cotrcltrcl - Mathbox for Richard Penner

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Theorem cotrcltrcl 43155

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 43150	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑_𝑟𝑖))
2		dftrcl3 43150	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dftrcl3 43150	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘))
4		nnex 12249	. 2 ⊢ ℕ ∈ V
5		unidm 4151	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2737	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 11241	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7428	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
9	7, 8	iunxsn 5094	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
10		ovex 7453	. . . . . . . 8 ⊢ (𝑑↑_𝑟𝑗) ∈ V
11	4, 10	iunex 7972	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
12		relexp1g 15006	. . . . . . 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 7428	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟𝑘))
15	14	cbviunv 5043	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
16	13, 15	eqtri 2756	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
17	9, 16	eqtri 2756	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
18	17	eqcomi 2737	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
19		1nn 12254	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4812	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 5010	. . . 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 4014	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
24		iunss 5048	. . . 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 7428	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
26	25	sseq1d 4011	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
27		oveq2 7428	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦))
28	27	sseq1d 4011	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
29		oveq2 7428	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)))
30	29	sseq1d 4011	. . . . 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 7428	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
32	31	sseq1d 4011	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
33	16	eqimssi 4040	. . . . 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 15005	. . . . . . . 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 586	. . . . . . 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 5858	. . . . . . . . 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 5863	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
40		trclfvcotrg 14996	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑_𝑟𝑘) = (𝑑↑_𝑟𝑘))
42	41	iuneq2d 5025	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
43		ovex 7453	. . . . . . . . . . . . . 14 ⊢ (𝑑↑_𝑟𝑘) ∈ V
44	4, 43	iunex 7972	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 7005	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
46	45	elv 3477	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
47	46, 46	coeq12i 5866	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
48	40, 47, 46	3sstr3i 4022	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
49	39, 48	eqsstri 4014	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
50	38, 49	sstrdi 3992	. . . . . . 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 12261	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
54	24, 53	mprgbir 3065	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
55		iuneq1 5012	. . . 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 4016	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑_𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 43136	1 ⊢ (t+ ∘ t+) = t+

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∪ cun 3945 ⊆ wss 3947 {csn 4629 ∪ ciun 4996 ∘ ccom 5682 ‘cfv 6548 (class class class)co 7420 1c1 11140 + caddc 11142 ℕcn 12243 t+ctcl 14965 ↑_𝑟crelexp 14999

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-trcl 14967 df-relexp 15000

This theorem is referenced by: cortrcltrcl 43170 cotrclrtrcl 43174

W3C validator