Theorem cotrcltrcl 41303

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 41298	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖))
2		dftrcl3 41298	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dftrcl3 41298	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘))
4		nnex 11977	. 2 ⊢ ℕ ∈ V
5		unidm 4087	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2747	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 10969	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7285	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
9	7, 8	iunxsn 5022	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
10		ovex 7310	. . . . . . . 8 ⊢ (𝑑↑𝑟𝑗) ∈ V
11	4, 10	iunex 7811	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
12		relexp1g 14735	. . . . . . 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 7285	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘))
15	14	cbviunv 4972	. . . . . 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 11982	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4743	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 4940	. . . 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 3956	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
24		iunss 4977	. . . 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 7285	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
26	25	sseq1d 3953	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
27		oveq2 7285	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦))
28	27	sseq1d 3953	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
29		oveq2 7285	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)))
30	29	sseq1d 3953	. . . . 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 7285	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
32	31	sseq1d 3953	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
33	16	eqimssi 3980	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
34		simpl 483	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → 𝑦 ∈ ℕ)
35		relexpsucnnr 14734	. . . . . . . 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 5766	. . . . . . . . 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 482	. . . . . . . 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 5771	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
40		trclfvcotrg 14725	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7284	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑𝑟𝑘) = (𝑑↑𝑟𝑘))
42	41	iuneq2d 4955	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
43		ovex 7310	. . . . . . . . . . . . . 14 ⊢ (𝑑↑𝑟𝑘) ∈ V
44	4, 43	iunex 7811	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 6877	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
46	45	elv 3437	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
47	46, 46	coeq12i 5774	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
48	40, 47, 46	3sstr3i 3964	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
49	39, 48	eqsstri 3956	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
50	38, 49	sstrdi 3934	. . . . . . 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 3960	. . . . . 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 413	. . . . 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 11989	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
54	24, 53	mprgbir 3079	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
55		iuneq1 4942	. . . 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 3958	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 41284	1 ⊢ (t+ ∘ t+) = t+

Syntax hints:   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3431   ∪ cun 3886   ⊆ wss 3888  {csn 4563  ∪ ciun 4926   ∘ ccom 5595  ‘cfv 6435  (class class class)co 7277  1c1 10870   + caddc 10872  ℕcn 11971  t+ctcl 14694  ↑_𝑟crelexp 14728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588  ax-cnex 10925  ax-resscn 10926  ax-1cn 10927  ax-icn 10928  ax-addcl 10929  ax-addrcl 10930  ax-mulcl 10931  ax-mulrcl 10932  ax-mulcom 10933  ax-addass 10934  ax-mulass 10935  ax-distr 10936  ax-i2m1 10937  ax-1ne0 10938  ax-1rid 10939  ax-rnegex 10940  ax-rrecex 10941  ax-cnre 10942  ax-pre-lttri 10943  ax-pre-lttrn 10944  ax-pre-ltadd 10945  ax-pre-mulgt0 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6204  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-2nd 7832  df-frecs 8095  df-wrecs 8126  df-recs 8200  df-rdg 8239  df-er 8496  df-en 8732  df-dom 8733  df-sdom 8734  df-pnf 11009  df-mnf 11010  df-xr 11011  df-ltxr 11012  df-le 11013  df-sub 11205  df-neg 11206  df-nn 11972  df-2 12034  df-n0 12232  df-z 12318  df-uz 12581  df-seq 13720  df-trcl 14696  df-relexp 14729

This theorem is referenced by:  cortrcltrcl  41318  cotrclrtrcl  41322