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Mirrors > Home > MPE Home > Th. List > numclwwlkovh0 | Structured version Visualization version GIF version |
Description: Value of operation π», mapping a vertex π£ and an integer π 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 7 in [Huneke] p. 2. (Contributed by AV, 1-May-2022.) |
Ref | Expression |
---|---|
numclwwlkovh.h | β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) |
Ref | Expression |
---|---|
numclwwlkovh0 | β’ ((π β π β§ π β (β€β₯β2)) β (ππ»π) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7423 | . . 3 β’ ((π£ = π β§ π = π) β (π£(ClWWalksNOnβπΊ)π) = (π(ClWWalksNOnβπΊ)π)) | |
2 | oveq1 7421 | . . . . . 6 β’ (π = π β (π β 2) = (π β 2)) | |
3 | 2 | adantl 481 | . . . . 5 β’ ((π£ = π β§ π = π) β (π β 2) = (π β 2)) |
4 | 3 | fveq2d 6895 | . . . 4 β’ ((π£ = π β§ π = π) β (π€β(π β 2)) = (π€β(π β 2))) |
5 | simpl 482 | . . . 4 β’ ((π£ = π β§ π = π) β π£ = π) | |
6 | 4, 5 | neeq12d 2997 | . . 3 β’ ((π£ = π β§ π = π) β ((π€β(π β 2)) β π£ β (π€β(π β 2)) β π)) |
7 | 1, 6 | rabeqbidv 3444 | . 2 β’ ((π£ = π β§ π = π) β {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£} = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π}) |
8 | numclwwlkovh.h | . 2 β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) | |
9 | ovex 7447 | . . 3 β’ (π(ClWWalksNOnβπΊ)π) β V | |
10 | 9 | rabex 5328 | . 2 β’ {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π} β V |
11 | 7, 8, 10 | ovmpoa 7570 | 1 β’ ((π β π β§ π β (β€β₯β2)) β (ππ»π) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wne 2935 {crab 3427 βcfv 6542 (class class class)co 7414 β cmpo 7416 β cmin 11466 2c2 12289 β€β₯cuz 12844 ClWWalksNOncclwwlknon 29884 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 |
This theorem is referenced by: numclwwlkovh 30170 numclwwlk3lem2lem 30180 numclwwlk3lem2 30181 |
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