cotrclrcl - Mathbox for Richard Penner

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Theorem cotrclrcl 43983

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.

Step	Hyp	Ref	Expression
1		dftrcl3 43961	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑_𝑟𝑖))
2		dfrcl4 43917	. 2 ⊢ r* = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ {0, 1} (𝑏↑_𝑟𝑗))
3		dfrtrcl3 43974	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ₀ (𝑐↑_𝑟𝑘))
4		nnex 12151	. 2 ⊢ ℕ ∈ V
5		prex 5382	. 2 ⊢ {0, 1} ∈ V
6		df-n0 12402	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
7		df-pr 4583	. . . . . 6 ⊢ {0, 1} = ({0} ∪ {1})
8	7	equncomi 4112	. . . . 5 ⊢ {0, 1} = ({1} ∪ {0})
9	8	uneq2i 4117	. . . 4 ⊢ (ℕ ∪ {0, 1}) = (ℕ ∪ ({1} ∪ {0}))
10		unass 4124	. . . 4 ⊢ ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ ({1} ∪ {0}))
11		1nn 12156	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4764	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn2 4141	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ (ℕ ∪ {1}) = ℕ)
15	13, 14	mpbi 230	. . . . 5 ⊢ (ℕ ∪ {1}) = ℕ
16	15	uneq1i 4116	. . . 4 ⊢ ((ℕ ∪ {1}) ∪ {0}) = (ℕ ∪ {0})
17	9, 10, 16	3eqtr2ri 2766	. . 3 ⊢ (ℕ ∪ {0}) = (ℕ ∪ {0, 1})
18	6, 17	eqtri 2759	. 2 ⊢ ℕ₀ = (ℕ ∪ {0, 1})
19		oveq2 7366	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑖))
20	19	cbviunv 4994	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ ℕ (𝑑↑_𝑟𝑖)
21		ss2iun 4965	. . . 4 ⊢ (∀𝑖 ∈ ℕ (𝑑↑_𝑟𝑖) ⊆ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) → ∪ 𝑖 ∈ ℕ (𝑑↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
22		1ex 11128	. . . . . . . 8 ⊢ 1 ∈ V
23	22	prid2 4720	. . . . . . 7 ⊢ 1 ∈ {0, 1}
24		oveq2 7366	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟1))
25		relexp1g 14949	. . . . . . . . . 10 ⊢ (𝑑 ∈ V → (𝑑↑_𝑟1) = 𝑑)
26	25	elv 3445	. . . . . . . . 9 ⊢ (𝑑↑_𝑟1) = 𝑑
27	24, 26	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑗 = 1 → (𝑑↑_𝑟𝑗) = 𝑑)
28	27	ssiun2s 5004	. . . . . . 7 ⊢ (1 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗))
29	23, 28	ax-mp 5	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)
30	29	a1i 11	. . . . 5 ⊢ (𝑖 ∈ ℕ → 𝑑 ⊆ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗))
31		ovex 7391	. . . . . . 7 ⊢ (𝑑↑_𝑟𝑗) ∈ V
32	5, 31	iunex 7912	. . . . . 6 ⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) ∈ V
33	32	a1i 11	. . . . 5 ⊢ (𝑖 ∈ ℕ → ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) ∈ V)
34		nnnn0 12408	. . . . 5 ⊢ (𝑖 ∈ ℕ → 𝑖 ∈ ℕ₀)
35	30, 33, 34	relexpss1d 43946	. . . 4 ⊢ (𝑖 ∈ ℕ → (𝑑↑_𝑟𝑖) ⊆ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
36	21, 35	mprg 3057	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (𝑑↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
37	20, 36	eqsstri 3980	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
38		oveq2 7366	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟1))
39		relexp1g 14949	. . . . . . 7 ⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) ∈ V → (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗))
40	32, 39	ax-mp 5	. . . . . 6 ⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)
41		oveq2 7366	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟𝑘))
42	41	cbviunv 4994	. . . . . 6 ⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) = ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘)
43	40, 42	eqtri 2759	. . . . 5 ⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘)
44	38, 43	eqtrdi 2787	. . . 4 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘))
45	44	ssiun2s 5004	. . 3 ⊢ (1 ∈ ℕ → ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
46	11, 45	ax-mp 5	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
47		iunss 5000	. . . 4 ⊢ (∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ↔ ∀𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
48		iuneq1 4963	. . . . . . . 8 ⊢ ({0, 1} = ({0} ∪ {1}) → ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) = ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑_𝑟𝑗))
49	7, 48	ax-mp 5	. . . . . . 7 ⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) = ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑_𝑟𝑗)
50		iunxun 5049	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ({0} ∪ {1})(𝑑↑_𝑟𝑗) = (∪ 𝑗 ∈ {0} (𝑑↑_𝑟𝑗) ∪ ∪ 𝑗 ∈ {1} (𝑑↑_𝑟𝑗))
51		c0ex 11126	. . . . . . . . 9 ⊢ 0 ∈ V
52		oveq2 7366	. . . . . . . . 9 ⊢ (𝑗 = 0 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟0))
53	51, 52	iunxsn 5046	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {0} (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟0)
54	22, 24	iunxsn 5046	. . . . . . . 8 ⊢ ∪ 𝑗 ∈ {1} (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟1)
55	53, 54	uneq12i 4118	. . . . . . 7 ⊢ (∪ 𝑗 ∈ {0} (𝑑↑_𝑟𝑗) ∪ ∪ 𝑗 ∈ {1} (𝑑↑_𝑟𝑗)) = ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))
56	49, 50, 55	3eqtri 2763	. . . . . 6 ⊢ ∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗) = ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))
57	56	oveq1i 7368	. . . . 5 ⊢ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑖)
58		oveq2 7366	. . . . . . 7 ⊢ (𝑥 = 1 → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) = (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟1))
59	58	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = 1 → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ↔ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)))
60		oveq2 7366	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) = (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦))
61	60	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ↔ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)))
62		oveq2 7366	. . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) = (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)))
63	62	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ↔ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)))
64		oveq2 7366	. . . . . . 7 ⊢ (𝑥 = 𝑖 → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) = (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑖))
65	64	sseq1d 3965	. . . . . 6 ⊢ (𝑥 = 𝑖 → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ↔ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)))
66		ovex 7391	. . . . . . . . 9 ⊢ (𝑑↑_𝑟0) ∈ V
67		ovex 7391	. . . . . . . . 9 ⊢ (𝑑↑_𝑟1) ∈ V
68	66, 67	unex 7689	. . . . . . . 8 ⊢ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1)) ∈ V
69		relexp1g 14949	. . . . . . . 8 ⊢ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1)) ∈ V → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟1) = ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1)))
70	68, 69	ax-mp 5	. . . . . . 7 ⊢ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟1) = ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))
71		0nn0 12416	. . . . . . . . 9 ⊢ 0 ∈ ℕ₀
72		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟0))
73	72	ssiun2s 5004	. . . . . . . . 9 ⊢ (0 ∈ ℕ₀ → (𝑑↑_𝑟0) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
74	71, 73	ax-mp 5	. . . . . . . 8 ⊢ (𝑑↑_𝑟0) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
75		1nn0 12417	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
76		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑘 = 1 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟1))
77	76	ssiun2s 5004	. . . . . . . . 9 ⊢ (1 ∈ ℕ₀ → (𝑑↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
78	75, 77	ax-mp 5	. . . . . . . 8 ⊢ (𝑑↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
79	74, 78	unssi 4143	. . . . . . 7 ⊢ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
80	70, 79	eqsstri 3980	. . . . . 6 ⊢ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
81		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)) → 𝑦 ∈ ℕ)
82		relexpsucnnr 14948	. . . . . . . . 9 ⊢ ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1)) ∈ V ∧ 𝑦 ∈ ℕ) → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)) = ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))))
83	68, 81, 82	sylancr 587	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)) → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)) = ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))))
84		coss1 5804	. . . . . . . . . 10 ⊢ ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))) ⊆ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))))
85		coundi 6205	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))) = ((∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟0)) ∪ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)))
86		relexp0g 14945	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ V → (𝑑↑_𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
87	86	elv 3445	. . . . . . . . . . . . . . 15 ⊢ (𝑑↑_𝑟0) = ( I ↾ (dom 𝑑 ∪ ran 𝑑))
88	87	coeq2i 5809	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟0)) = (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
89		coiun1 43893	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑)))
90		coires1 6223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
91	90	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → ((𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)))
92	91	iuneq2i 4968	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ ( I ↾ (dom 𝑑 ∪ ran 𝑑))) = ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
93	88, 89, 92	3eqtri 2763	. . . . . . . . . . . . 13 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟0)) = ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑))
94		ss2iun 4965	. . . . . . . . . . . . . 14 ⊢ (∀𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑_𝑟𝑘) → ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
95		resss 5960	. . . . . . . . . . . . . . 15 ⊢ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑_𝑟𝑘)
96	95	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ (𝑑↑_𝑟𝑘))
97	94, 96	mprg 3057	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ↾ (dom 𝑑 ∪ ran 𝑑)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
98	93, 97	eqsstri 3980	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟0)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
99		coiun1 43893	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) = ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1))
100		iunss2 5005	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ ℕ₀ ∃𝑖 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖) → ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ∪ 𝑖 ∈ ℕ₀ (𝑑↑_𝑟𝑖))
101		peano2nn0 12441	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ₀ → (𝑘 + 1) ∈ ℕ₀)
102		sbcel1v 3806	. . . . . . . . . . . . . . . . . . 19 ⊢ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ₀ ↔ (𝑘 + 1) ∈ ℕ₀)
103	101, 102	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ₀ → [(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ₀)
104		vex 3444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑑 ∈ V
105		relexpaddss 43959	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ 1 ∈ ℕ₀ ∧ 𝑑 ∈ V) → ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟(𝑘 + 1)))
106	75, 104, 105	mp3an23 1455	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ₀ → ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟(𝑘 + 1)))
107		ovex 7391	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 + 1) ∈ V
108		csbconstg 3868	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 + 1) ∈ V → ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) = ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)))
109	107, 108	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) = ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1))
110		csbov2g 7406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 + 1) ∈ V → ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖) = (𝑑↑_𝑟⦋(𝑘 + 1) / 𝑖⦌𝑖))
111		csbvarg 4386	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 + 1) ∈ V → ⦋(𝑘 + 1) / 𝑖⦌𝑖 = (𝑘 + 1))
112	111	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 + 1) ∈ V → (𝑑↑_𝑟⦋(𝑘 + 1) / 𝑖⦌𝑖) = (𝑑↑_𝑟(𝑘 + 1)))
113	110, 112	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 + 1) ∈ V → ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖) = (𝑑↑_𝑟(𝑘 + 1)))
114	107, 113	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖) = (𝑑↑_𝑟(𝑘 + 1))
115	106, 109, 114	3sstr4g 3987	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ₀ → ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖))
116		sbcssg 4474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 + 1) ∈ V → ([(𝑘 + 1) / 𝑖]((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖) ↔ ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖)))
117	107, 116	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ([(𝑘 + 1) / 𝑖]((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖) ↔ ⦋(𝑘 + 1) / 𝑖⦌((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ⦋(𝑘 + 1) / 𝑖⦌(𝑑↑_𝑟𝑖))
118	115, 117	sylibr 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ₀ → [(𝑘 + 1) / 𝑖]((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖))
119		sbcan 3790	. . . . . . . . . . . . . . . . . 18 ⊢ ([(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ₀ ∧ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖)) ↔ ([(𝑘 + 1) / 𝑖]𝑖 ∈ ℕ₀ ∧ [(𝑘 + 1) / 𝑖]((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖)))
120	103, 118, 119	sylanbrc 583	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ℕ₀ → [(𝑘 + 1) / 𝑖](𝑖 ∈ ℕ₀ ∧ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖)))
121	120	spesbcd 3833	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ₀ → ∃𝑖(𝑖 ∈ ℕ₀ ∧ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖)))
122		df-rex 3061	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑖 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖) ↔ ∃𝑖(𝑖 ∈ ℕ₀ ∧ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖)))
123	121, 122	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ₀ → ∃𝑖 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ (𝑑↑_𝑟𝑖))
124	100, 123	mprg 3057	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑘 ∈ ℕ₀ ((𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ∪ 𝑖 ∈ ℕ₀ (𝑑↑_𝑟𝑖)
125	99, 124	eqsstri 3980	. . . . . . . . . . . . 13 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ∪ 𝑖 ∈ ℕ₀ (𝑑↑_𝑟𝑖)
126		oveq2 7366	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 → (𝑑↑_𝑟𝑖) = (𝑑↑_𝑟𝑘))
127	126	cbviunv 4994	. . . . . . . . . . . . 13 ⊢ ∪ 𝑖 ∈ ℕ₀ (𝑑↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
128	125, 127	sseqtri 3982	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
129	98, 128	unssi 4143	. . . . . . . . . . 11 ⊢ ((∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟0)) ∪ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ (𝑑↑_𝑟1))) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
130	85, 129	eqsstri 3980	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
131	84, 130	sstrdi 3946	. . . . . . . . 9 ⊢ ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
132	131	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)) → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ∘ ((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
133	83, 132	eqsstrd 3968	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)) → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
134	133	ex 412	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)))
135	59, 61, 63, 65, 80, 134	nnind 12163	. . . . 5 ⊢ (𝑖 ∈ ℕ → (((𝑑↑_𝑟0) ∪ (𝑑↑_𝑟1))↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
136	57, 135	eqsstrid 3972	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘))
137	47, 136	mprgbir 3058	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘)
138		iuneq1 4963	. . . 4 ⊢ (ℕ₀ = (ℕ ∪ {0, 1}) → ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑↑_𝑟𝑘))
139	18, 138	ax-mp 5	. . 3 ⊢ ∪ 𝑘 ∈ ℕ₀ (𝑑↑_𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑↑_𝑟𝑘)
140	137, 139	sseqtri 3982	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ {0, 1} (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ {0, 1})(𝑑↑_𝑟𝑘)
141	1, 2, 3, 4, 5, 18, 37, 46, 140	comptiunov2i 43947	1 ⊢ (t+ ∘ r) = t

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∃wrex 3060 Vcvv 3440 [wsbc 3740 ⦋csb 3849 ∪ cun 3899 ⊆ wss 3901 {csn 4580 {cpr 4582 ∪ ciun 4946 I cid 5518 dom cdm 5624 ran crn 5625 ↾ cres 5626 ∘ ccom 5628 (class class class)co 7358 0cc0 11026 1c1 11027 + caddc 11029 ℕcn 12145 ℕ₀cn0 12401 t+ctcl 14908 t*crtcl 14909 ↑_𝑟crelexp 14942 r*crcl 43913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-n0 12402 df-z 12489 df-uz 12752 df-seq 13925 df-trcl 14910 df-rtrcl 14911 df-relexp 14943 df-rcl 43914

This theorem is referenced by: cortrclrcl 43984 cotrclrtrcl 43985 cortrclrtrcl 43986

W3C validator