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Theorem cotrcltrcl 38518
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 38513 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dftrcl3 38513 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dftrcl3 38513 . 2 t+ = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ (𝑐𝑟𝑘))
4 nnex 11314 . 2 ℕ ∈ V
5 unidm 3962 . . 3 (ℕ ∪ ℕ) = ℕ
65eqcomi 2822 . 2 ℕ = (ℕ ∪ ℕ)
7 1ex 10324 . . . . . 6 1 ∈ V
8 oveq2 6885 . . . . . 6 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
97, 8iunxsn 4802 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
10 ovex 6909 . . . . . . . 8 (𝑑𝑟𝑗) ∈ V
114, 10iunex 7380 . . . . . . 7 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
12 relexp1g 13992 . . . . . . 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 6885 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
1514cbviunv 4758 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1613, 15eqtri 2835 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
179, 16eqtri 2835 . . . 4 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1817eqcomi 2822 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
19 1nn 11319 . . . . 5 1 ∈ ℕ
20 snssi 4536 . . . . 5 (1 ∈ ℕ → {1} ⊆ ℕ)
2119, 20ax-mp 5 . . . 4 {1} ⊆ ℕ
22 iunss1 4731 . . . 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 3839 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
25 iunss 4760 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
26 oveq2 6885 . . . . . 6 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
2726sseq1d 3836 . . . . 5 (𝑥 = 1 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
28 oveq2 6885 . . . . . 6 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦))
2928sseq1d 3836 . . . . 5 (𝑥 = 𝑦 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
30 oveq2 6885 . . . . . 6 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)))
3130sseq1d 3836 . . . . 5 (𝑥 = (𝑦 + 1) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
32 oveq2 6885 . . . . . 6 (𝑥 = 𝑖 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3332sseq1d 3836 . . . . 5 (𝑥 = 𝑖 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
3416eqimssi 3863 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
35 simpl 470 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
36 relexpsucnnr 13991 . . . . . . . 8 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3711, 35, 36sylancr 577 . . . . . . 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 5486 . . . . . . . . 9 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3938adantl 469 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
4015coeq2i 5491 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
41 trclfvcotrg 13983 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
42 vex 3401 . . . . . . . . . . . 12 𝑑 ∈ V
43 oveq1 6884 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑐𝑟𝑘) = (𝑑𝑟𝑘))
4443iuneq2d 4746 . . . . . . . . . . . . 13 (𝑐 = 𝑑 𝑘 ∈ ℕ (𝑐𝑟𝑘) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
45 ovex 6909 . . . . . . . . . . . . . 14 (𝑑𝑟𝑘) ∈ V
464, 45iunex 7380 . . . . . . . . . . . . 13 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∈ V
4744, 3, 46fvmpt 6506 . . . . . . . . . . . 12 (𝑑 ∈ V → (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4842, 47ax-mp 5 . . . . . . . . . . 11 (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4948, 48coeq12i 5494 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5041, 49, 483sstr3i 3847 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5140, 50eqsstri 3839 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5239, 51syl6ss 3817 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5337, 52eqsstrd 3843 . . . . . 6 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5453ex 399 . . . . 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 11326 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5625, 55mprgbir 3122 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
57 iuneq1 4733 . . . 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 3841 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
601, 2, 3, 4, 4, 6, 24, 24, 59comptiunov2i 38499 1 (t+ ∘ t+) = t+
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wcel 2157  Vcvv 3398  cun 3774  wss 3776  {csn 4377   ciun 4719  ccom 5322  cfv 6104  (class class class)co 6877  1c1 10225   + caddc 10227  cn 11308  t+ctcl 13952  𝑟crelexp 13986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-2 11367  df-n0 11563  df-z 11647  df-uz 11908  df-seq 13028  df-trcl 13954  df-relexp 13987
This theorem is referenced by:  cortrcltrcl  38533  cotrclrtrcl  38537
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