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Mirrors > Home > MPE Home > Th. List > Mathboxes > trclfvcom | Structured version Visualization version GIF version |
Description: The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.) |
Ref | Expression |
---|---|
trclfvcom | ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3484 | . 2 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
2 | relexpsucnnr 15024 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ) → (𝑅↑𝑟(𝑛 + 1)) = ((𝑅↑𝑟𝑛) ∘ 𝑅)) | |
3 | relexpsucnnl 15029 | . . . . 5 ⊢ ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ) → (𝑅↑𝑟(𝑛 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑛))) | |
4 | 2, 3 | eqtr3d 2768 | . . . 4 ⊢ ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ) → ((𝑅↑𝑟𝑛) ∘ 𝑅) = (𝑅 ∘ (𝑅↑𝑟𝑛))) |
5 | 4 | iuneq2dv 5019 | . . 3 ⊢ (𝑅 ∈ V → ∪ 𝑛 ∈ ℕ ((𝑅↑𝑟𝑛) ∘ 𝑅) = ∪ 𝑛 ∈ ℕ (𝑅 ∘ (𝑅↑𝑟𝑛))) |
6 | oveq1 7422 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛)) | |
7 | 6 | iuneq2d 5024 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
8 | dftrcl3 43423 | . . . . . 6 ⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛)) | |
9 | nnex 12263 | . . . . . . 7 ⊢ ℕ ∈ V | |
10 | ovex 7448 | . . . . . . 7 ⊢ (𝑅↑𝑟𝑛) ∈ V | |
11 | 9, 10 | iunex 7973 | . . . . . 6 ⊢ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∈ V |
12 | 7, 8, 11 | fvmpt 7000 | . . . . 5 ⊢ (𝑅 ∈ V → (t+‘𝑅) = ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) |
13 | 12 | coeq1d 5860 | . . . 4 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ 𝑅) = (∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∘ 𝑅)) |
14 | coiun1 43355 | . . . 4 ⊢ (∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛) ∘ 𝑅) = ∪ 𝑛 ∈ ℕ ((𝑅↑𝑟𝑛) ∘ 𝑅) | |
15 | 13, 14 | eqtrdi 2782 | . . 3 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ 𝑅) = ∪ 𝑛 ∈ ℕ ((𝑅↑𝑟𝑛) ∘ 𝑅)) |
16 | 12 | coeq2d 5861 | . . . 4 ⊢ (𝑅 ∈ V → (𝑅 ∘ (t+‘𝑅)) = (𝑅 ∘ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛))) |
17 | coiun 6259 | . . . 4 ⊢ (𝑅 ∘ ∪ 𝑛 ∈ ℕ (𝑅↑𝑟𝑛)) = ∪ 𝑛 ∈ ℕ (𝑅 ∘ (𝑅↑𝑟𝑛)) | |
18 | 16, 17 | eqtrdi 2782 | . . 3 ⊢ (𝑅 ∈ V → (𝑅 ∘ (t+‘𝑅)) = ∪ 𝑛 ∈ ℕ (𝑅 ∘ (𝑅↑𝑟𝑛))) |
19 | 5, 15, 18 | 3eqtr4d 2776 | . 2 ⊢ (𝑅 ∈ V → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅))) |
20 | 1, 19 | syl 17 | 1 ⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3464 ∪ ciun 4995 ∘ ccom 5678 ‘cfv 6545 (class class class)co 7415 1c1 11149 + caddc 11151 ℕcn 12257 t+ctcl 14984 ↑𝑟crelexp 15018 |
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 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-int 4949 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12258 df-2 12320 df-n0 12518 df-z 12604 df-uz 12868 df-seq 14015 df-trcl 14986 df-relexp 15019 |
This theorem is referenced by: trclfvdecoml 43432 |
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