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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 26972. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) |
Ref | Expression |
---|---|
numclwwlk8 | ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwwlksndivn 26957 | . 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → 𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺))) | |
2 | dvdszrcl 14988 | . . 3 ⊢ (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) → (𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ)) | |
3 | prmnn 15388 | . . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
4 | 3 | adantl 482 | . . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℕ) |
5 | simpr 477 | . . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) → (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) | |
6 | dvdsval3 14987 | . . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) → (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) ↔ ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)) | |
7 | 4, 5, 6 | syl2anr 495 | . . . . . 6 ⊢ (((𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ)) → (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) ↔ ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)) |
8 | 7 | biimpd 219 | . . . . 5 ⊢ (((𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) ∧ (𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ)) → (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)) |
9 | 8 | ex 450 | . . . 4 ⊢ ((𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) → ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0))) |
10 | 9 | com23 86 | . . 3 ⊢ ((𝑃 ∈ ℤ ∧ (#‘(𝑃 ClWWalksN 𝐺)) ∈ ℤ) → (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) → ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0))) |
11 | 2, 10 | mpcom 38 | . 2 ⊢ (𝑃 ∥ (#‘(𝑃 ClWWalksN 𝐺)) → ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)) |
12 | 1, 11 | mpcom 38 | 1 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕcn 11020 ℤcz 11377 mod cmo 12668 #chash 13117 ∥ cdvds 14983 ℙcprime 15385 FinUSGraph cfusgr 26208 ClWWalksN cclwwlksn 26876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-substr 13303 df-reps 13306 df-csh 13535 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-phi 15471 df-edg 25940 df-umgr 25978 df-usgr 26046 df-fusgr 26209 df-clwwlks 26877 df-clwwlksn 26878 |
This theorem is referenced by: frgrreggt1 27251 |
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