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Theorem cotrcltrcl 40237
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 40232 . 2 t+ = (𝑎 ∈ V ↦ 𝑖 ∈ ℕ (𝑎𝑟𝑖))
2 dftrcl3 40232 . 2 t+ = (𝑏 ∈ V ↦ 𝑗 ∈ ℕ (𝑏𝑟𝑗))
3 dftrcl3 40232 . 2 t+ = (𝑐 ∈ V ↦ 𝑘 ∈ ℕ (𝑐𝑟𝑘))
4 nnex 11621 . 2 ℕ ∈ V
5 unidm 4104 . . 3 (ℕ ∪ ℕ) = ℕ
65eqcomi 2830 . 2 ℕ = (ℕ ∪ ℕ)
7 1ex 10614 . . . . . 6 1 ∈ V
8 oveq2 7138 . . . . . 6 (𝑖 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
97, 8iunxsn 4986 . . . . 5 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1)
10 ovex 7163 . . . . . . . 8 (𝑑𝑟𝑗) ∈ V
114, 10iunex 7644 . . . . . . 7 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V
12 relexp1g 14364 . . . . . . 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 7138 . . . . . . 7 (𝑗 = 𝑘 → (𝑑𝑟𝑗) = (𝑑𝑟𝑘))
1514cbviunv 4938 . . . . . 6 𝑗 ∈ ℕ (𝑑𝑟𝑗) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1613, 15eqtri 2844 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
179, 16eqtri 2844 . . . 4 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
1817eqcomi 2830 . . 3 𝑘 ∈ ℕ (𝑑𝑟𝑘) = 𝑖 ∈ {1} ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
19 1nn 11626 . . . 4 1 ∈ ℕ
20 snssi 4714 . . . 4 (1 ∈ ℕ → {1} ⊆ ℕ)
21 iunss1 4906 . . . 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 3977 . 2 𝑘 ∈ ℕ (𝑑𝑟𝑘) ⊆ 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖)
24 iunss 4942 . . . 4 ( 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ∀𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
25 oveq2 7138 . . . . . 6 (𝑥 = 1 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1))
2625sseq1d 3974 . . . . 5 (𝑥 = 1 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
27 oveq2 7138 . . . . . 6 (𝑥 = 𝑦 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦))
2827sseq1d 3974 . . . . 5 (𝑥 = 𝑦 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
29 oveq2 7138 . . . . . 6 (𝑥 = (𝑦 + 1) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)))
3029sseq1d 3974 . . . . 5 (𝑥 = (𝑦 + 1) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
31 oveq2 7138 . . . . . 6 (𝑥 = 𝑖 → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) = ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖))
3231sseq1d 3974 . . . . 5 (𝑥 = 𝑖 → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑥) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) ↔ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)))
3316eqimssi 4001 . . . . 5 ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟1) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
34 simpl 486 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → 𝑦 ∈ ℕ)
35 relexpsucnnr 14363 . . . . . . . 8 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗) ∈ V ∧ 𝑦 ∈ ℕ) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) = (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3611, 34, 35sylancr 590 . . . . . . 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 5699 . . . . . . . . 9 (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3837adantl 485 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)))
3915coeq2i 5704 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
40 trclfvcotrg 14355 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) ⊆ (t+‘𝑑)
41 oveq1 7137 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑐𝑟𝑘) = (𝑑𝑟𝑘))
4241iuneq2d 4921 . . . . . . . . . . . . 13 (𝑐 = 𝑑 𝑘 ∈ ℕ (𝑐𝑟𝑘) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
43 ovex 7163 . . . . . . . . . . . . . 14 (𝑑𝑟𝑘) ∈ V
444, 43iunex 7644 . . . . . . . . . . . . 13 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∈ V
4542, 3, 44fvmpt 6741 . . . . . . . . . . . 12 (𝑑 ∈ V → (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4645elv 3476 . . . . . . . . . . 11 (t+‘𝑑) = 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4746, 46coeq12i 5707 . . . . . . . . . 10 ((t+‘𝑑) ∘ (t+‘𝑑)) = ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
4840, 47, 463sstr3i 3985 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
4939, 48eqsstri 3977 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑑𝑟𝑘) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
5038, 49sstrdi 3955 . . . . . . 7 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → (( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ∘ 𝑗 ∈ ℕ (𝑑𝑟𝑗)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5136, 50eqsstrd 3981 . . . . . 6 ((𝑦 ∈ ℕ ∧ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑦) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)) → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟(𝑦 + 1)) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5251ex 416 . . . . 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 11633 . . . 4 (𝑖 ∈ ℕ → ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘))
5424, 53mprgbir 3141 . . 3 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ ℕ (𝑑𝑟𝑘)
55 iuneq1 4908 . . . 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 3979 . 2 𝑖 ∈ ℕ ( 𝑗 ∈ ℕ (𝑑𝑟𝑗)↑𝑟𝑖) ⊆ 𝑘 ∈ (ℕ ∪ ℕ)(𝑑𝑟𝑘)
581, 2, 3, 4, 4, 6, 23, 23, 57comptiunov2i 40218 1 (t+ ∘ t+) = t+
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cun 3908  wss 3910  {csn 4540   ciun 4892  ccom 5532  cfv 6328  (class class class)co 7130  1c1 10515   + caddc 10517  cn 11615  t+ctcl 14324  𝑟crelexp 14358
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 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-n0 11876  df-z 11960  df-uz 12222  df-seq 13353  df-trcl 14326  df-relexp 14359
This theorem is referenced by:  cortrcltrcl  40252  cotrclrtrcl  40256
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