cotrcltrcl - Mathbox for Richard Penner

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Theorem cotrcltrcl 41222

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 41217	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑_𝑟𝑖))
2		dftrcl3 41217	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dftrcl3 41217	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘))
4		nnex 11909	. 2 ⊢ ℕ ∈ V
5		unidm 4082	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2747	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 10902	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7263	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
9	7, 8	iunxsn 5016	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
10		ovex 7288	. . . . . . . 8 ⊢ (𝑑↑_𝑟𝑗) ∈ V
11	4, 10	iunex 7784	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
12		relexp1g 14665	. . . . . . 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 7263	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟𝑘))
15	14	cbviunv 4966	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
16	13, 15	eqtri 2766	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
17	9, 16	eqtri 2766	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
18	17	eqcomi 2747	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
19		1nn 11914	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4738	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 4935	. . . 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 3951	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
24		iunss 4971	. . . 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 7263	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
26	25	sseq1d 3948	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
27		oveq2 7263	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦))
28	27	sseq1d 3948	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
29		oveq2 7263	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟(𝑦 + 1)))
30	29	sseq1d 3948	. . . . 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 7263	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
32	31	sseq1d 3948	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)))
33	16	eqimssi 3975	. . . . 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 14664	. . . . . . . 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 5753	. . . . . . . . 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 5758	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
40		trclfvcotrg 14655	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7262	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑_𝑟𝑘) = (𝑑↑_𝑟𝑘))
42	41	iuneq2d 4950	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑_𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
43		ovex 7288	. . . . . . . . . . . . . 14 ⊢ (𝑑↑_𝑟𝑘) ∈ V
44	4, 43	iunex 7784	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 6857	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
46	45	elv 3428	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
47	46, 46	coeq12i 5761	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
48	40, 47, 46	3sstr3i 3959	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
49	39, 48	eqsstri 3951	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
50	38, 49	sstrdi 3929	. . . . . . 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 3955	. . . . . 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 11921	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
54	24, 53	mprgbir 3078	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
55		iuneq1 4937	. . . 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 3953	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑_𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 41203	1 ⊢ (t+ ∘ t+) = t+

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ∪ cun 3881 ⊆ wss 3883 {csn 4558 ∪ ciun 4921 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 1c1 10803 + caddc 10805 ℕcn 11903 t+ctcl 14624 ↑_𝑟crelexp 14658

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-seq 13650 df-trcl 14626 df-relexp 14659

This theorem is referenced by: cortrcltrcl 41237 cotrclrtrcl 41241

W3C validator