Theorem trclfvdecomr 43300

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 3480	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2		oveq1 7426	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 5026	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))
4		dftrcl3 43292	. . . 4 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
5		nnex 12251	. . . . 5 ⊢ ℕ ∈ V
6		ovex 7452	. . . . 5 ⊢ (𝑅↑𝑟𝑛) ∈ V
7	5, 6	iunex 7973	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V
8	3, 4, 7	fvmpt 7004	. . 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 12898	. . . . . 6 ⊢ ℕ = (ℤ≥‘1)
11		2eluzge1 12911	. . . . . . 7 ⊢ 2 ∈ (ℤ≥‘1)
12		uzsplit 13608	. . . . . . 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 12371	. . . . . . . . 9 ⊢ (2 − 1) = 1
15	14	oveq2i 7430	. . . . . . . 8 ⊢ (1...(2 − 1)) = (1...1)
16		1z 12625	. . . . . . . . 9 ⊢ 1 ∈ ℤ
17		fzsn 13578	. . . . . . . . 9 ⊢ (1 ∈ ℤ → (1...1) = {1})
18	16, 17	ax-mp 5	. . . . . . . 8 ⊢ (1...1) = {1}
19	15, 18	eqtri 2753	. . . . . . 7 ⊢ (1...(2 − 1)) = {1}
20	19	uneq1i 4156	. . . . . 6 ⊢ ((1...(2 − 1)) ∪ (ℤ≥‘2)) = ({1} ∪ (ℤ≥‘2))
21	10, 13, 20	3eqtri 2757	. . . . 5 ⊢ ℕ = ({1} ∪ (ℤ≥‘2))
22		iuneq1 5013	. . . . 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 5098	. . . 4 ⊢ ∪ 𝑛 ∈ ({1} ∪ (ℤ≥‘2))(𝑅↑𝑟𝑛) = (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
25		1ex 11242	. . . . . 6 ⊢ 1 ∈ V
26		oveq2 7427	. . . . . 6 ⊢ (𝑛 = 1 → (𝑅↑𝑟𝑛) = (𝑅↑𝑟1))
27	25, 26	iunxsn 5095	. . . . 5 ⊢ ∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) = (𝑅↑𝑟1)
28	27	uneq1i 4156	. . . 4 ⊢ (∪ 𝑛 ∈ {1} (𝑅↑𝑟𝑛) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
29	23, 24, 28	3eqtri 2757	. . 3 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
30		relexp1g 15009	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅)
31		oveq1 7426	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑚) = (𝑅↑𝑟𝑚))
32	31	iuneq2d 5026	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚) = ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚))
33		dftrcl3 43292	. . . . . . . . 9 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑚 ∈ ℕ (𝑟↑𝑟𝑚))
34		ovex 7452	. . . . . . . . . 10 ⊢ (𝑅↑𝑟𝑚) ∈ V
35	5, 34	iunex 7973	. . . . . . . . 9 ⊢ ∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∈ V
36	32, 33, 35	fvmpt 7004	. . . . . . . 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 5864	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅))
39		coiun1 43224	. . . . . . 7 ⊢ (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅)
40		uz2m1nn 12940	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘2) → (𝑛 − 1) ∈ ℕ)
41	40	adantl 480	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑛 − 1) ∈ ℕ)
42		eluzp1p1 12883	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘1) → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
43	42, 10	eleq2s 2843	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘(1 + 1)))
44		1p1e2 12370	. . . . . . . . . . 11 ⊢ (1 + 1) = 2
45	44	fveq2i 6899	. . . . . . . . . 10 ⊢ (ℤ≥‘(1 + 1)) = (ℤ≥‘2)
46	43, 45	eleqtrdi 2835	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ≥‘2))
47	46	adantl 480	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ (ℤ≥‘2))
48		oveq2 7427	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 − 1) → (𝑅↑𝑟𝑚) = (𝑅↑𝑟(𝑛 − 1)))
49	48	coeq1d 5864	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 − 1) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
50	49	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
51		oveq2 7427	. . . . . . . . 9 ⊢ (𝑛 = (𝑚 + 1) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
52	51	3ad2ant3 1132	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅↑𝑟𝑛) = (𝑅↑𝑟(𝑚 + 1)))
53		relexpsucnnr 15008	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
54	53	eqcomd 2731	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑚 ∈ ℕ) → ((𝑅↑𝑟𝑚) ∘ 𝑅) = (𝑅↑𝑟(𝑚 + 1)))
55		relexpsucnnr 15008	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑛 − 1) ∈ ℕ) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
56	40, 55	sylan2 591	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑛 ∈ (ℤ≥‘2)) → (𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅))
57		eluzelcn 12867	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘2) → 𝑛 ∈ ℂ)
58		npcan1 11671	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59		oveq2 7427	. . . . . . . . . . . 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 2727	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘2) → ((𝑅↑𝑟((𝑛 − 1) + 1)) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅↑𝑟𝑛) = ((𝑅↑𝑟(𝑛 − 1)) ∘ 𝑅)))
62	61	adantl 480	. . . . . . . . 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 43234	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑚 ∈ ℕ ((𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
65	39, 64	eqtrid 2777	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (∪ 𝑚 ∈ ℕ (𝑅↑𝑟𝑚) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
66	38, 65	eqtrd 2765	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛))
67	66	eqcomd 2731	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
68	30, 67	uneq12d 4161	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1) ∪ ∪ 𝑛 ∈ (ℤ≥‘2)(𝑅↑𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
69	29, 68	eqtrid 2777	. 2 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
70	9, 69	eqtrd 2765	1 ⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∪ cun 3942  {csn 4630  ∪ ciun 4997   ∘ ccom 5682  ‘cfv 6549  (class class class)co 7419  ℂcc 11138  1c1 11141   + caddc 11143   − cmin 11476  ℕcn 12245  2c2 12300  ℤcz 12591  ℤ_≥cuz 12855  ...cfz 13519  t+ctcl 14968  ↑_𝑟crelexp 15002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-seq 14003  df-trcl 14970  df-relexp 15003

This theorem is referenced by:  trclfvdecoml  43301  dmtrclfvRP  43302  frege124d  43333  frege131d  43336