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Mirrors > Home > MPE Home > Th. List > numclwwlk2lem3 | Structured version Visualization version GIF version |
Description: In a friendship graph, the size of the set of walks of length π starting with a fixed vertex π and ending not at this vertex equals the size of the set of all closed walks of length (π + 2) starting at this vertex π and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Proof shortened by AV, 3-Nov-2022.) |
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£)}) |
numclwwlk.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 |
---|---|
numclwwlk2lem3 | β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (β―β(πππ)) = (β―β(ππ»(π + 2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovexd 7449 | . . 3 β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (ππ»(π + 2)) β V) | |
2 | numclwwlk.v | . . . 4 β’ π = (VtxβπΊ) | |
3 | numclwwlk.q | . . . 4 β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) | |
4 | numclwwlk.h | . . . 4 β’ π» = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) | |
5 | eqid 2727 | . . . 4 β’ (β β (ππ»(π + 2)) β¦ (β prefix (π + 1))) = (β β (ππ»(π + 2)) β¦ (β prefix (π + 1))) | |
6 | 2, 3, 4, 5 | numclwlk2lem2f1o 30176 | . . 3 β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (β β (ππ»(π + 2)) β¦ (β prefix (π + 1))):(ππ»(π + 2))β1-1-ontoβ(πππ)) |
7 | 1, 6 | hasheqf1od 14336 | . 2 β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (β―β(ππ»(π + 2))) = (β―β(πππ))) |
8 | 7 | eqcomd 2733 | 1 β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (β―β(πππ)) = (β―β(ππ»(π + 2)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2935 {crab 3427 Vcvv 3469 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 β cmpo 7416 0cc0 11130 1c1 11131 + caddc 11133 β cmin 11466 βcn 12234 2c2 12289 β€β₯cuz 12844 β―chash 14313 lastSclsw 14536 prefix cpfx 14644 Vtxcvtx 28796 WWalksN cwwlksn 29624 ClWWalksNOncclwwlknon 29884 FriendGraph cfrgr 30055 |
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-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 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-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-rp 12999 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-lsw 14537 df-concat 14545 df-s1 14570 df-substr 14615 df-pfx 14645 df-wwlks 29628 df-wwlksn 29629 df-clwwlk 29779 df-clwwlkn 29822 df-clwwlknon 29885 df-frgr 30056 |
This theorem is referenced by: numclwwlk2 30178 |
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