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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 2818 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | fusgrvtxfi 27028 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → (Vtx‘𝐺) ∈ Fin) |
3 | 2 | adantr 481 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → (Vtx‘𝐺) ∈ Fin) |
4 | wwlksnfi 27611 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 WWalksN 𝐺) ∈ Fin) |
6 | fusgrusgr 27031 | . . . 4 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph) | |
7 | usgruspgr 26890 | . . . 4 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USPGraph) |
9 | wlknwwlksnen 27594 | . . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺)) | |
10 | 8, 9 | sylan 580 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺)) |
11 | enfii 8723 | . 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 584 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 ≈ cen 8494 Fincfn 8497 ℕ0cn0 11885 ♯chash 13678 Vtxcvtx 26708 USPGraphcuspgr 26860 USGraphcusgr 26861 FinUSGraphcfusgr 27025 Walkscwlks 27305 WWalksN cwwlksn 27531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-edg 26760 df-uhgr 26770 df-upgr 26794 df-uspgr 26862 df-usgr 26863 df-fusgr 27026 df-wlks 27308 df-wwlks 27535 df-wwlksn 27536 |
This theorem is referenced by: clwlknon2num 28074 numclwlk1lem2 28076 |
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