Theorem corcltrcl 42475

Description: The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)

Assertion

Ref	Expression
corcltrcl	⊢ (r* ∘ t+) = t*

Proof of Theorem corcltrcl

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrcl4 42412	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑𝑟𝑖))
2		dftrcl3 42456	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dfrtrcl3 42469	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑐↑𝑟𝑘))
4		prex 5431	. 2 ⊢ {0, 1} ∈ V
5		nnex 12214	. 2 ⊢ ℕ ∈ V
6		df-n0 12469	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
7		uncom 4152	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4630	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4158	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4165	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 12219	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4810	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4179	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
15	13, 14	mpbi 229	. . . . 5 ⊢ ({1} ∪ ℕ) = ℕ
16	15	uneq2i 4159	. . . 4 ⊢ ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
17	9, 10, 16	3eqtrri 2765	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2764	. 2 ⊢ ℕ0 = ({0, 1} ∪ ℕ)
19		oveq2 7413	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑖))
20	19	cbviunv 5042	. . 3 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖)
21		ss2iun 5014	. . . 4 ⊢ (∀𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) → ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
22		relexp1g 14969	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑𝑟1) = 𝑑)
23	22	elv 3480	. . . . . . 7 ⊢ (𝑑↑𝑟1) = 𝑑
24		oveq2 7413	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟1))
25	24	ssiun2s 5050	. . . . . . . 8 ⊢ (1 ∈ ℕ → (𝑑↑𝑟1) ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗))
26	11, 25	ax-mp 5	. . . . . . 7 ⊢ (𝑑↑𝑟1) ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
27	23, 26	eqsstrri 4016	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗))
29		ovex 7438	. . . . . . 7 ⊢ (𝑑↑𝑟𝑗) ∈ V
30	5, 29	iunex 7951	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V)
32		0nn0 12483	. . . . . . 7 ⊢ 0 ∈ ℕ0
33		1nn0 12484	. . . . . . 7 ⊢ 1 ∈ ℕ0
34		prssi 4823	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
35	32, 33, 34	mp2an 690	. . . . . 6 ⊢ {0, 1} ⊆ ℕ0
36	35	sseli 3977	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ0)
37	28, 31, 36	relexpss1d 42441	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
38	21, 37	mprg 3067	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
39	20, 38	eqsstri 4015	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
40		oveq2 7413	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑗))
41	40	cbviunv 5042	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
42		relexp1g 14969	. . . . 5 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗))
43	30, 42	ax-mp 5	. . . 4 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
44	41, 43	eqtr4i 2763	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
45		1ex 11206	. . . . 5 ⊢ 1 ∈ V
46	45	prid2 4766	. . . 4 ⊢ 1 ∈ {0, 1}
47		oveq2 7413	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
48	47	ssiun2s 5050	. . . 4 ⊢ (1 ∈ {0, 1} → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
49	46, 48	ax-mp 5	. . 3 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
50	44, 49	eqsstri 4015	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
51		c0ex 11204	. . . . . 6 ⊢ 0 ∈ V
52	51	prid1 4765	. . . . 5 ⊢ 0 ∈ {0, 1}
53		oveq2 7413	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟0))
54	53	ssiun2s 5050	. . . . 5 ⊢ (0 ∈ {0, 1} → (𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘))
55	52, 54	ax-mp 5	. . . 4 ⊢ (𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘)
56		ssid 4003	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
57		unss12 4181	. . . 4 ⊢ (((𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∧ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) → ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)))
58	55, 56, 57	mp2an 690	. . 3 ⊢ ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
59		iuneq1 5012	. . . . 5 ⊢ ({0, 1} = ({0} ∪ {1}) → ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
60	8, 59	ax-mp 5	. . . 4 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
61		iunxun 5096	. . . 4 ⊢ ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ∪ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
62		oveq2 7413	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0))
63	51, 62	iunxsn 5093	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0)
64		vex 3478	. . . . . . 7 ⊢ 𝑑 ∈ V
65		nnssnn0 12471	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
66		inelcm 4463	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
67	46, 11, 66	mp2an 690	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
68		iunrelexp0 42438	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ0 ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0))
69	64, 65, 67, 68	mp3an 1461	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0)
70	63, 69	eqtri 2760	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (𝑑↑𝑟0)
71	45, 47	iunxsn 5093	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
72	43, 41	eqtr4i 2763	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
73	71, 72	eqtri 2760	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
74	70, 73	uneq12i 4160	. . . 4 ⊢ (∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ∪ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)) = ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
75	60, 61, 74	3eqtri 2764	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
76		iunxun 5096	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
77	58, 75, 76	3sstr4i 4024	. 2 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑𝑟𝑘)
78	1, 2, 3, 4, 5, 18, 39, 50, 77	comptiunov2i 42442	1 ⊢ (r* ∘ t+) = t*

Syntax hints:   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  Vcvv 3474   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  {csn 4627  {cpr 4629  ∪ ciun 4996   ∘ ccom 5679  (class class class)co 7405  0cc0 11106  1c1 11107  ℕcn 12208  ℕ₀cn0 12468  t+ctcl 14928  t*crtcl 14929  ↑_𝑟crelexp 14962  r*crcl 42408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-seq 13963  df-trcl 14930  df-rtrcl 14931  df-relexp 14963  df-rcl 42409

This theorem is referenced by:  cortrcltrcl  42476  corclrtrcl  42477