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Mirrors > Home > MPE Home > Th. List > qerclwwlknfi | Structured version Visualization version GIF version |
Description: The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation ∼ is finite. (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 |
---|---|
qerclwwlknfi | ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erclwwlkn.w | . . . 4 ⊢ 𝑊 = (𝑁 ClWWalksN 𝐺) | |
2 | clwwlknfi 28159 | . . . 4 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin) | |
3 | 1, 2 | eqeltrid 2844 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → 𝑊 ∈ Fin) |
4 | pwfi 8881 | . . 3 ⊢ (𝑊 ∈ Fin ↔ 𝒫 𝑊 ∈ Fin) | |
5 | 3, 4 | sylib 221 | . 2 ⊢ ((Vtx‘𝐺) ∈ Fin → 𝒫 𝑊 ∈ Fin) |
6 | erclwwlkn.r | . . . . 5 ⊢ ∼ = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
7 | 1, 6 | erclwwlkn 28186 | . . . 4 ⊢ ∼ Er 𝑊 |
8 | 7 | a1i 11 | . . 3 ⊢ ((Vtx‘𝐺) ∈ Fin → ∼ Er 𝑊) |
9 | 8 | qsss 8483 | . 2 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ⊆ 𝒫 𝑊) |
10 | 5, 9 | ssfid 8927 | 1 ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∃wrex 3065 𝒫 cpw 4529 {copab 5131 ‘cfv 6400 (class class class)co 7234 Er wer 8411 / cqs 8413 Fincfn 8649 0cc0 10758 ...cfz 13124 cyclShift ccsh 14385 Vtxcvtx 27118 ClWWalksN cclwwlkn 28138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 ax-pre-sup 10836 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-oadd 8229 df-er 8414 df-ec 8416 df-qs 8420 df-map 8533 df-pm 8534 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-sup 9087 df-inf 9088 df-dju 9546 df-card 9584 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-div 11519 df-nn 11860 df-2 11922 df-n0 12120 df-xnn0 12192 df-z 12206 df-uz 12468 df-rp 12616 df-fz 13125 df-fzo 13268 df-fl 13396 df-mod 13474 df-seq 13606 df-exp 13667 df-hash 13929 df-word 14102 df-concat 14158 df-substr 14238 df-pfx 14268 df-csh 14386 df-clwwlk 28096 df-clwwlkn 28139 |
This theorem is referenced by: fusgrhashclwwlkn 28193 clwwlkndivn 28194 |
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