Theorem corcltrcl 44879

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 44816	. 2 ⊢ r* = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ {0, 1} (𝑎↑𝑟𝑖))
2		dftrcl3 44860	. 2 ⊢ t+ = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ ℕ (𝑏↑𝑟𝑗))
3		dfrtrcl3 44873	. 2 ⊢ t* = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑐↑𝑟𝑘))
4		prex 5399	. 2 ⊢ {0, 1} ∈ V
5		nnex 12351	. 2 ⊢ ℕ ∈ V
6		df-n0 12617	. . 3 ⊢ ℕ0 = (ℕ ∪ {0})
7		uncom 4159	. . 3 ⊢ (ℕ ∪ {0}) = ({0} ∪ ℕ)
8		df-pr 4590	. . . . 5 ⊢ {0, 1} = ({0} ∪ {1})
9	8	uneq1i 4165	. . . 4 ⊢ ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10		unass 4172	. . . 4 ⊢ (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11		1nn 12356	. . . . . . 7 ⊢ 1 ∈ ℕ
12		snssi 4745	. . . . . . 7 ⊢ (1 ∈ ℕ → {1} ⊆ ℕ)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ {1} ⊆ ℕ
14		ssequn1 4186	. . . . . 6 ⊢ ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
15	13, 14	mpbi 233	. . . . 5 ⊢ ({1} ∪ ℕ) = ℕ
16	15	uneq2i 4166	. . . 4 ⊢ ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
17	9, 10, 16	3eqtrri 2791	. . 3 ⊢ ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
18	6, 7, 17	3eqtri 2790	. 2 ⊢ ℕ0 = ({0, 1} ∪ ℕ)
19		oveq2 7435	. . . 4 ⊢ (𝑘 = 𝑖 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑖))
20	19	cbviunv 4999	. . 3 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) = ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖)
21		ss2iun 4972	. . . 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 15191	. . . . . . . 8 ⊢ (𝑑 ∈ V → (𝑑↑𝑟1) = 𝑑)
23	22	elv 3458	. . . . . . 7 ⊢ (𝑑↑𝑟1) = 𝑑
24		oveq2 7435	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝑑↑𝑟𝑗) = (𝑑↑𝑟1))
25	24	ssiun2s 5009	. . . . . . . 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 3959	. . . . . 6 ⊢ 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
28	27	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑑 ⊆ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗))
29		ovex 7460	. . . . . . 7 ⊢ (𝑑↑𝑟𝑗) ∈ V
30	5, 29	iunex 7987	. . . . . 6 ⊢ ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V
31	30	a1i 11	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗) ∈ V)
32		0nn0 12631	. . . . . . 7 ⊢ 0 ∈ ℕ0
33		1nn0 12632	. . . . . . 7 ⊢ 1 ∈ ℕ0
34		prssi 4779	. . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
35	32, 33, 34	mp2an 705	. . . . . 6 ⊢ {0, 1} ⊆ ℕ0
36	35	sseli 3908	. . . . 5 ⊢ (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ0)
37	28, 31, 36	relexpss1d 44845	. . . 4 ⊢ (𝑖 ∈ {0, 1} → (𝑑↑𝑟𝑖) ⊆ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
38	21, 37	mprg 3085	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (𝑑↑𝑟𝑖) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
39	20, 38	eqsstri 3958	. 2 ⊢ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
40		oveq2 7435	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟𝑗))
41	40	cbviunv 4999	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = ∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)
42		relexp1g 15191	. . . . 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 2789	. . 3 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
45		1elpr01 11314	. . . 4 ⊢ 1 ∈ {0, 1}
46		oveq2 7435	. . . . 5 ⊢ (𝑖 = 1 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1))
47	46	ssiun2s 5009	. . . 4 ⊢ (1 ∈ {0, 1} → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
48	45, 47	ax-mp 5	. . 3 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
49	44, 48	eqsstri 3958	. 2 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
50		c0ex 11310	. . . . . 6 ⊢ 0 ∈ V
51	50	prid1 4722	. . . . 5 ⊢ 0 ∈ {0, 1}
52		oveq2 7435	. . . . . 6 ⊢ (𝑘 = 0 → (𝑑↑𝑟𝑘) = (𝑑↑𝑟0))
53	52	ssiun2s 5009	. . . . 5 ⊢ (0 ∈ {0, 1} → (𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘))
54	51, 53	ax-mp 5	. . . 4 ⊢ (𝑑↑𝑟0) ⊆ ∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘)
55		ssid 3934	. . . 4 ⊢ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘) ⊆ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
56		unss12 4188	. . . 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)))
57	54, 55, 56	mp2an 705	. . 3 ⊢ ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)) ⊆ (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
58		iuneq1 4970	. . . . 5 ⊢ ({0, 1} = ({0} ∪ {1}) → ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
59	8, 58	ax-mp 5	. . . 4 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)
60		iunxun 5056	. . . 4 ⊢ ∪ 𝑖 ∈ ({0} ∪ {1})(∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ∪ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖))
61		oveq2 7435	. . . . . . 7 ⊢ (𝑖 = 0 → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0))
62	50, 61	iunxsn 5053	. . . . . 6 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0)
63		vex 3457	. . . . . . 7 ⊢ 𝑑 ∈ V
64		nnssnn0 12619	. . . . . . 7 ⊢ ℕ ⊆ ℕ0
65		inelcm 4420	. . . . . . . 8 ⊢ ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
66	45, 11, 65	mp2an 705	. . . . . . 7 ⊢ ({0, 1} ∩ ℕ) ≠ ∅
67		iunrelexp0 44842	. . . . . . 7 ⊢ ((𝑑 ∈ V ∧ ℕ ⊆ ℕ0 ∧ ({0, 1} ∩ ℕ) ≠ ∅) → (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0))
68	63, 64, 66, 67	mp3an 1490	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟0) = (𝑑↑𝑟0)
69	62, 68	eqtri 2786	. . . . 5 ⊢ ∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (𝑑↑𝑟0)
70		1ex 11313	. . . . . . 7 ⊢ 1 ∈ V
71	70, 46	iunxsn 5053	. . . . . 6 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1)
72	43, 41	eqtr4i 2789	. . . . . 6 ⊢ (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟1) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
73	71, 72	eqtri 2786	. . . . 5 ⊢ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘)
74	69, 73	uneq12i 4167	. . . 4 ⊢ (∪ 𝑖 ∈ {0} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ∪ ∪ 𝑖 ∈ {1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖)) = ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
75	59, 60, 74	3eqtri 2790	. . 3 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) = ((𝑑↑𝑟0) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
76		iunxun 5056	. . 3 ⊢ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑𝑟𝑘) = (∪ 𝑘 ∈ {0, 1} (𝑑↑𝑟𝑘) ∪ ∪ 𝑘 ∈ ℕ (𝑑↑𝑟𝑘))
77	57, 75, 76	3sstr4i 3963	. 2 ⊢ ∪ 𝑖 ∈ {0, 1} (∪ 𝑗 ∈ ℕ (𝑑↑𝑟𝑗)↑𝑟𝑖) ⊆ ∪ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑↑𝑟𝑘)
78	1, 2, 3, 4, 5, 18, 39, 49, 77	comptiunov2i 44846	1 ⊢ (r* ∘ t+) = t*

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  Vcvv 3453   ⊆ wss 3899   ∩ cin 4091   ∪ cun 4151  ∅c0 4280  {csn 4587  {cpr 4589  ∪ ciun 4953   ∘ ccom 5658  (class class class)co 7427  0cc0 11210  1c1 11211  ℕcn 12345  ℕ₀cn0 12616  t+ctcl 15150  t*crtcl 15151  ↑_𝑟crelexp 15184  r*crcl 44812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-z 12704  df-uz 12976  df-seq 14155  df-trcl 15152  df-rtrcl 15153  df-relexp 15185  df-rcl 44813

This theorem is used by:  cortrcltrcl  44880  corclrtrcl  44881