![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clwwlkndivn | Structured version Visualization version GIF version |
Description: The size of the set of closed walks (defined as words) of length 𝑁 is divisible by 𝑁 if 𝑁 is a prime number. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 2-May-2021.) |
Ref | Expression |
---|---|
clwwlkndivn | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (♯‘(𝑁 ClWWalksN 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | fusgrvtxfi 29145 | . . . . . 6 ⊢ (𝐺 ∈ FinUSGraph → (Vtx‘𝐺) ∈ Fin) |
3 | 2 | adantr 480 | . . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (Vtx‘𝐺) ∈ Fin) |
4 | eqid 2728 | . . . . . 6 ⊢ (𝑁 ClWWalksN 𝐺) = (𝑁 ClWWalksN 𝐺) | |
5 | eqid 2728 | . . . . . 6 ⊢ {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} = {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))} | |
6 | 4, 5 | qerclwwlknfi 29896 | . . . . 5 ⊢ ((Vtx‘𝐺) ∈ Fin → ((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}) ∈ Fin) |
7 | hashcl 14348 | . . . . 5 ⊢ (((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}) ∈ Fin → (♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) ∈ ℕ0) | |
8 | 3, 6, 7 | 3syl 18 | . . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) ∈ ℕ0) |
9 | 8 | nn0zd 12615 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) ∈ ℤ) |
10 | prmz 16646 | . . . 4 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℤ) | |
11 | 10 | adantl 481 | . . 3 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℤ) |
12 | dvdsmul2 16256 | . . 3 ⊢ (((♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ ((♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) · 𝑁)) | |
13 | 9, 11, 12 | syl2anc 583 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ ((♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) · 𝑁)) |
14 | 4, 5 | fusgrhashclwwlkn 29902 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘(𝑁 ClWWalksN 𝐺)) = ((♯‘((𝑁 ClWWalksN 𝐺) / {〈𝑡, 𝑢〉 ∣ (𝑡 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))})) · 𝑁)) |
15 | 13, 14 | breqtrrd 5176 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (♯‘(𝑁 ClWWalksN 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∃wrex 3067 class class class wbr 5148 {copab 5210 ‘cfv 6548 (class class class)co 7420 / cqs 8724 Fincfn 8964 0cc0 11139 · cmul 11144 ℕ0cn0 12503 ℤcz 12589 ...cfz 13517 ♯chash 14322 cyclShift ccsh 14771 ∥ cdvds 16231 ℙcprime 16642 Vtxcvtx 28822 FinUSGraphcfusgr 29142 ClWWalksN cclwwlkn 29847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-rp 13008 df-ico 13363 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-hash 14323 df-word 14498 df-lsw 14546 df-concat 14554 df-substr 14624 df-pfx 14654 df-reps 14752 df-csh 14772 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-dvds 16232 df-gcd 16470 df-prm 16643 df-phi 16735 df-edg 28874 df-umgr 28909 df-usgr 28977 df-fusgr 29143 df-clwwlk 29805 df-clwwlkn 29848 |
This theorem is referenced by: clwlksndivn 29909 numclwwlk8 30215 |
Copyright terms: Public domain | W3C validator |