Theorem cotrcltrcl 42462

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 42457	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖))
2		dftrcl3 42457	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dftrcl3 42457	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘))
4		nnex 12215	. 2 ⊢ ℕ ∈ V
5		unidm 4152	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2742	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 11207	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 7414	. . . . . 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 7439	. . . . . . . 8 ⊢ (𝑑↑𝑟𝑗) ∈ V
11	4, 10	iunex 7952	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
12		relexp1g 14970	. . . . . . 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 7414	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘))
15	14	cbviunv 5043	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
16	13, 15	eqtri 2761	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
17	9, 16	eqtri 2761	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
18	17	eqcomi 2742	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
19		1nn 12220	. . . 4 ⊢ 1 ∈ ℕ
20		snssi 4811	. . . 4 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21		iunss1 5011	. . . 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 4016	. 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 7414	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
26	25	sseq1d 4013	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
27		oveq2 7414	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦))
28	27	sseq1d 4013	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
29		oveq2 7414	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)))
30	29	sseq1d 4013	. . . . 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 7414	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
32	31	sseq1d 4013	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
33	16	eqimssi 4042	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
34		simpl 484	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → 𝑦 ∈ ℕ)
35		relexpsucnnr 14969	. . . . . . . 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 588	. . . . . . 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 5854	. . . . . . . . 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 483	. . . . . . . 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 5859	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
40		trclfvcotrg 14960	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41		oveq1 7413	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑𝑟𝑘) = (𝑑↑𝑟𝑘))
42	41	iuneq2d 5026	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
43		ovex 7439	. . . . . . . . . . . . . 14 ⊢ (𝑑↑𝑟𝑘) ∈ V
44	4, 43	iunex 7952	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∈ V
45	42, 3, 44	fvmpt 6996	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
46	45	elv 3481	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
47	46, 46	coeq12i 5862	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
48	40, 47, 46	3sstr3i 4024	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
49	39, 48	eqsstri 4016	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
50	38, 49	sstrdi 3994	. . . . . . 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 4020	. . . . . 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 414	. . . . 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 12227	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
54	24, 53	mprgbir 3069	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
55		iuneq1 5013	. . . 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 4018	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
58	1, 2, 3, 4, 4, 6, 23, 23, 57	comptiunov2i 42443	1 ⊢ (t+ ∘ t+) = t+

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∪ cun 3946   ⊆ wss 3948  {csn 4628  ∪ ciun 4997   ∘ ccom 5680  ‘cfv 6541  (class class class)co 7406  1c1 11108   + caddc 11110  ℕcn 12209  t+ctcl 14929  ↑_𝑟crelexp 14963

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-seq 13964  df-trcl 14931  df-relexp 14964

This theorem is referenced by:  cortrcltrcl  42477  cotrclrtrcl  42481