cotrcltrcl - Mathbox for Richard Penner

	Mathbox for Richard Penner		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cotrcltrcl		Structured version Visualization version GIF version

Theorem cotrcltrcl 43738

Description: The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)

**Assertion**
Ref	Expression
cotrcltrcl	⊢ (t+ ∘ t+) = t+

Proof of Theorem cotrcltrcl 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 43733	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑_𝑟𝑖))
2		dftrcl3 43733	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dftrcl3 43733	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘))
4		nnex 12272	. 2 ⊢ ℕ ∈ V
5		unidm 4157	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2746	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 11257	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7439	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
9	7, 8	iunxsn 5091	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
10		ovex 7464	. . . . . . . 8 ⊢ (𝑑↑_𝑟𝑗) ∈ V
11	4, 10	iunex 7993	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
12		relexp1g 15065	. . . . . . 7 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
14		oveq2 7439	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟𝑘))
15	14	cbviunv 5040	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
16	13, 15	eqtri 2765	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
17	9, 16	eqtri 2765	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
18	17	eqcomi 2746	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
19		1nn 12277	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4808	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 5006	. . . 4 ⊢ ({1} ⊆ ℕ → ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
22	19, 20, 21	mp2b 10	. . 3 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
23	18, 22	eqsstri 4030	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
24		iunss 5045	. . . 4 ⊢ (∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ ∀𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
25		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
26	25	sseq1d 4015	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
27		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦))
28	27	sseq1d 4015	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
29		oveq2 7439	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)))
30	29	sseq1d 4015	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
31		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
32	31	sseq1d 4015	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
33	16	eqimssi 4044	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
34		simpl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) → 𝑦 ∈ ℕ)
35		relexpsucnnr 15064	. . . . . . . 8 ⊢ ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)))
36	11, 34, 35	sylancr 587	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)))
37		coss1 5866	. . . . . . . . 9 ⊢ ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)))
38	37	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)))
39	15	coeq2i 5871	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
40		trclfvcotrg 15055	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑_𝑟𝑘) = (𝑑↑_𝑟𝑘))
42	41	iuneq2d 5022	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
43		ovex 7464	. . . . . . . . . . . . . 14 ⊢ (𝑑↑_𝑟𝑘) ∈ V
44	4, 43	iunex 7993	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 7016	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
46	45	elv 3485	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
47	46, 46	coeq12i 5874	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
48	40, 47, 46	3sstr3i 4034	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
49	39, 48	eqsstri 4030	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
50	38, 49	sstrdi 3996	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
51	36, 50	eqsstrd 4018	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
52	51	ex 412	. . . . 5 ⊢ (𝑦 ∈ ℕ → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
53	26, 28, 30, 32, 33, 52	nnind 12284	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
54	24, 53	mprgbir 3068	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
55		iuneq1 5008	. . . 4 ⊢ (ℕ = (ℕ ∪ ℕ) → ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑_𝑟𝑘))
56	6, 55	ax-mp 5	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑_𝑟𝑘)
57	54, 56	sseqtri 4032	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑_𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 43719	1 ⊢ (t+ ∘ t+) = t+

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ∪ ciun 4991 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 ℕcn 12266 t+ctcl 15024 ↑_𝑟crelexp 15058

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-trcl 15026 df-relexp 15059

This theorem is referenced by: cortrcltrcl 43753 cotrclrtrcl 43757

W3C validator