Theorem trclfvdecomr 43071

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 3488	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		oveq1 7421	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 5020	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
4		dftrcl3 43063	. . . 4 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
5		nnex 12234	. . . . 5 ⊢ ℕ ∈ V
6		ovex 7447	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
7	5, 6	iunex 7964	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
8	3, 4, 7	fvmpt 6999	. . 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 12881	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
11		2eluzge1 12894	. . . . . . 7 ⊢ 2 ∈ (ℤ≥‘1)
12		uzsplit 13591	. . . . . . 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 12354	. . . . . . . . 9 ⊢ (2 − 1) = 1
15	14	oveq2i 7425	. . . . . . . 8 ⊢ (1...(2 − 1)) = (1...1)
16		1z 12608	. . . . . . . . 9 ⊢ 1 ∈ ℤ
17		fzsn 13561	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1...1) = {1})
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (1...1) = {1}
19	15, 18	eqtri 2755	. . . . . . 7 ⊢ (1...(2 − 1)) = {1}
20	19	uneq1i 4155	. . . . . 6 ⊢ ((1...(2 − 1)) ∪ (ℤ≥‘2)) = ({1} ∪ (ℤ≥‘2))
21	10, 13, 20	3eqtri 2759	. . . . 5 ⊢ ℕ = ({1} ∪ (ℤ≥‘2))
22		iuneq1 5007	. . . . 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 5091	. . . 4 ⊢ ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛) = (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
25		1ex 11226	. . . . . 6 ⊢ 1 ∈ V
26		oveq2 7422	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
27	25, 26	iunxsn 5088	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)
28	27	uneq1i 4155	. . . 4 ⊢ (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
29	23, 24, 28	3eqtri 2759	. . 3 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
30		relexp1g 14991	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31		oveq1 7421	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑚) = (𝑅↑𝑟𝑚))
32	31	iuneq2d 5020	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
33		dftrcl3 43063	. . . . . . . . 9 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚))
34		ovex 7447	. . . . . . . . . 10 ⊢ (𝑅↑𝑟𝑚) ∈ V
35	5, 34	iunex 7964	. . . . . . . . 9 ⊢ ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∈ V
36	32, 33, 35	fvmpt 6999	. . . . . . . 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 5858	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅))
39		coiun1 42995	. . . . . . 7 ⊢ (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅)
40		uz2m1nn 12923	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 − 1) ∈ ℕ)
41	40	adantl 481	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑛 − 1) ∈ ℕ)
42		eluzp1p1 12866	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘1) → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
43	42, 10	eleq2s 2846	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
44		1p1e2 12353	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
45	44	fveq2i 6894	. . . . . . . . . 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 7422	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → (𝑅↑𝑟𝑚) = (𝑅↑𝑟(𝑛 − 1)))
49	48	coeq1d 5858	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 1) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
50	49	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
51		oveq2 7422	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
52	51	3ad2ant3 1133	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
53		relexpsucnnr 14990	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
54	53	eqcomd 2733	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = (𝑅↑𝑟(𝑚 + 1)))
55		relexpsucnnr 14990	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑛 − 1) ∈ ℕ) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
56	40, 55	sylan2 592	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
57		eluzelcn 12850	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℂ)
58		npcan1 11655	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59		oveq2 7422	. . . . . . . . . . . 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 2729	. . . . . . . . . 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 231	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
64	41, 47, 50, 52, 54, 63	cbviuneq12dv 43005	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
65	39, 64	eqtrid 2779	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
66	38, 65	eqtrd 2767	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
67	66	eqcomd 2733	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
68	30, 67	uneq12d 4160	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
69	29, 68	eqtrid 2779	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
70	9, 69	eqtrd 2767	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∪ cun 3942  {csn 4624  ∪ ciun 4991   ∘ ccom 5676  ‘cfv 6542  (class class class)co 7414  ℂcc 11122  1c1 11125   + caddc 11127   − cmin 11460  ℕcn 12228  2c2 12283  ℤcz 12574  ℤ_≥cuz 12838  ...cfz 13502  t+ctcl 14950  ↑_𝑟crelexp 14984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-n0 12489  df-z 12575  df-uz 12839  df-fz 13503  df-seq 13985  df-trcl 14952  df-relexp 14985

This theorem is referenced by:  trclfvdecoml  43072  dmtrclfvRP  43073  frege124d  43104  frege131d  43107