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Theorem trclfvdecomr 36833
Description: The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
Assertion
Ref Expression
trclfvdecomr (𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Proof of Theorem trclfvdecomr
Dummy variables 𝑚 𝑛 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . 3 (𝑅𝑉𝑅 ∈ V)
2 oveq1 6534 . . . . 5 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
32iuneq2d 4478 . . . 4 (𝑟 = 𝑅 𝑛 ∈ ℕ (𝑟𝑟𝑛) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
4 dftrcl3 36825 . . . 4 t+ = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))
5 nnex 10876 . . . . 5 ℕ ∈ V
6 ovex 6555 . . . . 5 (𝑅𝑟𝑛) ∈ V
75, 6iunex 7017 . . . 4 𝑛 ∈ ℕ (𝑅𝑟𝑛) ∈ V
83, 4, 7fvmpt 6176 . . 3 (𝑅 ∈ V → (t+‘𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
91, 8syl 17 . 2 (𝑅𝑉 → (t+‘𝑅) = 𝑛 ∈ ℕ (𝑅𝑟𝑛))
10 nnuz 11558 . . . . . 6 ℕ = (ℤ‘1)
11 2eluzge1 11569 . . . . . . 7 2 ∈ (ℤ‘1)
12 uzsplit 12239 . . . . . . 7 (2 ∈ (ℤ‘1) → (ℤ‘1) = ((1...(2 − 1)) ∪ (ℤ‘2)))
1311, 12ax-mp 5 . . . . . 6 (ℤ‘1) = ((1...(2 − 1)) ∪ (ℤ‘2))
14 2m1e1 10985 . . . . . . . . 9 (2 − 1) = 1
1514oveq2i 6538 . . . . . . . 8 (1...(2 − 1)) = (1...1)
16 1z 11243 . . . . . . . . 9 1 ∈ ℤ
17 fzsn 12212 . . . . . . . . 9 (1 ∈ ℤ → (1...1) = {1})
1816, 17ax-mp 5 . . . . . . . 8 (1...1) = {1}
1915, 18eqtri 2632 . . . . . . 7 (1...(2 − 1)) = {1}
2019uneq1i 3725 . . . . . 6 ((1...(2 − 1)) ∪ (ℤ‘2)) = ({1} ∪ (ℤ‘2))
2110, 13, 203eqtri 2636 . . . . 5 ℕ = ({1} ∪ (ℤ‘2))
22 iuneq1 4465 . . . . 5 (ℕ = ({1} ∪ (ℤ‘2)) → 𝑛 ∈ ℕ (𝑅𝑟𝑛) = 𝑛 ∈ ({1} ∪ (ℤ‘2))(𝑅𝑟𝑛))
2321, 22ax-mp 5 . . . 4 𝑛 ∈ ℕ (𝑅𝑟𝑛) = 𝑛 ∈ ({1} ∪ (ℤ‘2))(𝑅𝑟𝑛)
24 iunxun 4536 . . . 4 𝑛 ∈ ({1} ∪ (ℤ‘2))(𝑅𝑟𝑛) = ( 𝑛 ∈ {1} (𝑅𝑟𝑛) ∪ 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
25 1ex 9892 . . . . . 6 1 ∈ V
26 oveq2 6535 . . . . . 6 (𝑛 = 1 → (𝑅𝑟𝑛) = (𝑅𝑟1))
2725, 26iunxsn 4534 . . . . 5 𝑛 ∈ {1} (𝑅𝑟𝑛) = (𝑅𝑟1)
2827uneq1i 3725 . . . 4 ( 𝑛 ∈ {1} (𝑅𝑟𝑛) ∪ 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛)) = ((𝑅𝑟1) ∪ 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
2923, 24, 283eqtri 2636 . . 3 𝑛 ∈ ℕ (𝑅𝑟𝑛) = ((𝑅𝑟1) ∪ 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
30 relexp1g 13563 . . . 4 (𝑅𝑉 → (𝑅𝑟1) = 𝑅)
31 oveq1 6534 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑟𝑟𝑚) = (𝑅𝑟𝑚))
3231iuneq2d 4478 . . . . . . . . 9 (𝑟 = 𝑅 𝑚 ∈ ℕ (𝑟𝑟𝑚) = 𝑚 ∈ ℕ (𝑅𝑟𝑚))
33 dftrcl3 36825 . . . . . . . . 9 t+ = (𝑟 ∈ V ↦ 𝑚 ∈ ℕ (𝑟𝑟𝑚))
34 ovex 6555 . . . . . . . . . 10 (𝑅𝑟𝑚) ∈ V
355, 34iunex 7017 . . . . . . . . 9 𝑚 ∈ ℕ (𝑅𝑟𝑚) ∈ V
3632, 33, 35fvmpt 6176 . . . . . . . 8 (𝑅 ∈ V → (t+‘𝑅) = 𝑚 ∈ ℕ (𝑅𝑟𝑚))
371, 36syl 17 . . . . . . 7 (𝑅𝑉 → (t+‘𝑅) = 𝑚 ∈ ℕ (𝑅𝑟𝑚))
3837coeq1d 5193 . . . . . 6 (𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = ( 𝑚 ∈ ℕ (𝑅𝑟𝑚) ∘ 𝑅))
39 coiun1 36757 . . . . . . 7 ( 𝑚 ∈ ℕ (𝑅𝑟𝑚) ∘ 𝑅) = 𝑚 ∈ ℕ ((𝑅𝑟𝑚) ∘ 𝑅)
40 uz2m1nn 11598 . . . . . . . . 9 (𝑛 ∈ (ℤ‘2) → (𝑛 − 1) ∈ ℕ)
4140adantl 481 . . . . . . . 8 ((𝑅𝑉𝑛 ∈ (ℤ‘2)) → (𝑛 − 1) ∈ ℕ)
42 eluzp1p1 11548 . . . . . . . . . . 11 (𝑚 ∈ (ℤ‘1) → (𝑚 + 1) ∈ (ℤ‘(1 + 1)))
4342, 10eleq2s 2706 . . . . . . . . . 10 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ‘(1 + 1)))
44 1p1e2 10984 . . . . . . . . . . 11 (1 + 1) = 2
4544fveq2i 6091 . . . . . . . . . 10 (ℤ‘(1 + 1)) = (ℤ‘2)
4643, 45syl6eleq 2698 . . . . . . . . 9 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ (ℤ‘2))
4746adantl 481 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑚 + 1) ∈ (ℤ‘2))
48 oveq2 6535 . . . . . . . . . 10 (𝑚 = (𝑛 − 1) → (𝑅𝑟𝑚) = (𝑅𝑟(𝑛 − 1)))
4948coeq1d 5193 . . . . . . . . 9 (𝑚 = (𝑛 − 1) → ((𝑅𝑟𝑚) ∘ 𝑅) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅))
50493ad2ant3 1077 . . . . . . . 8 ((𝑅𝑉𝑛 ∈ (ℤ‘2) ∧ 𝑚 = (𝑛 − 1)) → ((𝑅𝑟𝑚) ∘ 𝑅) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅))
51 oveq2 6535 . . . . . . . . 9 (𝑛 = (𝑚 + 1) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
52513ad2ant3 1077 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ ∧ 𝑛 = (𝑚 + 1)) → (𝑅𝑟𝑛) = (𝑅𝑟(𝑚 + 1)))
53 relexpsucnnr 13562 . . . . . . . . 9 ((𝑅𝑉𝑚 ∈ ℕ) → (𝑅𝑟(𝑚 + 1)) = ((𝑅𝑟𝑚) ∘ 𝑅))
5453eqcomd 2616 . . . . . . . 8 ((𝑅𝑉𝑚 ∈ ℕ) → ((𝑅𝑟𝑚) ∘ 𝑅) = (𝑅𝑟(𝑚 + 1)))
55 relexpsucnnr 13562 . . . . . . . . . 10 ((𝑅𝑉 ∧ (𝑛 − 1) ∈ ℕ) → (𝑅𝑟((𝑛 − 1) + 1)) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅))
5640, 55sylan2 490 . . . . . . . . 9 ((𝑅𝑉𝑛 ∈ (ℤ‘2)) → (𝑅𝑟((𝑛 − 1) + 1)) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅))
57 eluzelcn 11534 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ‘2) → 𝑛 ∈ ℂ)
58 npcan1 10307 . . . . . . . . . . . 12 (𝑛 ∈ ℂ → ((𝑛 − 1) + 1) = 𝑛)
59 oveq2 6535 . . . . . . . . . . . 12 (((𝑛 − 1) + 1) = 𝑛 → (𝑅𝑟((𝑛 − 1) + 1)) = (𝑅𝑟𝑛))
6057, 58, 593syl 18 . . . . . . . . . . 11 (𝑛 ∈ (ℤ‘2) → (𝑅𝑟((𝑛 − 1) + 1)) = (𝑅𝑟𝑛))
6160eqeq1d 2612 . . . . . . . . . 10 (𝑛 ∈ (ℤ‘2) → ((𝑅𝑟((𝑛 − 1) + 1)) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅𝑟𝑛) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅)))
6261adantl 481 . . . . . . . . 9 ((𝑅𝑉𝑛 ∈ (ℤ‘2)) → ((𝑅𝑟((𝑛 − 1) + 1)) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅) ↔ (𝑅𝑟𝑛) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅)))
6356, 62mpbid 221 . . . . . . . 8 ((𝑅𝑉𝑛 ∈ (ℤ‘2)) → (𝑅𝑟𝑛) = ((𝑅𝑟(𝑛 − 1)) ∘ 𝑅))
6441, 47, 50, 52, 54, 63cbviuneq12dv 36767 . . . . . . 7 (𝑅𝑉 𝑚 ∈ ℕ ((𝑅𝑟𝑚) ∘ 𝑅) = 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
6539, 64syl5eq 2656 . . . . . 6 (𝑅𝑉 → ( 𝑚 ∈ ℕ (𝑅𝑟𝑚) ∘ 𝑅) = 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
6638, 65eqtrd 2644 . . . . 5 (𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛))
6766eqcomd 2616 . . . 4 (𝑅𝑉 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛) = ((t+‘𝑅) ∘ 𝑅))
6830, 67uneq12d 3730 . . 3 (𝑅𝑉 → ((𝑅𝑟1) ∪ 𝑛 ∈ (ℤ‘2)(𝑅𝑟𝑛)) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
6929, 68syl5eq 2656 . 2 (𝑅𝑉 𝑛 ∈ ℕ (𝑅𝑟𝑛) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
709, 69eqtrd 2644 1 (𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cun 3538  {csn 4125   ciun 4450  ccom 5032  cfv 5790  (class class class)co 6527  cc 9791  1c1 9794   + caddc 9796  cmin 10118  cn 10870  2c2 10920  cz 11213  cuz 11522  ...cfz 12155  t+ctcl 13521  𝑟crelexp 13557
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 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
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 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-er 7607  df-en 7820  df-dom 7821  df-sdom 7822  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-seq 12622  df-trcl 13523  df-relexp 13558
This theorem is referenced by:  trclfvdecoml  36834  dmtrclfvRP  36835  frege124d  36866  frege131d  36869
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