V)
35 nnnn0 11176 . . . . 5 (𝑖 ∈ ℕ → 𝑖 ∈ ℕ0)
3631, 34, 35relexpss1d 37016 . . . 4 (𝑖 ∈ ℕ → (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
3721, 36mprg 2910 . . 3 𝑖 ∈ ℕ (𝑑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
3820, 37eqsstri 3598 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
39 oveq2 6557 . . . . 5 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1))
40 relexp1g 13614 . . . . . . 7 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) ∈ V → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ {0, 1} (𝑑𝑟𝑗))
4133, 40ax-mp 5 . . . . . 6 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)
42 oveq2 6557 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
4342cbviunv 4495 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4441, 43eqtri 2632 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
4539, 44syl6eq 2660 . . . 4 (𝑖 = 1 → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ {0, 1} (𝑑𝑟𝑘))
4645ssiun2s 4500 . . 3 (1 ∈ ℕ → 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖))
4711, 46ax-mp 5 . 2 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖)
48 iunss 4497 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
49 iuneq1 4470 . . . . . . . 8 ({0, 1} = ({0} ∪ {1}) → 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗))
507, 49ax-mp 5 . . . . . . 7 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗)
51 iunxun 4541 . . . . . . 7 𝑗 ∈ ({0} ∪ {1})(𝑑𝑟𝑗) = ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗))
52 c0ex 9913 . . . . . . . . 9 0 ∈ V
53 oveq2 6557 . . . . . . . . 9 (𝑗 = 0 → (𝑑𝑟𝑗) = (𝑑𝑟0))
5452, 53iunxsn 4539 . . . . . . . 8 𝑗 ∈ {0} (𝑑𝑟𝑗) = (𝑑𝑟0)
5522, 24iunxsn 4539 . . . . . . . 8 𝑗 ∈ {1} (𝑑𝑟𝑗) = (𝑑𝑟1)
5654, 55uneq12i 3727 . . . . . . 7 ( 𝑗 ∈ {0} (𝑑𝑟𝑗) ∪ 𝑗 ∈ {1} (𝑑𝑟𝑗)) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5750, 51, 563eqtri 2636 . . . . . 6 𝑗 ∈ {0, 1} (𝑑𝑟𝑗) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
5857oveq1i 6559 . . . . 5 ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖)
59 oveq2 6557 . . . . . . 7 (𝑥 = 1 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1))
6059sseq1d 3595 . . . . . 6 (𝑥 = 1 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
61 oveq2 6557 . . . . . . 7 (𝑥 = 𝑦 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦))
6261sseq1d 3595 . . . . . 6 (𝑥 = 𝑦 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
63 oveq2 6557 . . . . . . 7 (𝑥 = (𝑦 + 1) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)))
6463sseq1d 3595 . . . . . 6 (𝑥 = (𝑦 + 1) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
65 oveq2 6557 . . . . . . 7 (𝑥 = 𝑖 → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) = (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖))
6665sseq1d 3595 . . . . . 6 (𝑥 = 𝑖 → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ↔ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
67 ovex 6577 . . . . . . . . 9 (𝑑𝑟0) ∈ V
68 ovex 6577 . . . . . . . . 9 (𝑑𝑟1) ∈ V
6967, 68unex 6854 . . . . . . . 8 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V
70 relexp1g 13614 . . . . . . . 8 (((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1)))
7169, 70ax-mp 5 . . . . . . 7 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) = ((𝑑𝑟0) ∪ (𝑑𝑟1))
72 0nn0 11184 . . . . . . . . 9 0 ∈ ℕ0
73 oveq2 6557 . . . . . . . . . 10 (𝑘 = 0 → (𝑑𝑟𝑘) = (𝑑𝑟0))
7473ssiun2s 4500 . . . . . . . . 9 (0 ∈ ℕ0 → (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7572, 74ax-mp 5 . . . . . . . 8 (𝑑𝑟0) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
76 1nn0 11185 . . . . . . . . 9 1 ∈ ℕ0
77 oveq2 6557 . . . . . . . . . 10 (𝑘 = 1 → (𝑑𝑟𝑘) = (𝑑𝑟1))
7877ssiun2s 4500 . . . . . . . . 9 (1 ∈ ℕ0 → (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
7976, 78ax-mp 5 . . . . . . . 8 (𝑑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
8075, 79unssi 3750 . . . . . . 7 ((𝑑𝑟0) ∪ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
8171, 80eqsstri 3598 . . . . . 6 (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟1) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
82 simpl 472 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
83 relexpsucnnr 13613 . . . . . . . . 9 ((((𝑑𝑟0) ∪ (𝑑𝑟1)) ∈ V ∧ 𝑦 ∈ ℕ) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
8469, 82, 83sylancr 694 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) = ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
85 coss1 5199 . . . . . . . . . 10 ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))))
86 coundi 5553 . . . . . . . . . . 11 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) = (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)))
87 relexp0g 13610 . . . . . . . . . . . . . . . 16 (𝑑 ∈ V → (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
8825, 87ax-mp 5 . . . . . . . . . . . . . . 15 (𝑑𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑))
8988coeq2i 5204 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
90 coiun1 36963 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
91 coires1 5570 . . . . . . . . . . . . . . . 16 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
9291a1i 11 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)))
9392iuneq2i 4475 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
9489, 90, 933eqtri 2636 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
95 ss2iun 4472 . . . . . . . . . . . . . 14 (∀𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
96 resss 5342 . . . . . . . . . . . . . . 15 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘)
9796a1i 11 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑𝑟𝑘))
9895, 97mprg 2910 . . . . . . . . . . . . 13 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
9994, 98eqsstri 3598 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
100 coiun1 36963 . . . . . . . . . . . . . 14 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) = 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1))
101 iunss2 4501 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ ℕ0𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) → 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖))
102 peano2nn0 11210 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
103 sbcel1v 3462 . . . . . . . . . . . . . . . . . . 19 ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0 ↔ (𝑘 + 1) ∈ ℕ0)
104102, 103sylibr 223 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0)
105 relexpaddss 37029 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℕ0 ∧ 1 ∈ ℕ0𝑑 ∈ V) → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
10676, 25, 105mp3an23 1408 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℕ0 → ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟(𝑘 + 1)))
107 ovex 6577 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 + 1) ∈ V
108 csbconstg 3512 . . . . . . . . . . . . . . . . . . . . 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 6589 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖(𝑑𝑟𝑖) = (𝑑𝑟(𝑘 + 1) / 𝑖𝑖))
111 csbvarg 3955 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 + 1) ∈ V → (𝑘 + 1) / 𝑖𝑖 = (𝑘 + 1))
112111oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 + 1) ∈ V → (𝑑𝑟(𝑘 + 1) / 𝑖𝑖) = (𝑑𝑟(𝑘 + 1)))
113110, 112eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0(𝑘 + 1) / 𝑖((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑘 + 1) / 𝑖(𝑑𝑟𝑖))
116 sbcssg 4035 . . . . . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
119 sbcan 3445 . . . . . . . . . . . . . . . . . 18 ([(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)) ↔ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ0[(𝑘 + 1) / 𝑖]((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
120104, 118, 119sylanbrc 695 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0[(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
121120spesbcd 3488 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 → ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
122 df-rex 2902 . . . . . . . . . . . . . . . 16 (∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖) ↔ ∃𝑖(𝑖 ∈ ℕ0 ∧ ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖)))
123121, 122sylibr 223 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ0 → ∃𝑖 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ (𝑑𝑟𝑖))
124101, 123mprg 2910 . . . . . . . . . . . . . 14 𝑘 ∈ ℕ0 ((𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
125100, 124eqsstri 3598 . . . . . . . . . . . . 13 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑖 ∈ ℕ0 (𝑑𝑟𝑖)
126 oveq2 6557 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (𝑑𝑟𝑖) = (𝑑𝑟𝑘))
127126cbviunv 4495 . . . . . . . . . . . . 13 𝑖 ∈ ℕ0 (𝑑𝑟𝑖) = 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
128125, 127sseqtri 3600 . . . . . . . . . . . 12 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
12999, 128unssi 3750 . . . . . . . . . . 11 (( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟0)) ∪ ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13086, 129eqsstri 3598 . . . . . . . . . 10 ( 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) ∘ ((𝑑𝑟0) ∪ (𝑑𝑟1))) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
13185, 130syl6ss 3580 . . . . . . . . 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))
13384, 132eqsstrd 3602 . . . . . . 7 ((𝑦 ∈ ℕ ∧ (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
134133ex 449 . . . . . 6 (𝑦 ∈ ℕ → ((((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘) → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)))
13560, 62, 64, 66, 81, 134nnind 10915 . . . . 5 (𝑖 ∈ ℕ → (((𝑑𝑟0) ∪ (𝑑𝑟1))↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13658, 135syl5eqss 3612 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘))
13748, 136mprgbir 2911 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ0 (𝑑𝑟𝑘)
138 iuneq1 4470 . . . 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 3600 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ {0, 1} (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑𝑟𝑘)
1411, 2, 3, 4, 5, 18, 38, 47, 140comptiunov2i 37017 1 (t+ ∘ r*) = t*
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  [wsbc 3402  ⦋csb 3499   ∪ cun 3538   ⊆ wss 3540  {csn 4125  {cpr 4127  ∪ ciun 4455   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ0cn0 11169  t+ctcl 13572  t*crtcl 13573  ↑𝑟crelexp 13608  r*crcl 36983 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-trcl 13574  df-rtrcl 13575  df-relexp 13609  df-rcl 36984 This theorem is referenced by:  cortrclrcl  37054  cotrclrtrcl  37055  cortrclrtrcl  37056
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