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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 29870. (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 7420 | . . 3 β’ ((π£ = π β§ π = π) β (π£(ClWWalksNOnβπΊ)π) = (π(ClWWalksNOnβπΊ)π)) | |
2 | fvoveq1 7434 | . . . . 5 β’ (π = π β (π€β(π β 2)) = (π€β(π β 2))) | |
3 | 2 | adantl 480 | . . . 4 β’ ((π£ = π β§ π = π) β (π€β(π β 2)) = (π€β(π β 2))) |
4 | simpl 481 | . . . 4 β’ ((π£ = π β§ π = π) β π£ = π) | |
5 | 3, 4 | eqeq12d 2746 | . . 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 3447 | . 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 7444 | . . 3 β’ (π(ClWWalksNOnβπΊ)π) β V | |
9 | 8 | rabex 5331 | . 2 β’ {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π} β V |
10 | 6, 7, 9 | ovmpoa 7565 | 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 394 = wceq 1539 β wcel 2104 {crab 3430 βcfv 6542 (class class class)co 7411 β cmpo 7413 β cmin 11448 2c2 12271 β€β₯cuz 12826 ClWWalksNOncclwwlknon 29607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: 2clwwlk2 29868 2clwwlkel 29869 extwwlkfab 29872 numclwwlk3lem2lem 29903 numclwwlk3lem2 29904 |
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