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Theorem hashclwwlkn 26363

Description: The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for ∼ over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)

**Hypotheses**
Ref	Expression
erclwwlkn.w	⊢ 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
erclwwlkn.r	⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

**Assertion**
Ref	Expression
hashclwwlkn	⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (#‘𝑊) = ((#‘(𝑊 / ∼ )) · 𝑁))

Distinct variable groups: 𝑡,𝐸,𝑢 𝑡,𝑁,𝑢 𝑛,𝑉,𝑡,𝑢 𝑡,𝑊,𝑢 𝑛,𝑁 𝑛,𝑊 𝑛,𝐸 Allowed substitution hints: ∼ (𝑢,𝑡,𝑛)

Proof of Theorem hashclwwlkn Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑉 ∈ Fin → 𝑉 ∈ Fin)
2		usgrav 25867	. . . 4 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3	2	simprd 478	. . 3 ⊢ (𝑉 USGrph 𝐸 → 𝐸 ∈ V)
4		prmnn 15226	. . . 4 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ)
5	4	nnnn0d 11228	. . 3 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℕ₀)
6		erclwwlkn.w	. . . 4 ⊢ 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
7		erclwwlkn.r	. . . 4 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
8	6, 7	hashclwwlkn0 26358	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → (#‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(#‘𝑥))
9	1, 3, 5, 8	syl3an 1360	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (#‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(#‘𝑥))
10	6, 7	usghashecclwwlk 26362	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (𝑥 ∈ (𝑊 / ∼ ) → (#‘𝑥) = 𝑁))
11	10	3adant1 1072	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (𝑥 ∈ (𝑊 / ∼ ) → (#‘𝑥) = 𝑁))
12	11	imp 444	. . 3 ⊢ (((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) ∧ 𝑥 ∈ (𝑊 / ∼ )) → (#‘𝑥) = 𝑁)
13	12	sumeq2dv 14281	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → Σ𝑥 ∈ (𝑊 / ∼ )(#‘𝑥) = Σ𝑥 ∈ (𝑊 / ∼ )𝑁)
14	6, 7	qerclwwlknfi 26357	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ₀) → (𝑊 / ∼ ) ∈ Fin)
15	1, 3, 5, 14	syl3an 1360	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (𝑊 / ∼ ) ∈ Fin)
16	4	nncnd 10913	. . . 4 ⊢ (𝑁 ∈ ℙ → 𝑁 ∈ ℂ)
17	16	3ad2ant3 1077	. . 3 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → 𝑁 ∈ ℂ)
18		fsumconst 14364	. . 3 ⊢ (((𝑊 / ∼ ) ∈ Fin ∧ 𝑁 ∈ ℂ) → Σ𝑥 ∈ (𝑊 / ∼ )𝑁 = ((#‘(𝑊 / ∼ )) · 𝑁))
19	15, 17, 18	syl2anc 691	. 2 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → Σ𝑥 ∈ (𝑊 / ∼ )𝑁 = ((#‘(𝑊 / ∼ )) · 𝑁))
20	9, 13, 19	3eqtrd 2648	1 ⊢ ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℙ) → (#‘𝑊) = ((#‘(𝑊 / ∼ )) · 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 {copab 4642 ‘cfv 5804 (class class class)co 6549 / cqs 7628 Fincfn 7841 ℂcc 9813 0cc0 9815 · cmul 9820 ℕ₀cn0 11169 ...cfz 12197 #chash 12979 cyclShift ccsh 13385 Σcsu 14264 ℙcprime 15223 USGrph cusg 25859 ClWWalksN cclwwlkn 26277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-substr 13158 df-reps 13161 df-csh 13386 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-phi 15309 df-usgra 25862 df-clwwlk 26279 df-clwwlkn 26280

This theorem is referenced by: clwwlkndivn 26364

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