Theorem trclfvdecomr 44179

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 3453	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		oveq1 7370	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 4959	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
4		dftrcl3 44171	. . . 4 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
5		nnex 12178	. . . . 5 ⊢ ℕ ∈ V
6		ovex 7396	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
7	5, 6	iunex 7917	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
8	3, 4, 7	fvmpt 6942	. . 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 12825	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
11		2eluzge1 12830	. . . . . . 7 ⊢ 2 ∈ (ℤ≥‘1)
12		uzsplit 13548	. . . . . . 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 12300	. . . . . . . . 9 ⊢ (2 − 1) = 1
15	14	oveq2i 7374	. . . . . . . 8 ⊢ (1...(2 − 1)) = (1...1)
16		1z 12555	. . . . . . . . 9 ⊢ 1 ∈ ℤ
17		fzsn 13518	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1...1) = {1})
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (1...1) = {1}
19	15, 18	eqtri 2763	. . . . . . 7 ⊢ (1...(2 − 1)) = {1}
20	19	uneq1i 4101	. . . . . 6 ⊢ ((1...(2 − 1)) ∪ (ℤ≥‘2)) = ({1} ∪ (ℤ≥‘2))
21	10, 13, 20	3eqtri 2767	. . . . 5 ⊢ ℕ = ({1} ∪ (ℤ≥‘2))
22		iuneq1 4945	. . . . 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 5030	. . . 4 ⊢ ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛) = (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
25		1ex 11138	. . . . . 6 ⊢ 1 ∈ V
26		oveq2 7371	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
27	25, 26	iunxsn 5027	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)
28	27	uneq1i 4101	. . . 4 ⊢ (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
29	23, 24, 28	3eqtri 2767	. . 3 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
30		relexp1g 14986	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31		oveq1 7370	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑚) = (𝑅↑𝑟𝑚))
32	31	iuneq2d 4959	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
33		dftrcl3 44171	. . . . . . . . 9 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚))
34		ovex 7396	. . . . . . . . . 10 ⊢ (𝑅↑𝑟𝑚) ∈ V
35	5, 34	iunex 7917	. . . . . . . . 9 ⊢ ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∈ V
36	32, 33, 35	fvmpt 6942	. . . . . . . 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 5810	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅))
39		coiun1 44103	. . . . . . 7 ⊢ (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅)
40		uz2m1nn 12871	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 − 1) ∈ ℕ)
41	40	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑛 − 1) ∈ ℕ)
42		eluzp1p1 12814	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘1) → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
43	42, 10	eleq2s 2858	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
44		1p1e2 12299	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
45	44	fveq2i 6837	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 + 1)) = (ℤ≥‘2)
46	43, 45	eleqtrdi 2850	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘2))
47	46	adantl 482	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ (ℤ≥‘2))
48		oveq2 7371	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → (𝑅↑𝑟𝑚) = (𝑅↑𝑟(𝑛 − 1)))
49	48	coeq1d 5810	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 1) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
50	49	3ad2ant3 1141	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
51		oveq2 7371	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
52	51	3ad2ant3 1141	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
53		relexpsucnnr 14985	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
54	53	eqcomd 2746	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = (𝑅↑𝑟(𝑚 + 1)))
55		relexpsucnnr 14985	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑛 − 1) ∈ ℕ) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
56	40, 55	sylan2 599	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
57		eluzelcn 12798	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℂ)
58		npcan1 11573	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59		oveq2 7371	. . . . . . . . . . . 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 2742	. . . . . . . . . 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 233	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
64	41, 47, 50, 52, 54, 63	cbviuneq12dv 44113	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
65	39, 64	eqtrid 2787	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
66	38, 65	eqtrd 2775	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
67	66	eqcomd 2746	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
68	30, 67	uneq12d 4106	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
69	29, 68	eqtrid 2787	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
70	9, 69	eqtrd 2775	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888  {csn 4562  ∪ ciun 4928   ∘ ccom 5629  ‘cfv 6492  (class class class)co 7363  ℂcc 11034  1c1 11037   + caddc 11039   − cmin 11375  ℕcn 12172  2c2 12234  ℤcz 12522  ℤ_≥cuz 12786  ...cfz 13459  t+ctcl 14945  ↑_𝑟crelexp 14979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-seq 13962  df-trcl 14947  df-relexp 14980

This theorem is referenced by:  trclfvdecoml  44180  dmtrclfvRP  44181  frege124d  44212  frege131d  44215