corcltrcl - Mathbox for Richard Penner

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Theorem corcltrcl 43701

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 43638	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑_𝑟𝑖))
2		dftrcl3 43682	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dfrtrcl3 43695	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ₀ (𝑐↑_𝑟𝑘))
4		prex 5387	. 2 ⊢ {0, 1} ∈ V
5		nnex 12168	. 2 ⊢ ℕ ∈ V
6		df-n0 12419	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
7		uncom 4117	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4588	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4123	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4131	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 12173	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4768	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4145	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
15	13, 14	mpbi 230	. . . . 5 ⊢ ({1} ∪ ℕ) = ℕ
16	15	uneq2i 4124	. . . 4 ⊢ ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
17	9, 10, 16	3eqtrri 2757	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2756	. 2 ⊢ ℕ₀ = ({0, 1} ∪ ℕ)
19		oveq2 7377	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑖))
20	19	cbviunv 4999	. . 3 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ {0, 1} (𝑑↑_𝑟𝑖)
21		ss2iun 4970	. . . 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 14968	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑_𝑟1) = 𝑑)
23	22	elv 3449	. . . . . . 7 ⊢ (𝑑↑_𝑟1) = 𝑑
24		oveq2 7377	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟1))
25	24	ssiun2s 5007	. . . . . . . 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 3991	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗))
29		ovex 7402	. . . . . . 7 ⊢ (𝑑↑_𝑟𝑗) ∈ V
30	5, 29	iunex 7926	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V)
32		0nn0 12433	. . . . . . 7 ⊢ 0 ∈ ℕ₀
33		1nn0 12434	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34		prssi 4781	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
35	32, 33, 34	mp2an 692	. . . . . 6 ⊢ {0, 1} ⊆ ℕ₀
36	35	sseli 3939	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ₀)
37	28, 31, 36	relexpss1d 43667	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑_𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
38	21, 37	mprg 3050	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
39	20, 38	eqsstri 3990	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
40		oveq2 7377	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑗))
41	40	cbviunv 4999	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
42		relexp1g 14968	. . . . 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 2755	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
45		1ex 11146	. . . . 5 ⊢ 1 ∈ V
46	45	prid2 4723	. . . 4 ⊢ 1 ∈ {0, 1}
47		oveq2 7377	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
48	47	ssiun2s 5007	. . . 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 3990	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
51		c0ex 11144	. . . . . 6 ⊢ 0 ∈ V
52	51	prid1 4722	. . . . 5 ⊢ 0 ∈ {0, 1}
53		oveq2 7377	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟0))
54	53	ssiun2s 5007	. . . . 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 3966	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
57		unss12 4147	. . . 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 4968	. . . . 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 5053	. . . 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 7377	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0))
63	51, 62	iunxsn 5050	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0)
64		vex 3448	. . . . . . 7 ⊢ 𝑑 ∈ V
65		nnssnn0 12421	. . . . . . 7 ⊢ ℕ ⊆ ℕ₀
66		inelcm 4424	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
67	46, 11, 66	mp2an 692	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
68		iunrelexp0 43664	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ₀ ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0))
69	64, 65, 67, 68	mp3an 1463	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0)
70	63, 69	eqtri 2752	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (𝑑↑_𝑟0)
71	45, 47	iunxsn 5050	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
72	43, 41	eqtr4i 2755	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
73	71, 72	eqtri 2752	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
74	70, 73	uneq12i 4125	. . . 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 2756	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ((𝑑↑_𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
76		iunxun 5053	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑_𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
77	58, 75, 76	3sstr4i 3995	. 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 43668	1 ⊢ (r* ∘ t+) = t*

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {csn 4585 {cpr 4587 ∪ ciun 4951 ∘ ccom 5635 (class class class)co 7369 0cc0 11044 1c1 11045 ℕcn 12162 ℕ₀cn0 12418 t+ctcl 14927 t*crtcl 14928 ↑_𝑟crelexp 14961 r*crcl 43634

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-n0 12419 df-z 12506 df-uz 12770 df-seq 13943 df-trcl 14929 df-rtrcl 14930 df-relexp 14962 df-rcl 43635

This theorem is referenced by: cortrcltrcl 43702 corclrtrcl 43703

W3C validator