Theorem cotrcltrcl 38017

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 38012	. 2 ⊢ t+ = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ ℕ (𝑎↑𝑟𝑖))
2		dftrcl3 38012	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dftrcl3 38012	. 2 ⊢ t+ = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘))
4		nnex 11026	. 2 ⊢ ℕ ∈ V
5		unidm 3756	. . 3 ⊢ (ℕ ∪ ℕ) = ℕ
6	5	eqcomi 2631	. 2 ⊢ ℕ = (ℕ ∪ ℕ)
7		1ex 10035	. . . . . 6 ⊢ 1 ∈ V
8		oveq2 6658	. . . . . 6 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
9	7, 8	iunxsn 4603	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
10		ovex 6678	. . . . . . . 8 ⊢ (𝑑↑𝑟𝑗) ∈ V
11	4, 10	iunex 7147	. . . . . . 7 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
12		relexp1g 13766	. . . . . . 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 6658	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟𝑘))
15	14	cbviunv 4559	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
16	13, 15	eqtri 2644	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
17	9, 16	eqtri 2644	. . . 4 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
18	17	eqcomi 2631	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
19		1nn 11031	. . . . 5 ⊢ 1 ∈ ℕ
20		snssi 4339	. . . . 5 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
21	19, 20	ax-mp 5	. . . 4 ⊢ {1} ⊆ ℕ
22		iunss1 4532	. . . 4 ⊢ ({1} ⊆ ℕ → ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
23	21, 22	ax-mp 5	. . 3 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
24	18, 23	eqsstri 3635	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
25		iunss 4561	. . . 4 ⊢ (∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
26		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
27	26	sseq1d 3632	. . . . 5 ⊢ (𝑥 = 1 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
28		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦))
29	28	sseq1d 3632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
30		oveq2 6658	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)))
31	30	sseq1d 3632	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
32		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑖 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
33	32	sseq1d 3632	. . . . 5 ⊢ (𝑥 = 𝑖 → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑥) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ↔ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
34	16	eqimssi 3659	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
35		simpl 473	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → 𝑦 ∈ ℕ)
36		relexpsucnnr 13765	. . . . . . . 8 ⊢ ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)) = ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)))
37	11, 35, 36	sylancr 695	. . . . . . 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)))
38		coss1 5277	. . . . . . . . 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)))
39	38	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)))
40	15	coeq2i 5282	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
41		trclfvcotrg 13757	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
42		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑑 ∈ V
43		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑐↑𝑟𝑘) = (𝑑↑𝑟𝑘))
44	43	iuneq2d 4547	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → ∪ 𝑘 ∈ ℕ (𝑐↑𝑟𝑘) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
45		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑑↑𝑟𝑘) ∈ V
46	4, 45	iunex 7147	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∈ V
47	44, 3, 46	fvmpt 6282	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ V → (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
48	42, 47	ax-mp 5	. . . . . . . . . . 11 ⊢ (t+‘𝑑) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
49	48, 48	coeq12i 5285	. . . . . . . . . 10 ⊢ ((t+‘𝑑) ∘ (t+‘𝑑)) = (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
50	41, 49, 48	3sstr3i 3643	. . . . . . . . 9 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
51	40, 50	eqsstri 3635	. . . . . . . 8 ⊢ (∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ∘ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
52	39, 51	syl6ss 3615	. . . . . . 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))
53	37, 52	eqsstrd 3639	. . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
54	53	ex 450	. . . . 5 ⊢ (𝑦 ∈ ℕ → ((∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑦) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
55	27, 29, 31, 33, 34, 54	nnind 11038	. . . 4 ⊢ (𝑖 ∈ ℕ → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
56	25, 55	mprgbir 2927	. . 3 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
57		iuneq1 4534	. . . 4 ⊢ (ℕ = (ℕ ∪ ℕ) → ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘))
58	6, 57	ax-mp 5	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
59	56, 58	sseqtri 3637	. 2 ⊢ ∪ 𝑖 ∈ ℕ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑↑𝑟𝑘)
60	1, 2, 3, 4, 4, 6, 24, 24, 59	comptiunov2i 37998	1 ⊢ (t+ ∘ t+) = t+

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ⊆ wss 3574  {csn 4177  ∪ ciun 4520   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939  ℕcn 11020  t+ctcl 13724  ↑_𝑟crelexp 13760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-trcl 13726  df-relexp 13761

This theorem is referenced by:  cortrcltrcl  38032  cotrclrtrcl  38036