Theorem trclfvdecomr 41336

Description: The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)

Assertion

Ref	Expression
trclfvdecomr	⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Proof of Theorem trclfvdecomr

Dummy variables 𝑚 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3450	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		oveq1 7282	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 4953	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
4		dftrcl3 41328	. . . 4 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
5		nnex 11979	. . . . 5 ⊢ ℕ ∈ V
6		ovex 7308	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
7	5, 6	iunex 7811	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
8	3, 4, 7	fvmpt 6875	. . 3 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
9	1, 8	syl 17	. 2 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
10		nnuz 12621	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
11		2eluzge1 12634	. . . . . . 7 ⊢ 2 ∈ (ℤ≥‘1)
12		uzsplit 13328	. . . . . . 7 ⊢ (2 ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1...(2 − 1)) ∪ (ℤ≥‘2)))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ (ℤ≥‘1) = ((1...(2 − 1)) ∪ (ℤ≥‘2))
14		2m1e1 12099	. . . . . . . . 9 ⊢ (2 − 1) = 1
15	14	oveq2i 7286	. . . . . . . 8 ⊢ (1...(2 − 1)) = (1...1)
16		1z 12350	. . . . . . . . 9 ⊢ 1 ∈ ℤ
17		fzsn 13298	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1...1) = {1})
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (1...1) = {1}
19	15, 18	eqtri 2766	. . . . . . 7 ⊢ (1...(2 − 1)) = {1}
20	19	uneq1i 4093	. . . . . 6 ⊢ ((1...(2 − 1)) ∪ (ℤ≥‘2)) = ({1} ∪ (ℤ≥‘2))
21	10, 13, 20	3eqtri 2770	. . . . 5 ⊢ ℕ = ({1} ∪ (ℤ≥‘2))
22		iuneq1 4940	. . . . 5 ⊢ (ℕ = ({1} ∪ (ℤ≥‘2)) → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛))
23	21, 22	ax-mp 5	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛)
24		iunxun 5023	. . . 4 ⊢ ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛) = (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
25		1ex 10971	. . . . . 6 ⊢ 1 ∈ V
26		oveq2 7283	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
27	25, 26	iunxsn 5020	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)
28	27	uneq1i 4093	. . . 4 ⊢ (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
29	23, 24, 28	3eqtri 2770	. . 3 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
30		relexp1g 14737	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑚) = (𝑅↑𝑟𝑚))
32	31	iuneq2d 4953	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
33		dftrcl3 41328	. . . . . . . . 9 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚))
34		ovex 7308	. . . . . . . . . 10 ⊢ (𝑅↑𝑟𝑚) ∈ V
35	5, 34	iunex 7811	. . . . . . . . 9 ⊢ ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∈ V
36	32, 33, 35	fvmpt 6875	. . . . . . . 8 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
37	1, 36	syl 17	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
38	37	coeq1d 5770	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅))
39		coiun1 41260	. . . . . . 7 ⊢ (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅)
40		uz2m1nn 12663	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 − 1) ∈ ℕ)
41	40	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑛 − 1) ∈ ℕ)
42		eluzp1p1 12610	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘1) → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
43	42, 10	eleq2s 2857	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
44		1p1e2 12098	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
45	44	fveq2i 6777	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 + 1)) = (ℤ≥‘2)
46	43, 45	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘2))
47	46	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ (ℤ≥‘2))
48		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → (𝑅↑𝑟𝑚) = (𝑅↑𝑟(𝑛 − 1)))
49	48	coeq1d 5770	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 1) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
50	49	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
51		oveq2 7283	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
52	51	3ad2ant3 1134	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
53		relexpsucnnr 14736	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
54	53	eqcomd 2744	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = (𝑅↑𝑟(𝑚 + 1)))
55		relexpsucnnr 14736	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑛 − 1) ∈ ℕ) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
56	40, 55	sylan2 593	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
57		eluzelcn 12594	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℂ)
58		npcan1 11400	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59		oveq2 7283	. . . . . . . . . . . 12 ⊢ (((𝑛 − 1) + 1) = 𝑛 → (𝑅↑𝑟((𝑛 − 1) + 1)) = (𝑅↑𝑟𝑛))
60	57, 58, 59	3syl 18	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑅↑𝑟((𝑛 − 1) + 1)) = (𝑅↑𝑟𝑛))
61	60	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘2) → ((𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅)))
62	61	adantl 482	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → ((𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅)))
63	56, 62	mpbid 231	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
64	41, 47, 50, 52, 54, 63	cbviuneq12dv 41270	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
65	39, 64	eqtrid 2790	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
66	38, 65	eqtrd 2778	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
67	66	eqcomd 2744	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
68	30, 67	uneq12d 4098	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
69	29, 68	eqtrid 2790	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
70	9, 69	eqtrd 2778	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885  {csn 4561  ∪ ciun 4924   ∘ ccom 5593  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  1c1 10872   + caddc 10874   − cmin 11205  ℕcn 11973  2c2 12028  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239  t+ctcl 14696  ↑_𝑟crelexp 14730

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 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

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 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-seq 13722  df-trcl 14698  df-relexp 14731

This theorem is referenced by:  trclfvdecoml  41337  dmtrclfvRP  41338  frege124d  41369  frege131d  41372