Theorem cotrcltrcl 44301

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 44296	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖))
2		dftrcl3 44296	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dftrcl3 44296	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘))
4		nnex 12216	. 2 ⊢ ℕ ∈ V
5		unidm 4110	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2771	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 11176	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7404	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
9	7, 8	iunxsn 5048	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
10		ovex 7429	. . . . . . . 8 ⊢ (𝑑↑𝑟𝑗) ∈ V
11	4, 10	iunex 7949	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
12		relexp1g 15039	. . . . . . 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 7404	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘))
15	14	cbviunv 4996	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
16	13, 15	eqtri 2785	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
17	9, 16	eqtri 2785	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
18	17	eqcomi 2771	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
19		1nn 12221	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4744	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 4964	. . . 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 3982	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
24		iunss 5002	. . . 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 7404	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
26	25	sseq1d 3967	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
27		oveq2 7404	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦))
28	27	sseq1d 3967	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
29		oveq2 7404	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)))
30	29	sseq1d 3967	. . . . 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 7404	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
32	31	sseq1d 3967	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
33	16	eqimssi 3996	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
34		simpl 486	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → 𝑦 ∈ ℕ)
35		relexpsucnnr 15038	. . . . . . . 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 596	. . . . . . 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 5827	. . . . . . . . 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 485	. . . . . . . 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 5832	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
40		trclfvcotrg 15029	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7403	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑𝑟𝑘) = (𝑑↑𝑟𝑘))
42	41	iuneq2d 4980	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
43		ovex 7429	. . . . . . . . . . . . . 14 ⊢ (𝑑↑𝑟𝑘) ∈ V
44	4, 43	iunex 7949	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 6975	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
46	45	elv 3459	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
47	46, 46	coeq12i 5835	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
48	40, 47, 46	3sstr3i 3986	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
49	39, 48	eqsstri 3982	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
50	38, 49	sstrdi 3948	. . . . . . 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 3970	. . . . . 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 416	. . . . 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 12228	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
54	24, 53	mprgbir 3083	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
55		iuneq1 4966	. . . 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 3984	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 44282	1 ⊢ (t+ ∘ t+) = t+

Syntax hints:   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∪ cun 3902   ⊆ wss 3904  {csn 4582  ∪ ciun 4949   ∘ ccom 5651  ‘cfv 6521  (class class class)co 7396  1c1 11074   + caddc 11076  ℕcn 12210  t+ctcl 14998  ↑_𝑟crelexp 15032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-n0 12482  df-z 12569  df-uz 12840  df-seq 14015  df-trcl 15000  df-relexp 15033

This theorem is referenced by:  cortrcltrcl  44316  cotrclrtrcl  44320