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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 29027 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 7344 | . . . 4 β’ (π = π β (π WWalksN πΊ) = (π WWalksN πΊ)) | |
2 | 1 | adantl 482 | . . 3 β’ ((π£ = π β§ π = π) β (π WWalksN πΊ) = (π WWalksN πΊ)) |
3 | eqeq2 2748 | . . . . 5 β’ (π£ = π β ((π€β0) = π£ β (π€β0) = π)) | |
4 | neeq2 3004 | . . . . 5 β’ (π£ = π β ((lastSβπ€) β π£ β (lastSβπ€) β π)) | |
5 | 3, 4 | anbi12d 631 | . . . 4 β’ (π£ = π β (((π€β0) = π£ β§ (lastSβπ€) β π£) β ((π€β0) = π β§ (lastSβπ€) β π))) |
6 | 5 | adantr 481 | . . 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 3420 | . 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 7370 | . . 3 β’ (π WWalksN πΊ) β V | |
10 | 9 | rabex 5276 | . 2 β’ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)} β V |
11 | 7, 8, 10 | ovmpoa 7490 | 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 396 = wceq 1540 β wcel 2105 β wne 2940 {crab 3403 βcfv 6479 (class class class)co 7337 β cmpo 7339 0cc0 10972 βcn 12074 lastSclsw 14365 Vtxcvtx 27655 WWalksN cwwlksn 28479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 |
This theorem is referenced by: numclwwlkqhash 29027 numclwwlk2lem1 29028 numclwlk2lem2f 29029 numclwlk2lem2f1o 29031 |
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