Theorem corcltrcl 43757

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 43694	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑𝑟𝑖))
2		dftrcl3 43738	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dfrtrcl3 43751	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑐↑𝑟𝑘))
4		prex 5436	. 2 ⊢ {0, 1} ∈ V
5		nnex 12273	. 2 ⊢ ℕ ∈ V
6		df-n0 12529	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
7		uncom 4157	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4628	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4163	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4171	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 12278	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4807	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4185	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
15	13, 14	mpbi 230	. . . . 5 ⊢ ({1} ∪ ℕ) = ℕ
16	15	uneq2i 4164	. . . 4 ⊢ ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
17	9, 10, 16	3eqtrri 2769	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2768	. 2 ⊢ ℕ0 = ({0, 1} ∪ ℕ)
19		oveq2 7440	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑖))
20	19	cbviunv 5039	. . 3 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖)
21		ss2iun 5009	. . . 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 15066	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑𝑟1) = 𝑑)
23	22	elv 3484	. . . . . . 7 ⊢ (𝑑↑𝑟1) = 𝑑
24		oveq2 7440	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟1))
25	24	ssiun2s 5047	. . . . . . . 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 4030	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗))
29		ovex 7465	. . . . . . 7 ⊢ (𝑑↑𝑟𝑗) ∈ V
30	5, 29	iunex 7994	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V)
32		0nn0 12543	. . . . . . 7 ⊢ 0 ∈ ℕ0
33		1nn0 12544	. . . . . . 7 ⊢ 1 ∈ ℕ0
34		prssi 4820	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
35	32, 33, 34	mp2an 692	. . . . . 6 ⊢ {0, 1} ⊆ ℕ0
36	35	sseli 3978	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ0)
37	28, 31, 36	relexpss1d 43723	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
38	21, 37	mprg 3066	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
39	20, 38	eqsstri 4029	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
40		oveq2 7440	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑗))
41	40	cbviunv 5039	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
42		relexp1g 15066	. . . . 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 2767	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
45		1ex 11258	. . . . 5 ⊢ 1 ∈ V
46	45	prid2 4762	. . . 4 ⊢ 1 ∈ {0, 1}
47		oveq2 7440	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
48	47	ssiun2s 5047	. . . 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 4029	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
51		c0ex 11256	. . . . . 6 ⊢ 0 ∈ V
52	51	prid1 4761	. . . . 5 ⊢ 0 ∈ {0, 1}
53		oveq2 7440	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟0))
54	53	ssiun2s 5047	. . . . 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 4005	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
57		unss12 4187	. . . 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 692	. . 3 ⊢ ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
59		iuneq1 5007	. . . . 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 5093	. . . 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 7440	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0))
63	51, 62	iunxsn 5090	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0)
64		vex 3483	. . . . . . 7 ⊢ 𝑑 ∈ V
65		nnssnn0 12531	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
66		inelcm 4464	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
67	46, 11, 66	mp2an 692	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
68		iunrelexp0 43720	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ0 ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0))
69	64, 65, 67, 68	mp3an 1462	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0)
70	63, 69	eqtri 2764	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (𝑑↑𝑟0)
71	45, 47	iunxsn 5090	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
72	43, 41	eqtr4i 2767	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
73	71, 72	eqtri 2764	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
74	70, 73	uneq12i 4165	. . . 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 2768	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
76		iunxun 5093	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
77	58, 75, 76	3sstr4i 4034	. 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 43724	1 ⊢ (r* ∘ t+) = t*

Syntax hints:   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {csn 4625  {cpr 4627  ∪ ciun 4990   ∘ ccom 5688  (class class class)co 7432  0cc0 11156  1c1 11157  ℕcn 12267  ℕ₀cn0 12528  t+ctcl 15025  t*crtcl 15026  ↑_𝑟crelexp 15059  r*crcl 43690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-seq 14044  df-trcl 15027  df-rtrcl 15028  df-relexp 15060  df-rcl 43691

This theorem is referenced by:  cortrcltrcl  43758  corclrtrcl  43759