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Theorem cotrcltrcl 43155
Description: The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
Assertion
Ref Expression
cotrcltrcl (t+ ∘ t+) = t+

Proof of Theorem cotrcltrcl
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑖 𝑗 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftrcl3 43150 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dftrcl3 43150 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dftrcl3 43150 . 2 t+ = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ (𝑐𝑟𝑘))
4 nnex 12249 . 2 ℕ ∈ V
5 unidm 4151 . . 3 (ℕ ∪ ℕ) = ℕ
65eqcomi 2737 . 2 ℕ = (ℕ ∪ ℕ)
7 1ex 11241 . . . . . 6 1 ∈ V
8 oveq2 7428 . . . . . 6 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
97, 8iunxsn 5094 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
10 ovex 7453 . . . . . . . 8 (𝑑𝑟𝑗) ∈ V
114, 10iunex 7972 . . . . . . 7 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
12 relexp1g 15006 . . . . . . 7 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗))
1311, 12ax-mp 5 . . . . . 6 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑗 ∈ ℕ (𝑑𝑟𝑗)
14 oveq2 7428 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
1514cbviunv 5043 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1613, 15eqtri 2756 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
179, 16eqtri 2756 . . . 4 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1817eqcomi 2737 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
19 1nn 12254 . . . 4 1 ∈ ℕ
20 snssi 4812 . . . 4 (1 ∈ ℕ → {1} ⊆ ℕ)
21 iunss1 5010 . . . 4 ({1} ⊆ ℕ → 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
2219, 20, 21mp2b 10 . . 3 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
2318, 22eqsstri 4014 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
24 iunss 5048 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
25 oveq2 7428 . . . . . 6 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
2625sseq1d 4011 . . . . 5 (𝑥 = 1 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
27 oveq2 7428 . . . . . 6 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦))
2827sseq1d 4011 . . . . 5 (𝑥 = 𝑦 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
29 oveq2 7428 . . . . . 6 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)))
3029sseq1d 4011 . . . . 5 (𝑥 = (𝑦 + 1) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
31 oveq2 7428 . . . . . 6 (𝑥 = 𝑖 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3231sseq1d 4011 . . . . 5 (𝑥 = 𝑖 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
3316eqimssi 4040 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
34 simpl 482 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
35 relexpsucnnr 15005 . . . . . . . 8 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3611, 34, 35sylancr 586 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
37 coss1 5858 . . . . . . . . 9 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3837adantl 481 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3915coeq2i 5863 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
40 trclfvcotrg 14996 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41 oveq1 7427 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑐𝑟𝑘) = (𝑑𝑟𝑘))
4241iuneq2d 5025 . . . . . . . . . . . . 13 (𝑐 = 𝑑 𝑘 ∈ ℕ (𝑐𝑟𝑘) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
43 ovex 7453 . . . . . . . . . . . . . 14 (𝑑𝑟𝑘) ∈ V
444, 43iunex 7972 . . . . . . . . . . . . 13 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∈ V
4542, 3, 44fvmpt 7005 . . . . . . . . . . . 12 (𝑑 ∈ V → (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4645elv 3477 . . . . . . . . . . 11 (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4746, 46coeq12i 5866 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4840, 47, 463sstr3i 4022 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4939, 48eqsstri 4014 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5038, 49sstrdi 3992 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5136, 50eqsstrd 4018 . . . . . 6 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5251ex 412 . . . . 5 (𝑦 ∈ ℕ → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
5326, 28, 30, 32, 33, 52nnind 12261 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5424, 53mprgbir 3065 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
55 iuneq1 5012 . . . 4 (ℕ = (ℕ ∪ ℕ) → 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘))
566, 55ax-mp 5 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
5754, 56sseqtri 4016 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
581, 2, 3, 4, 4, 6, 23, 23, 57comptiunov2i 43136 1 (t+ ∘ t+) = t+
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  cun 3945  wss 3947  {csn 4629   ciun 4996  ccom 5682  cfv 6548  (class class class)co 7420  1c1 11140   + caddc 11142  cn 12243  t+ctcl 14965  𝑟crelexp 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-seq 14000  df-trcl 14967  df-relexp 15000
This theorem is referenced by:  cortrcltrcl  43170  cotrclrtrcl  43174
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