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Theorem corcltrcl 38348
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.
StepHypRef Expression
1 dfrcl4 38285 . 2 r* = (𝑎 ∈ V ↦ 𝑖 ∈ {0, 1} (𝑎𝑟𝑖))
2 dftrcl3 38329 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dfrtrcl3 38342 . 2 t* = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑐𝑟𝑘))
4 prex 4939 . 2 {0, 1} ∈ V
5 nnex 11064 . 2 ℕ ∈ V
6 df-n0 11331 . . 3 0 = (ℕ ∪ {0})
7 uncom 3790 . . 3 (ℕ ∪ {0}) = ({0} ∪ ℕ)
8 df-pr 4213 . . . . 5 {0, 1} = ({0} ∪ {1})
98uneq1i 3796 . . . 4 ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10 unass 3803 . . . 4 (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11 1nn 11069 . . . . . . 7 1 ∈ ℕ
12 snssi 4371 . . . . . . 7 (1 ∈ ℕ → {1} ⊆ ℕ)
1311, 12ax-mp 5 . . . . . 6 {1} ⊆ ℕ
14 ssequn1 3816 . . . . . 6 ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
1513, 14mpbi 220 . . . . 5 ({1} ∪ ℕ) = ℕ
1615uneq2i 3797 . . . 4 ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
179, 10, 163eqtrri 2678 . . 3 ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
186, 7, 173eqtri 2677 . 2 0 = ({0, 1} ∪ ℕ)
19 oveq2 6698 . . . 4 (𝑘 = 𝑖 → (𝑑𝑟𝑘) = (𝑑𝑟𝑖))
2019cbviunv 4591 . . 3 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) = 𝑖 ∈ {0, 1} (𝑑𝑟𝑖)
21 ss2iun 4568 . . . 4 (∀𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) → 𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
22 vex 3234 . . . . . . . 8 𝑑 ∈ V
23 relexp1g 13810 . . . . . . . 8 (𝑑 ∈ V → (𝑑𝑟1) = 𝑑)
2422, 23ax-mp 5 . . . . . . 7 (𝑑𝑟1) = 𝑑
25 oveq2 6698 . . . . . . . . 9 (𝑗 = 1 → (𝑑𝑟𝑗) = (𝑑𝑟1))
2625ssiun2s 4596 . . . . . . . 8 (1 ∈ ℕ → (𝑑𝑟1) ⊆ 𝑗 ∈ ℕ (𝑑𝑟𝑗))
2711, 26ax-mp 5 . . . . . . 7 (𝑑𝑟1) ⊆ 𝑗 ∈ ℕ (𝑑𝑟𝑗)
2824, 27eqsstr3i 3669 . . . . . 6 𝑑 𝑗 ∈ ℕ (𝑑𝑟𝑗)
2928a1i 11 . . . . 5 (𝑖 ∈ {0, 1} → 𝑑 𝑗 ∈ ℕ (𝑑𝑟𝑗))
30 ovex 6718 . . . . . . 7 (𝑑𝑟𝑗) ∈ V
315, 30iunex 7189 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
3231a1i 11 . . . . 5 (𝑖 ∈ {0, 1} → 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V)
33 0nn0 11345 . . . . . . 7 0 ∈ ℕ0
34 1nn0 11346 . . . . . . 7 1 ∈ ℕ0
35 prssi 4385 . . . . . . 7 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
3633, 34, 35mp2an 708 . . . . . 6 {0, 1} ⊆ ℕ0
3736sseli 3632 . . . . 5 (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ0)
3829, 32, 37relexpss1d 38314 . . . 4 (𝑖 ∈ {0, 1} → (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3921, 38mprg 2955 . . 3 𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
4020, 39eqsstri 3668 . 2 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
41 oveq2 6698 . . . . 5 (𝑘 = 𝑗 → (𝑑𝑟𝑘) = (𝑑𝑟𝑗))
4241cbviunv 4591 . . . 4 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑗 ∈ ℕ (𝑑𝑟𝑗)
43 relexp1g 13810 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗))
4431, 43ax-mp 5 . . . 4 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗)
4542, 44eqtr4i 2676 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
46 1ex 10073 . . . . 5 1 ∈ V
4746prid2 4330 . . . 4 1 ∈ {0, 1}
48 oveq2 6698 . . . . 5 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
4948ssiun2s 4596 . . . 4 (1 ∈ {0, 1} → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
5047, 49ax-mp 5 . . 3 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
5145, 50eqsstri 3668 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
52 c0ex 10072 . . . . . 6 0 ∈ V
5352prid1 4329 . . . . 5 0 ∈ {0, 1}
54 oveq2 6698 . . . . . 6 (𝑘 = 0 → (𝑑𝑟𝑘) = (𝑑𝑟0))
5554ssiun2s 4596 . . . . 5 (0 ∈ {0, 1} → (𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘))
5653, 55ax-mp 5 . . . 4 (𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
57 ssid 3657 . . . 4 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
58 unss12 3818 . . . 4 (((𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∧ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
5956, 57, 58mp2an 708 . . 3 ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
60 iuneq1 4566 . . . . 5 ({0, 1} = ({0} ∪ {1}) → 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
618, 60ax-mp 5 . . . 4 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
62 iunxun 4637 . . . 4 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ∪ 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
63 oveq2 6698 . . . . . . 7 (𝑖 = 0 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0))
6452, 63iunxsn 4635 . . . . . 6 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0)
65 nnssnn0 11333 . . . . . . 7 ℕ ⊆ ℕ0
66 inelcm 4065 . . . . . . . 8 ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
6747, 11, 66mp2an 708 . . . . . . 7 ({0, 1} ∩ ℕ) ≠ ∅
68 iunrelexp0 38311 . . . . . . 7 ((𝑑 ∈ V ∧ ℕ ⊆ ℕ0 ∧ ({0, 1} ∩ ℕ) ≠ ∅) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0) = (𝑑𝑟0))
6922, 65, 67, 68mp3an 1464 . . . . . 6 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0) = (𝑑𝑟0)
7064, 69eqtri 2673 . . . . 5 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = (𝑑𝑟0)
7146, 48iunxsn 4635 . . . . . 6 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
7244, 42eqtr4i 2676 . . . . . 6 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
7371, 72eqtri 2673 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
7470, 73uneq12i 3798 . . . 4 ( 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ∪ 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)) = ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
7561, 62, 743eqtri 2677 . . 3 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
76 iunxun 4637 . . 3 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑𝑟𝑘) = ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
7759, 75, 763sstr4i 3677 . 2 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑𝑟𝑘)
781, 2, 3, 4, 5, 18, 40, 51, 77comptiunov2i 38315 1 (r* ∘ t+) = t*
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  wne 2823  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  {cpr 4212   ciun 4552  ccom 5147  (class class class)co 6690  0cc0 9974  1c1 9975  cn 11058  0cn0 11330  t+ctcl 13770  t*crtcl 13771  𝑟crelexp 13804  r*crcl 38281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-trcl 13772  df-rtrcl 13773  df-relexp 13805  df-rcl 38282
This theorem is referenced by:  cortrcltrcl  38349  corclrtrcl  38350
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