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Mirrors > Home > MPE Home > Th. List > 2clwwlk | Structured version Visualization version GIF version |
Description: Value of operation πΆ, mapping a vertex v and an integer n greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in [Huneke] p. 2. Such closed walks are "double loops" consisting of a closed (n-2)-walk v = v(0) ... v(n-2) = v and a closed 2-walk v = v(n-2) v(n-1) v(n) = v, see 2clwwlk2clwwlk 29603. (ππΆπ) is called the "set of double loops of length π on vertex π " in the following. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.) |
Ref | Expression |
---|---|
2clwwlk.c | β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) |
Ref | Expression |
---|---|
2clwwlk | β’ ((π β π β§ π β (β€β₯β2)) β (ππΆπ) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7418 | . . 3 β’ ((π£ = π β§ π = π) β (π£(ClWWalksNOnβπΊ)π) = (π(ClWWalksNOnβπΊ)π)) | |
2 | fvoveq1 7432 | . . . . 5 β’ (π = π β (π€β(π β 2)) = (π€β(π β 2))) | |
3 | 2 | adantl 483 | . . . 4 β’ ((π£ = π β§ π = π) β (π€β(π β 2)) = (π€β(π β 2))) |
4 | simpl 484 | . . . 4 β’ ((π£ = π β§ π = π) β π£ = π) | |
5 | 3, 4 | eqeq12d 2749 | . . 3 β’ ((π£ = π β§ π = π) β ((π€β(π β 2)) = π£ β (π€β(π β 2)) = π)) |
6 | 1, 5 | rabeqbidv 3450 | . 2 β’ ((π£ = π β§ π = π) β {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£} = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π}) |
7 | 2clwwlk.c | . 2 β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) | |
8 | ovex 7442 | . . 3 β’ (π(ClWWalksNOnβπΊ)π) β V | |
9 | 8 | rabex 5333 | . 2 β’ {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π} β V |
10 | 6, 7, 9 | ovmpoa 7563 | 1 β’ ((π β π β§ π β (β€β₯β2)) β (ππΆπ) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3433 βcfv 6544 (class class class)co 7409 β cmpo 7411 β cmin 11444 2c2 12267 β€β₯cuz 12822 ClWWalksNOncclwwlknon 29340 |
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-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 |
This theorem is referenced by: 2clwwlk2 29601 2clwwlkel 29602 extwwlkfab 29605 numclwwlk3lem2lem 29636 numclwwlk3lem2 29637 |
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