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Theorem clwlknon2num 29021
Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 28758, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
Hypothesis
Ref Expression
clwlknon2num.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
clwlknon2num ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋   𝑤,𝐾

Proof of Theorem clwlknon2num
StepHypRef Expression
1 rusgrusgr 28221 . . . . . 6 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
2 usgruspgr 27838 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph)
433ad2ant2 1133 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
65eleq2i 2828 . . . . . 6 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
76biimpi 215 . . . . 5 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
873ad2ant3 1134 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9 2nn 12148 . . . . 5 2 ∈ ℕ
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ)
11 clwwlknonclwlknonen 29016 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
124, 8, 10, 11syl3anc 1370 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
131anim2i 617 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 462 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 27975 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 233 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
17163adant3 1131 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ FinUSGraph)
18 2nn0 12352 . . . . . . 7 2 ∈ ℕ0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ0)
20 wlksnfi 28561 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
2117, 19, 20syl2anc 584 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
22 clwlkswks 28433 . . . . . . 7 (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
2322a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24 simp2l 1198 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) ∧ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st𝑤)) = 2)
2523, 24rabssrabd 4029 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2})
2621, 25ssfid 9133 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin)
275clwwlknonfin 28747 . . . . 5 (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28273ad2ant1 1132 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29 hashen 14163 . . . 4 (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin ∧ (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
3026, 28, 29syl2anc 584 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
3112, 30mpbird 256 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
327anim2i 617 . . . 4 ((𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
33323adant1 1129 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
34 clwwlknon2num 28758 . . 3 ((𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3533, 34syl 17 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3631, 35eqtrd 2776 1 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  {crab 3403  wss 3898   class class class wbr 5093  cfv 6480  (class class class)co 7338  1st c1st 7898  2nd c2nd 7899  cen 8802  Fincfn 8805  0cc0 10973  cn 12075  2c2 12130  0cn0 12335  chash 14146  Vtxcvtx 27656  USPGraphcuspgr 27808  USGraphcusgr 27809  FinUSGraphcfusgr 27973   RegUSGraph crusgr 28213  Walkscwlks 28253  ClWalkscclwlks 28427  ClWWalksNOncclwwlknon 28740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-om 7782  df-1st 7900  df-2nd 7901  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-2o 8369  df-oadd 8372  df-er 8570  df-map 8689  df-pm 8690  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-dju 9759  df-card 9797  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-nn 12076  df-2 12138  df-n0 12336  df-xnn0 12408  df-z 12422  df-uz 12685  df-rp 12833  df-xadd 12951  df-fz 13342  df-fzo 13485  df-seq 13824  df-exp 13885  df-hash 14147  df-word 14319  df-lsw 14367  df-concat 14375  df-s1 14401  df-substr 14453  df-pfx 14483  df-edg 27708  df-uhgr 27718  df-ushgr 27719  df-upgr 27742  df-umgr 27743  df-uspgr 27810  df-usgr 27811  df-fusgr 27974  df-nbgr 27990  df-vtxdg 28123  df-rgr 28214  df-rusgr 28215  df-wlks 28256  df-clwlks 28428  df-wwlks 28484  df-wwlksn 28485  df-clwwlk 28635  df-clwwlkn 28678  df-clwwlknon 28741
This theorem is referenced by:  numclwlk1lem1  29022
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