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Mirrors > Home > MPE Home > Th. List > numclwwlkovq | Structured version Visualization version GIF version |
Description: Value of operation π, mapping a vertex π£ and a positive integer π to the not closed walks v(0) ... v(n) of length π from a fixed vertex π£ = v(0). "Not closed" means v(n) =/= v(0). Remark: π β β0 would not be useful: numclwwlkqhash 30224 would not hold, because (πΎβ0) = 1! (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.) |
Ref | Expression |
---|---|
numclwwlk.v | β’ π = (VtxβπΊ) |
numclwwlk.q | β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) |
Ref | Expression |
---|---|
numclwwlkovq | β’ ((π β π β§ π β β) β (πππ) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7420 | . . . 4 β’ (π = π β (π WWalksN πΊ) = (π WWalksN πΊ)) | |
2 | 1 | adantl 480 | . . 3 β’ ((π£ = π β§ π = π) β (π WWalksN πΊ) = (π WWalksN πΊ)) |
3 | eqeq2 2737 | . . . . 5 β’ (π£ = π β ((π€β0) = π£ β (π€β0) = π)) | |
4 | neeq2 2994 | . . . . 5 β’ (π£ = π β ((lastSβπ€) β π£ β (lastSβπ€) β π)) | |
5 | 3, 4 | anbi12d 630 | . . . 4 β’ (π£ = π β (((π€β0) = π£ β§ (lastSβπ€) β π£) β ((π€β0) = π β§ (lastSβπ€) β π))) |
6 | 5 | adantr 479 | . . 3 β’ ((π£ = π β§ π = π) β (((π€β0) = π£ β§ (lastSβπ€) β π£) β ((π€β0) = π β§ (lastSβπ€) β π))) |
7 | 2, 6 | rabeqbidv 3437 | . 2 β’ ((π£ = π β§ π = π) β {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)} = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
8 | numclwwlk.q | . 2 β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) | |
9 | ovex 7446 | . . 3 β’ (π WWalksN πΊ) β V | |
10 | 9 | rabex 5330 | . 2 β’ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} β V |
11 | 7, 8, 10 | ovmpoa 7570 | 1 β’ ((π β π β§ π β β) β (πππ) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 {crab 3419 βcfv 6543 (class class class)co 7413 β cmpo 7415 0cc0 11133 βcn 12237 lastSclsw 14539 Vtxcvtx 28848 WWalksN cwwlksn 29676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 |
This theorem is referenced by: numclwwlkqhash 30224 numclwwlk2lem1 30225 numclwlk2lem2f 30226 numclwlk2lem2f1o 30228 |
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