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Mirrors > Home > MPE Home > Th. List > wlksnfi | Structured version Visualization version GIF version |
Description: The number of walks of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 20-Apr-2021.) |
Ref | Expression |
---|---|
wlksnfi | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | fusgrvtxfi 26616 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → (Vtx‘𝐺) ∈ Fin) |
3 | 2 | adantr 474 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → (Vtx‘𝐺) ∈ Fin) |
4 | wwlksnfi 27228 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) ∈ Fin) |
6 | fusgrusgr 26619 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) | |
7 | usgruspgr 26477 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph) |
9 | wlknwwlksnen 27194 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺)) | |
10 | 8, 9 | sylan 577 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺)) |
11 | enfii 8446 | . 2 ⊢ (((𝑁 WWalksN 𝐺) ∈ Fin ∧ {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺)) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin) | |
12 | 5, 10, 11 | syl2anc 581 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3121 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 1st c1st 7426 ≈ cen 8219 Fincfn 8222 ℕ0cn0 11618 ♯chash 13410 Vtxcvtx 26294 USPGraphcuspgr 26447 USGraphcusgr 26448 FinUSGraphcfusgr 26613 Walkscwlks 26894 WWalksN cwwlksn 27125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-ifp 1092 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-n0 11619 df-xnn0 11691 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-exp 13155 df-hash 13411 df-word 13575 df-edg 26346 df-uhgr 26356 df-upgr 26380 df-uspgr 26449 df-usgr 26450 df-fusgr 26614 df-wlks 26897 df-wwlks 27129 df-wwlksn 27130 |
This theorem is referenced by: clwlknon2num 27771 numclwlk1lem2 27773 |
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