![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clwwlknfi | Structured version Visualization version GIF version |
Description: If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) |
Ref | Expression |
---|---|
clwwlknfi | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnnn0 11588 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
2 | clwwlkn 27330 | . . . . 5 ⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} | |
3 | wrdnfi 13568 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) | |
4 | clwwlksswrd 27280 | . . . . . . 7 ⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | |
5 | rabss2 3881 | . . . . . . 7 ⊢ ((ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) | |
6 | 4, 5 | mp1i 13 | . . . . . 6 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ⊆ {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 𝑁}) |
7 | 3, 6 | ssfid 8425 | . . . . 5 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁} ∈ Fin) |
8 | 2, 7 | syl5eqel 2882 | . . . 4 ⊢ (((Vtx‘𝐺) ∈ Fin ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
9 | 8 | expcom 403 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
10 | 1, 9 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
11 | df-nel 3075 | . . . . . . 7 ⊢ (𝑁 ∉ ℕ ↔ ¬ 𝑁 ∈ ℕ) | |
12 | 11 | biimpri 220 | . . . . . 6 ⊢ (¬ 𝑁 ∈ ℕ → 𝑁 ∉ ℕ) |
13 | 12 | olcd 901 | . . . . 5 ⊢ (¬ 𝑁 ∈ ℕ → (𝐺 ∉ V ∨ 𝑁 ∉ ℕ)) |
14 | clwwlkneq0 27335 | . . . . 5 ⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = ∅) |
16 | 0fin 8430 | . . . 4 ⊢ ∅ ∈ Fin | |
17 | 15, 16 | syl6eqel 2886 | . . 3 ⊢ (¬ 𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
18 | 17 | a1d 25 | . 2 ⊢ (¬ 𝑁 ∈ ℕ → ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)) |
19 | 10, 18 | pm2.61i 177 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ∉ wnel 3074 {crab 3093 Vcvv 3385 ⊆ wss 3769 ∅c0 4115 ‘cfv 6101 (class class class)co 6878 Fincfn 8195 ℕcn 11312 ℕ0cn0 11580 ♯chash 13370 Word cword 13534 Vtxcvtx 26231 ClWWalkscclwwlk 27274 ClWWalksN cclwwlkn 27326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-2o 7800 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-cda 9278 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-xnn0 11653 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-seq 13056 df-exp 13115 df-hash 13371 df-word 13535 df-clwwlk 27275 df-clwwlkn 27328 |
This theorem is referenced by: qerclwwlknfi 27391 hashclwwlkn0 27392 clwwlknonfin 27432 numclwwlk3lemOLD 27766 |
Copyright terms: Public domain | W3C validator |