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Mirrors > Home > MPE Home > Th. List > numclwwlk8 | Structured version Visualization version GIF version |
Description: The size of the set of closed walks of length 𝑃, 𝑃 prime, is divisible by 𝑃. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p", see also clwlksndivn 28025. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) (Proof shortened by AV, 2-Mar-2022.) |
Ref | Expression |
---|---|
numclwwlk8 | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmnn 16117 | . 2 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
2 | clwwlkndivn 28019 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → 𝑃 ∥ (♯‘(𝑃 ClWWalksN 𝐺))) | |
3 | dvdsmod0 15707 | . 2 ⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ∥ (♯‘(𝑃 ClWWalksN 𝐺))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) | |
4 | 1, 2, 3 | syl2an2 686 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ‘cfv 6339 (class class class)co 7172 0cc0 10617 ℕcn 11718 mod cmo 13330 ♯chash 13784 ∥ cdvds 15701 ℙcprime 16114 FinUSGraphcfusgr 27260 ClWWalksN cclwwlkn 27963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-2o 8134 df-oadd 8137 df-er 8322 df-ec 8324 df-qs 8328 df-map 8441 df-pm 8442 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-sup 8981 df-inf 8982 df-oi 9049 df-dju 9405 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-n0 11979 df-xnn0 12051 df-z 12065 df-uz 12327 df-rp 12475 df-ico 12829 df-fz 12984 df-fzo 13127 df-fl 13255 df-mod 13331 df-seq 13463 df-exp 13524 df-hash 13785 df-word 13958 df-lsw 14006 df-concat 14014 df-substr 14094 df-pfx 14124 df-reps 14222 df-csh 14242 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-sum 15138 df-dvds 15702 df-gcd 15940 df-prm 16115 df-phi 16205 df-edg 26995 df-umgr 27030 df-usgr 27098 df-fusgr 27261 df-clwwlk 27921 df-clwwlkn 27964 |
This theorem is referenced by: frgrreggt1 28332 |
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