![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cottrcl | Structured version Visualization version GIF version |
Description: Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.) |
Ref | Expression |
---|---|
cottrcl | ⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 6011 | . . . . 5 ⊢ Rel (𝑅 ↾ V) | |
2 | ssttrcl 9710 | . . . . 5 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) |
4 | coss1 5856 | . . . 4 ⊢ ((𝑅 ↾ V) ⊆ t++(𝑅 ↾ V) → ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) |
6 | ttrcltr 9711 | . . 3 ⊢ (t++(𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) | |
7 | 5, 6 | sstri 3992 | . 2 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) ⊆ t++(𝑅 ↾ V) |
8 | ssv 4007 | . . . 4 ⊢ ran t++(𝑅 ↾ V) ⊆ V | |
9 | cores 6249 | . . . 4 ⊢ (ran t++(𝑅 ↾ V) ⊆ V → ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++(𝑅 ↾ V))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++(𝑅 ↾ V)) |
11 | ttrclresv 9712 | . . . 4 ⊢ t++(𝑅 ↾ V) = t++𝑅 | |
12 | 11 | coeq2i 5861 | . . 3 ⊢ (𝑅 ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++𝑅) |
13 | 10, 12 | eqtri 2761 | . 2 ⊢ ((𝑅 ↾ V) ∘ t++(𝑅 ↾ V)) = (𝑅 ∘ t++𝑅) |
14 | 7, 13, 11 | 3sstr3i 4025 | 1 ⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3475 ⊆ wss 3949 ran crn 5678 ↾ cres 5679 ∘ ccom 5681 Rel wrel 5682 t++cttrcl 9702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-ttrcl 9703 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |