Theorem cotrcltrcl 43721

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 43716	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖))
2		dftrcl3 43716	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dftrcl3 43716	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘))
4		nnex 12199	. 2 ⊢ ℕ ∈ V
5		unidm 4123	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2739	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 11177	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7398	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
9	7, 8	iunxsn 5058	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
10		ovex 7423	. . . . . . . 8 ⊢ (𝑑↑𝑟𝑗) ∈ V
11	4, 10	iunex 7950	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
12		relexp1g 14999	. . . . . . 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 7398	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘))
15	14	cbviunv 5007	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
16	13, 15	eqtri 2753	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
17	9, 16	eqtri 2753	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
18	17	eqcomi 2739	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
19		1nn 12204	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4775	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 4973	. . . 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 3996	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
24		iunss 5012	. . . 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 7398	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
26	25	sseq1d 3981	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
27		oveq2 7398	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦))
28	27	sseq1d 3981	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
29		oveq2 7398	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)))
30	29	sseq1d 3981	. . . . 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 7398	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
32	31	sseq1d 3981	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
33	16	eqimssi 4010	. . . . 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 14998	. . . . . . . 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 5822	. . . . . . . . 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 5827	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
40		trclfvcotrg 14989	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7397	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑𝑟𝑘) = (𝑑↑𝑟𝑘))
42	41	iuneq2d 4989	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
43		ovex 7423	. . . . . . . . . . . . . 14 ⊢ (𝑑↑𝑟𝑘) ∈ V
44	4, 43	iunex 7950	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 6971	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
46	45	elv 3455	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
47	46, 46	coeq12i 5830	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
48	40, 47, 46	3sstr3i 4000	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
49	39, 48	eqsstri 3996	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
50	38, 49	sstrdi 3962	. . . . . . 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 3984	. . . . . 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 12211	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
54	24, 53	mprgbir 3052	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
55		iuneq1 4975	. . . 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 3998	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 43702	1 ⊢ (t+ ∘ t+) = t+

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ∪ ciun 4958   ∘ ccom 5645  ‘cfv 6514  (class class class)co 7390  1c1 11076   + caddc 11078  ℕcn 12193  t+ctcl 14958  ↑_𝑟crelexp 14992

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-z 12537  df-uz 12801  df-seq 13974  df-trcl 14960  df-relexp 14993

This theorem is referenced by:  cortrcltrcl  43736  cotrclrtrcl  43740