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Theorem cotrcltrcl 37036
 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 37031 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dftrcl3 37031 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dftrcl3 37031 . 2 t+ = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ (𝑐𝑟𝑘))
4 nnex 10903 . 2 ℕ ∈ V
5 unidm 3718 . . 3 (ℕ ∪ ℕ) = ℕ
65eqcomi 2619 . 2 ℕ = (ℕ ∪ ℕ)
7 1ex 9914 . . . . . 6 1 ∈ V
8 oveq2 6557 . . . . . 6 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
97, 8iunxsn 4539 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
10 ovex 6577 . . . . . . . 8 (𝑑𝑟𝑗) ∈ V
114, 10iunex 7039 . . . . . . 7 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
12 relexp1g 13614 . . . . . . 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 6557 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
1514cbviunv 4495 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1613, 15eqtri 2632 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
179, 16eqtri 2632 . . . 4 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1817eqcomi 2619 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
19 1nn 10908 . . . . 5 1 ∈ ℕ
20 snssi 4280 . . . . 5 (1 ∈ ℕ → {1} ⊆ ℕ)
2119, 20ax-mp 5 . . . 4 {1} ⊆ ℕ
22 iunss1 4468 . . . 4 ({1} ⊆ ℕ → 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
2321, 22ax-mp 5 . . 3 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
2418, 23eqsstri 3598 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
25 iunss 4497 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
26 oveq2 6557 . . . . . 6 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
2726sseq1d 3595 . . . . 5 (𝑥 = 1 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
28 oveq2 6557 . . . . . 6 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦))
2928sseq1d 3595 . . . . 5 (𝑥 = 𝑦 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
30 oveq2 6557 . . . . . 6 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)))
3130sseq1d 3595 . . . . 5 (𝑥 = (𝑦 + 1) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
32 oveq2 6557 . . . . . 6 (𝑥 = 𝑖 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3332sseq1d 3595 . . . . 5 (𝑥 = 𝑖 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
3416eqimssi 3622 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
35 simpl 472 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
36 relexpsucnnr 13613 . . . . . . . 8 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3711, 35, 36sylancr 694 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
38 coss1 5199 . . . . . . . . 9 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3938adantl 481 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
4015coeq2i 5204 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
41 trclfvcotrg 13605 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
42 vex 3176 . . . . . . . . . . . 12 𝑑 ∈ V
43 oveq1 6556 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑐𝑟𝑘) = (𝑑𝑟𝑘))
4443iuneq2d 4483 . . . . . . . . . . . . 13 (𝑐 = 𝑑 𝑘 ∈ ℕ (𝑐𝑟𝑘) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
45 ovex 6577 . . . . . . . . . . . . . 14 (𝑑𝑟𝑘) ∈ V
464, 45iunex 7039 . . . . . . . . . . . . 13 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∈ V
4744, 3, 46fvmpt 6191 . . . . . . . . . . . 12 (𝑑 ∈ V → (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4842, 47ax-mp 5 . . . . . . . . . . 11 (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4948, 48coeq12i 5207 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5041, 49, 483sstr3i 3606 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5140, 50eqsstri 3598 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5239, 51syl6ss 3580 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5337, 52eqsstrd 3602 . . . . . 6 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5453ex 449 . . . . 5 (𝑦 ∈ ℕ → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
5527, 29, 31, 33, 34, 54nnind 10915 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5625, 55mprgbir 2911 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
57 iuneq1 4470 . . . 4 (ℕ = (ℕ ∪ ℕ) → 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘))
586, 57ax-mp 5 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
5956, 58sseqtri 3600 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
601, 2, 3, 4, 4, 6, 24, 24, 59comptiunov2i 37017 1 (t+ ∘ t+) = t+
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  {csn 4125  ∪ ciun 4455   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1c1 9816   + caddc 9818  ℕcn 10897  t+ctcl 13572  ↑𝑟crelexp 13608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-trcl 13574  df-relexp 13609 This theorem is referenced by:  cortrcltrcl  37051  cotrclrtrcl  37055
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