corcltrcl - Mathbox for Richard Penner

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Theorem corcltrcl 43169

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 43106	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑_𝑟𝑖))
2		dftrcl3 43150	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dfrtrcl3 43163	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ₀ (𝑐↑_𝑟𝑘))
4		prex 5434	. 2 ⊢ {0, 1} ∈ V
5		nnex 12248	. 2 ⊢ ℕ ∈ V
6		df-n0 12503	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
7		uncom 4152	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4632	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4158	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4166	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 12253	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4812	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4180	. . . . . 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 2761	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2760	. 2 ⊢ ℕ₀ = ({0, 1} ∪ ℕ)
19		oveq2 7428	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑖))
20	19	cbviunv 5043	. . 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 15005	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑_𝑟1) = 𝑑)
23	22	elv 3477	. . . . . . 7 ⊢ (𝑑↑_𝑟1) = 𝑑
24		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟1))
25	24	ssiun2s 5051	. . . . . . . 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 4015	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗))
29		ovex 7453	. . . . . . 7 ⊢ (𝑑↑_𝑟𝑗) ∈ V
30	5, 29	iunex 7972	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V)
32		0nn0 12517	. . . . . . 7 ⊢ 0 ∈ ℕ₀
33		1nn0 12518	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34		prssi 4825	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
35	32, 33, 34	mp2an 691	. . . . . 6 ⊢ {0, 1} ⊆ ℕ₀
36	35	sseli 3976	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ₀)
37	28, 31, 36	relexpss1d 43135	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑_𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
38	21, 37	mprg 3064	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
39	20, 38	eqsstri 4014	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
40		oveq2 7428	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑗))
41	40	cbviunv 5043	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
42		relexp1g 15005	. . . . 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 2759	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
45		1ex 11240	. . . . 5 ⊢ 1 ∈ V
46	45	prid2 4768	. . . 4 ⊢ 1 ∈ {0, 1}
47		oveq2 7428	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
48	47	ssiun2s 5051	. . . 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 4014	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
51		c0ex 11238	. . . . . 6 ⊢ 0 ∈ V
52	51	prid1 4767	. . . . 5 ⊢ 0 ∈ {0, 1}
53		oveq2 7428	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟0))
54	53	ssiun2s 5051	. . . . 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 4002	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
57		unss12 4182	. . . 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 691	. . 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 5097	. . . 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 7428	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0))
63	51, 62	iunxsn 5094	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0)
64		vex 3475	. . . . . . 7 ⊢ 𝑑 ∈ V
65		nnssnn0 12505	. . . . . . 7 ⊢ ℕ ⊆ ℕ₀
66		inelcm 4465	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
67	46, 11, 66	mp2an 691	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
68		iunrelexp0 43132	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ₀ ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0))
69	64, 65, 67, 68	mp3an 1458	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0)
70	63, 69	eqtri 2756	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (𝑑↑_𝑟0)
71	45, 47	iunxsn 5094	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
72	43, 41	eqtr4i 2759	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
73	71, 72	eqtri 2756	. . . . 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 2760	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ((𝑑↑_𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
76		iunxun 5097	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑_𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
77	58, 75, 76	3sstr4i 4023	. 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 43136	1 ⊢ (r* ∘ t+) = t*

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 ≠ wne 2937 Vcvv 3471 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 {csn 4629 {cpr 4631 ∪ ciun 4996 ∘ ccom 5682 (class class class)co 7420 0cc0 11138 1c1 11139 ℕcn 12242 ℕ₀cn0 12502 t+ctcl 14964 t*crtcl 14965 ↑_𝑟crelexp 14998 r*crcl 43102

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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 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 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-seq 13999 df-trcl 14966 df-rtrcl 14967 df-relexp 14999 df-rcl 43103

This theorem is referenced by: cortrcltrcl 43170 corclrtrcl 43171

W3C validator