Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  corcltrcl Structured version   Visualization version   GIF version

Theorem corcltrcl 40356
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 40293 . 2 r* = (𝑎 ∈ V ↦ 𝑖 ∈ {0, 1} (𝑎𝑟𝑖))
2 dftrcl3 40337 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dfrtrcl3 40350 . 2 t* = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ0 (𝑐𝑟𝑘))
4 prex 5320 . 2 {0, 1} ∈ V
5 nnex 11640 . 2 ℕ ∈ V
6 df-n0 11895 . . 3 0 = (ℕ ∪ {0})
7 uncom 4115 . . 3 (ℕ ∪ {0}) = ({0} ∪ ℕ)
8 df-pr 4553 . . . . 5 {0, 1} = ({0} ∪ {1})
98uneq1i 4121 . . . 4 ({0, 1} ∪ ℕ) = (({0} ∪ {1}) ∪ ℕ)
10 unass 4128 . . . 4 (({0} ∪ {1}) ∪ ℕ) = ({0} ∪ ({1} ∪ ℕ))
11 1nn 11645 . . . . . . 7 1 ∈ ℕ
12 snssi 4725 . . . . . . 7 (1 ∈ ℕ → {1} ⊆ ℕ)
1311, 12ax-mp 5 . . . . . 6 {1} ⊆ ℕ
14 ssequn1 4142 . . . . . 6 ({1} ⊆ ℕ ↔ ({1} ∪ ℕ) = ℕ)
1513, 14mpbi 233 . . . . 5 ({1} ∪ ℕ) = ℕ
1615uneq2i 4122 . . . 4 ({0} ∪ ({1} ∪ ℕ)) = ({0} ∪ ℕ)
179, 10, 163eqtrri 2852 . . 3 ({0} ∪ ℕ) = ({0, 1} ∪ ℕ)
186, 7, 173eqtri 2851 . 2 0 = ({0, 1} ∪ ℕ)
19 oveq2 7157 . . . 4 (𝑘 = 𝑖 → (𝑑𝑟𝑘) = (𝑑𝑟𝑖))
2019cbviunv 4951 . . 3 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) = 𝑖 ∈ {0, 1} (𝑑𝑟𝑖)
21 ss2iun 4923 . . . 4 (∀𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) → 𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
22 relexp1g 14385 . . . . . . . 8 (𝑑 ∈ V → (𝑑𝑟1) = 𝑑)
2322elv 3485 . . . . . . 7 (𝑑𝑟1) = 𝑑
24 oveq2 7157 . . . . . . . . 9 (𝑗 = 1 → (𝑑𝑟𝑗) = (𝑑𝑟1))
2524ssiun2s 4958 . . . . . . . 8 (1 ∈ ℕ → (𝑑𝑟1) ⊆ 𝑗 ∈ ℕ (𝑑𝑟𝑗))
2611, 25ax-mp 5 . . . . . . 7 (𝑑𝑟1) ⊆ 𝑗 ∈ ℕ (𝑑𝑟𝑗)
2723, 26eqsstrri 3988 . . . . . 6 𝑑 𝑗 ∈ ℕ (𝑑𝑟𝑗)
2827a1i 11 . . . . 5 (𝑖 ∈ {0, 1} → 𝑑 𝑗 ∈ ℕ (𝑑𝑟𝑗))
29 ovex 7182 . . . . . . 7 (𝑑𝑟𝑗) ∈ V
305, 29iunex 7664 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
3130a1i 11 . . . . 5 (𝑖 ∈ {0, 1} → 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V)
32 0nn0 11909 . . . . . . 7 0 ∈ ℕ0
33 1nn0 11910 . . . . . . 7 1 ∈ ℕ0
34 prssi 4738 . . . . . . 7 ((0 ∈ ℕ0 ∧ 1 ∈ ℕ0) → {0, 1} ⊆ ℕ0)
3532, 33, 34mp2an 691 . . . . . 6 {0, 1} ⊆ ℕ0
3635sseli 3949 . . . . 5 (𝑖 ∈ {0, 1} → 𝑖 ∈ ℕ0)
3728, 31, 36relexpss1d 40322 . . . 4 (𝑖 ∈ {0, 1} → (𝑑𝑟𝑖) ⊆ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3821, 37mprg 3147 . . 3 𝑖 ∈ {0, 1} (𝑑𝑟𝑖) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
3920, 38eqsstri 3987 . 2 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
40 oveq2 7157 . . . . 5 (𝑘 = 𝑗 → (𝑑𝑟𝑘) = (𝑑𝑟𝑗))
4140cbviunv 4951 . . . 4 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑗 ∈ ℕ (𝑑𝑟𝑗)
42 relexp1g 14385 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗))
4330, 42ax-mp 5 . . . 4 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗)
4441, 43eqtr4i 2850 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
45 1ex 10635 . . . . 5 1 ∈ V
4645prid2 4684 . . . 4 1 ∈ {0, 1}
47 oveq2 7157 . . . . 5 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
4847ssiun2s 4958 . . . 4 (1 ∈ {0, 1} → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
4946, 48ax-mp 5 . . 3 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
5044, 49eqsstri 3987 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
51 c0ex 10633 . . . . . 6 0 ∈ V
5251prid1 4683 . . . . 5 0 ∈ {0, 1}
53 oveq2 7157 . . . . . 6 (𝑘 = 0 → (𝑑𝑟𝑘) = (𝑑𝑟0))
5453ssiun2s 4958 . . . . 5 (0 ∈ {0, 1} → (𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘))
5552, 54ax-mp 5 . . . 4 (𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘)
56 ssid 3975 . . . 4 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
57 unss12 4144 . . . 4 (((𝑑𝑟0) ⊆ 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∧ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
5855, 56, 57mp2an 691 . . 3 ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
59 iuneq1 4921 . . . . 5 ({0, 1} = ({0} ∪ {1}) → 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
608, 59ax-mp 5 . . . 4 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
61 iunxun 5002 . . . 4 𝑖 ∈ ({0} ∪ {1})( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ∪ 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
62 oveq2 7157 . . . . . . 7 (𝑖 = 0 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0))
6351, 62iunxsn 4999 . . . . . 6 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0)
64 vex 3483 . . . . . . 7 𝑑 ∈ V
65 nnssnn0 11897 . . . . . . 7 ℕ ⊆ ℕ0
66 inelcm 4397 . . . . . . . 8 ((1 ∈ {0, 1} ∧ 1 ∈ ℕ) → ({0, 1} ∩ ℕ) ≠ ∅)
6746, 11, 66mp2an 691 . . . . . . 7 ({0, 1} ∩ ℕ) ≠ ∅
68 iunrelexp0 40319 . . . . . . 7 ((𝑑 ∈ V ∧ ℕ ⊆ ℕ0 ∧ ({0, 1} ∩ ℕ) ≠ ∅) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0) = (𝑑𝑟0))
6964, 65, 67, 68mp3an 1458 . . . . . 6 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟0) = (𝑑𝑟0)
7063, 69eqtri 2847 . . . . 5 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = (𝑑𝑟0)
7145, 47iunxsn 4999 . . . . . 6 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
7243, 41eqtr4i 2850 . . . . . 6 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
7371, 72eqtri 2847 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
7470, 73uneq12i 4123 . . . 4 ( 𝑖 ∈ {0} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ∪ 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)) = ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
7560, 61, 743eqtri 2851 . . 3 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ((𝑑𝑟0) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
76 iunxun 5002 . . 3 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑𝑟𝑘) = ( 𝑘 ∈ {0, 1} (𝑑𝑟𝑘) ∪ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
7758, 75, 763sstr4i 3996 . 2 𝑖 ∈ {0, 1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ({0, 1} ∪ ℕ)(𝑑𝑟𝑘)
781, 2, 3, 4, 5, 18, 39, 50, 77comptiunov2i 40323 1 (r* ∘ t+) = t*
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2115  wne 3014  Vcvv 3480  cun 3917  cin 3918  wss 3919  c0 4276  {csn 4550  {cpr 4552   ciun 4905  ccom 5546  (class class class)co 7149  0cc0 10535  1c1 10536  cn 11634  0cn0 11894  t+ctcl 14345  t*crtcl 14346  𝑟crelexp 14379  r*crcl 40289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-seq 13374  df-trcl 14347  df-rtrcl 14348  df-relexp 14380  df-rcl 40290
This theorem is referenced by:  cortrcltrcl  40357  corclrtrcl  40358
  Copyright terms: Public domain W3C validator