corcltrcl - Mathbox for Richard Penner

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Theorem corcltrcl 41317

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 41254	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑_𝑟𝑖))
2		dftrcl3 41298	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑_𝑟𝑗))
3		dfrtrcl3 41311	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ₀ (𝑐↑_𝑟𝑘))
4		prex 5359	. 2 ⊢ {0, 1} ∈ V
5		nnex 11979	. 2 ⊢ ℕ ∈ V
6		df-n0 12234	. . 3 ⊢ ℕ₀ = (ℕ ∪ {0})
7		uncom 4092	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4570	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4098	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4105	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 11984	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4747	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4119	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
15	13, 14	mpbi 229	. . . . 5 ⊢ ({1} ∪ ℕ) = ℕ
16	15	uneq2i 4099	. . . 4 ⊢ ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
17	9, 10, 16	3eqtrri 2773	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2772	. 2 ⊢ ℕ₀ = ({0, 1} ∪ ℕ)
19		oveq2 7279	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑖))
20	19	cbviunv 4975	. . 3 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) = ∪ 𝑖 ∈ {0, 1} (𝑑↑_𝑟𝑖)
21		ss2iun 4948	. . . 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 14735	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑_𝑟1) = 𝑑)
23	22	elv 3437	. . . . . . 7 ⊢ (𝑑↑_𝑟1) = 𝑑
24		oveq2 7279	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑_𝑟𝑗) = (𝑑↑_𝑟1))
25	24	ssiun2s 4983	. . . . . . . 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 3961	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗))
29		ovex 7304	. . . . . . 7 ⊢ (𝑑↑_𝑟𝑗) ∈ V
30	5, 29	iunex 7804	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗) ∈ V)
32		0nn0 12248	. . . . . . 7 ⊢ 0 ∈ ℕ₀
33		1nn0 12249	. . . . . . 7 ⊢ 1 ∈ ℕ₀
34		prssi 4760	. . . . . . 7 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ⊆ ℕ₀)
35	32, 33, 34	mp2an 689	. . . . . 6 ⊢ {0, 1} ⊆ ℕ₀
36	35	sseli 3922	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ₀)
37	28, 31, 36	relexpss1d 41283	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑_𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖))
38	21, 37	mprg 3080	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑_𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
39	20, 38	eqsstri 3960	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
40		oveq2 7279	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟𝑗))
41	40	cbviunv 4975	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)
42		relexp1g 14735	. . . . 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 2771	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
45		1ex 10972	. . . . 5 ⊢ 1 ∈ V
46	45	prid2 4705	. . . 4 ⊢ 1 ∈ {0, 1}
47		oveq2 7279	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1))
48	47	ssiun2s 4983	. . . 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 3960	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖)
51		c0ex 10970	. . . . . 6 ⊢ 0 ∈ V
52	51	prid1 4704	. . . . 5 ⊢ 0 ∈ {0, 1}
53		oveq2 7279	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑_𝑟𝑘) = (𝑑↑_𝑟0))
54	53	ssiun2s 4983	. . . . 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 3948	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
57		unss12 4121	. . . 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 689	. . 3 ⊢ ((𝑑↑_𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)) ⊆ (∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
59		iuneq1 4946	. . . . 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 5028	. . . 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 7279	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0))
63	51, 62	iunxsn 5025	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0)
64		vex 3435	. . . . . . 7 ⊢ 𝑑 ∈ V
65		nnssnn0 12236	. . . . . . 7 ⊢ ℕ ⊆ ℕ₀
66		inelcm 4404	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
67	46, 11, 66	mp2an 689	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
68		iunrelexp0 41280	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ₀ ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0))
69	64, 65, 67, 68	mp3an 1460	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟0) = (𝑑↑_𝑟0)
70	63, 69	eqtri 2768	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (𝑑↑_𝑟0)
71	45, 47	iunxsn 5025	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1)
72	43, 41	eqtr4i 2771	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
73	71, 72	eqtri 2768	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘)
74	70, 73	uneq12i 4100	. . . 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 2772	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑_𝑟𝑗)↑_𝑟𝑖) = ((𝑑↑_𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
76		iunxun 5028	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑_𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑_𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑_𝑟𝑘))
77	58, 75, 76	3sstr4i 3969	. 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 41284	1 ⊢ (r* ∘ t+) = t*

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2110 ≠ wne 2945 Vcvv 3431 ∪ cun 3890 ∩ cin 3891 ⊆ wss 3892 ∅c0 4262 {csn 4567 {cpr 4569 ∪ ciun 4930 ∘ ccom 5594 (class class class)co 7271 0cc0 10872 1c1 10873 ℕcn 11973 ℕ₀cn0 12233 t+ctcl 14694 t*crtcl 14695 ↑_𝑟crelexp 14728 r*crcl 41250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12582 df-seq 13720 df-trcl 14696 df-rtrcl 14697 df-relexp 14729 df-rcl 41251

This theorem is referenced by: cortrcltrcl 41318 corclrtrcl 41319

W3C validator