Theorem trclfvdecomr 43704

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 3459	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		oveq1 7360	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 4975	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
4		dftrcl3 43696	. . . 4 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
5		nnex 12152	. . . . 5 ⊢ ℕ ∈ V
6		ovex 7386	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
7	5, 6	iunex 7910	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
8	3, 4, 7	fvmpt 6934	. . 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 12796	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
11		2eluzge1 12801	. . . . . . 7 ⊢ 2 ∈ (ℤ≥‘1)
12		uzsplit 13517	. . . . . . 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 12267	. . . . . . . . 9 ⊢ (2 − 1) = 1
15	14	oveq2i 7364	. . . . . . . 8 ⊢ (1...(2 − 1)) = (1...1)
16		1z 12523	. . . . . . . . 9 ⊢ 1 ∈ ℤ
17		fzsn 13487	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1...1) = {1})
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (1...1) = {1}
19	15, 18	eqtri 2752	. . . . . . 7 ⊢ (1...(2 − 1)) = {1}
20	19	uneq1i 4117	. . . . . 6 ⊢ ((1...(2 − 1)) ∪ (ℤ≥‘2)) = ({1} ∪ (ℤ≥‘2))
21	10, 13, 20	3eqtri 2756	. . . . 5 ⊢ ℕ = ({1} ∪ (ℤ≥‘2))
22		iuneq1 4961	. . . . 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 5046	. . . 4 ⊢ ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛) = (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
25		1ex 11130	. . . . . 6 ⊢ 1 ∈ V
26		oveq2 7361	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
27	25, 26	iunxsn 5043	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)
28	27	uneq1i 4117	. . . 4 ⊢ (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
29	23, 24, 28	3eqtri 2756	. . 3 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
30		relexp1g 14951	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31		oveq1 7360	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑚) = (𝑅↑𝑟𝑚))
32	31	iuneq2d 4975	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
33		dftrcl3 43696	. . . . . . . . 9 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚))
34		ovex 7386	. . . . . . . . . 10 ⊢ (𝑅↑𝑟𝑚) ∈ V
35	5, 34	iunex 7910	. . . . . . . . 9 ⊢ ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∈ V
36	32, 33, 35	fvmpt 6934	. . . . . . . 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 5808	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅))
39		coiun1 43628	. . . . . . 7 ⊢ (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅)
40		uz2m1nn 12842	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 − 1) ∈ ℕ)
41	40	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑛 − 1) ∈ ℕ)
42		eluzp1p1 12781	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘1) → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
43	42, 10	eleq2s 2846	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
44		1p1e2 12266	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
45	44	fveq2i 6829	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 + 1)) = (ℤ≥‘2)
46	43, 45	eleqtrdi 2838	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘2))
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ (ℤ≥‘2))
48		oveq2 7361	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → (𝑅↑𝑟𝑚) = (𝑅↑𝑟(𝑛 − 1)))
49	48	coeq1d 5808	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 1) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
50	49	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
51		oveq2 7361	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
52	51	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
53		relexpsucnnr 14950	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
54	53	eqcomd 2735	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = (𝑅↑𝑟(𝑚 + 1)))
55		relexpsucnnr 14950	. . . . . . . . . 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 12765	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℂ)
58		npcan1 11563	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59		oveq2 7361	. . . . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘2) → ((𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅)))
62	61	adantl 481	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
64	41, 47, 50, 52, 54, 63	cbviuneq12dv 43638	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
65	39, 64	eqtrid 2776	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
66	38, 65	eqtrd 2764	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
67	66	eqcomd 2735	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
68	30, 67	uneq12d 4122	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
69	29, 68	eqtrid 2776	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
70	9, 69	eqtrd 2764	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∪ cun 3903  {csn 4579  ∪ ciun 4944   ∘ ccom 5627  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  1c1 11029   + caddc 11031   − cmin 11365  ℕcn 12146  2c2 12201  ℤcz 12489  ℤ_≥cuz 12753  ...cfz 13428  t+ctcl 14910  ↑_𝑟crelexp 14944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-seq 13927  df-trcl 14912  df-relexp 14945

This theorem is referenced by:  trclfvdecoml  43705  dmtrclfvRP  43706  frege124d  43737  frege131d  43740