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Mirrors > Home > MPE Home > Th. List > hashclwwlkn0 | Structured version Visualization version GIF version |
Description: The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to ∼. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.) |
Ref | Expression |
---|---|
erclwwlkn.w | ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) |
erclwwlkn.r | ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} |
Ref | Expression |
---|---|
hashclwwlkn0 | ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(♯‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlkn.w | . . . 4 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) | |
2 | erclwwlkn.r | . . . 4 ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
3 | 1, 2 | erclwwlkn 28415 | . . 3 ⊢ ∼ Er 𝑊 |
4 | 3 | a1i 11 | . 2 ⊢ ((Vtx‘𝐺) ∈ Fin → ∼ Er 𝑊) |
5 | clwwlknfi 28388 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) | |
6 | 1, 5 | eqeltrid 2844 | . 2 ⊢ ((Vtx‘𝐺) ∈ Fin → 𝑊 ∈ Fin) |
7 | 4, 6 | qshash 15520 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ∃wrex 3066 {copab 5140 ‘cfv 6430 (class class class)co 7268 Er wer 8469 / cqs 8471 Fincfn 8707 0cc0 10855 ...cfz 13221 ♯chash 14025 cyclShift ccsh 14482 Σcsu 15378 Vtxcvtx 27347 ClWWalksN cclwwlkn 28367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-disj 5044 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-oadd 8285 df-er 8472 df-ec 8474 df-qs 8478 df-map 8591 df-pm 8592 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-sup 9162 df-inf 9163 df-oi 9230 df-dju 9643 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-n0 12217 df-xnn0 12289 df-z 12303 df-uz 12565 df-rp 12713 df-fz 13222 df-fzo 13365 df-fl 13493 df-mod 13571 df-seq 13703 df-exp 13764 df-hash 14026 df-word 14199 df-concat 14255 df-substr 14335 df-pfx 14365 df-csh 14483 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-sum 15379 df-clwwlk 28325 df-clwwlkn 28368 |
This theorem is referenced by: fusgrhashclwwlkn 28422 |
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