Theorem numclwwlk2 30529

Description: Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgrnumwlkg 30126, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-May-2022.)

Hypotheses

Ref	Expression
numclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
numclwwlk.q	⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
numclwwlk.h	⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})

Assertion

Ref	Expression
numclwwlk2	⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))

Distinct variable groups:   𝑛,𝐺,𝑣,𝑤   𝑛,𝑁,𝑣,𝑤   𝑛,𝑉,𝑣   𝑛,𝑋,𝑣,𝑤   𝑤,𝐾   𝑤,𝑉

Allowed substitution hints:   𝑄(𝑤,𝑣,𝑛)   𝐻(𝑤,𝑣,𝑛)   𝐾(𝑣,𝑛)

Proof of Theorem numclwwlk2

Step	Hyp	Ref	Expression
1		eluzelcn 12848	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℂ)
2		2cnd 12293	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘3) → 2 ∈ ℂ)
3	1, 2	npcand 11543	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → ((𝑁 − 2) + 2) = 𝑁)
4	3	eqcomd 2767	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 = ((𝑁 − 2) + 2))
5	4	3ad2ant3 1147	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → 𝑁 = ((𝑁 − 2) + 2))
6	5	adantl 485	. . . 4 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑁 = ((𝑁 − 2) + 2))
7	6	oveq2d 7408	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑋𝐻𝑁) = (𝑋𝐻((𝑁 − 2) + 2)))
8	7	fveq2d 6867	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐻𝑁)) = (♯‘(𝑋𝐻((𝑁 − 2) + 2))))
9		simplr 778	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝐺 ∈ FriendGraph )
10		simpr2 1208	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → 𝑋 ∈ 𝑉)
11		uz3m2nn 12892	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘3) → (𝑁 − 2) ∈ ℕ)
12	11	3ad2ant3 1147	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑁 − 2) ∈ ℕ)
13	12	adantl 485	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑁 − 2) ∈ ℕ)
14		numclwwlk.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
15		numclwwlk.q	. . . 4 ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})
16		numclwwlk.h	. . . 4 ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
17	14, 15, 16	numclwwlk2lem3 30528	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ) → (♯‘(𝑋𝑄(𝑁 − 2))) = (♯‘(𝑋𝐻((𝑁 − 2) + 2))))
18	9, 10, 13, 17	syl3anc 1389	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝑄(𝑁 − 2))) = (♯‘(𝑋𝐻((𝑁 − 2) + 2))))
19		simpl 486	. . . 4 ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) → 𝐺 RegUSGraph 𝐾)
20		simp1 1148	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → 𝑉 ∈ Fin)
21	19, 20	anim12i 622	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin))
22	11	anim2i 626	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ))
23	22	3adant1 1142	. . . 4 ⊢ ((𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) → (𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ))
24	23	adantl 485	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ))
25	14, 15	numclwwlkqhash 30523	. . 3 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ (𝑁 − 2) ∈ ℕ)) → (♯‘(𝑋𝑄(𝑁 − 2))) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))
26	21, 24, 25	syl2anc 593	. 2 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝑄(𝑁 − 2))) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))
27	8, 18, 26	3eqtr2d 2802	1 ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3))) → (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  {crab 3413   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   ∈ cmpo 7394  Fincfn 8923  0cc0 11070   + caddc 11073   − cmin 11411  ℕcn 12207  2c2 12269  3c3 12270  ℤ_≥cuz 12836  ↑cexp 14071  ♯chash 14340  lastSclsw 14572  Vtxcvtx 29143   RegUSGraph crusgr 29703   WWalksN cwwlksn 29972  ClWWalksNOncclwwlknon 30235   FriendGraph cfrgr 30406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-inf2 9593  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-er 8673  df-map 8805  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-sup 9385  df-oi 9455  df-dju 9856  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-xnn0 12552  df-z 12566  df-uz 12837  df-rp 12991  df-xadd 13112  df-fz 13510  df-fzo 13657  df-seq 14012  df-exp 14072  df-hash 14341  df-word 14524  df-lsw 14573  df-concat 14581  df-s1 14607  df-substr 14652  df-pfx 14682  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-clim 15498  df-sum 15697  df-vtx 29145  df-iedg 29146  df-edg 29195  df-uhgr 29205  df-ushgr 29206  df-upgr 29229  df-umgr 29230  df-uspgr 29297  df-usgr 29298  df-fusgr 29464  df-nbgr 29480  df-vtxdg 29613  df-rgr 29704  df-rusgr 29705  df-wwlks 29976  df-wwlksn 29977  df-wwlksnon 29978  df-clwwlk 30130  df-clwwlkn 30173  df-clwwlknon 30236  df-frgr 30407

This theorem is referenced by:  numclwwlk3  30533