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Theorem clwlknon2num 27554
 Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 27277, 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 ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘{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 26694 . . . . . 6 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
2 usgruspgr 26294 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺RegUSGraph𝐾𝐺 ∈ USPGraph)
433ad2ant2 1127 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
65eleq2i 2841 . . . . . 6 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
76biimpi 206 . . . . 5 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
873ad2ant3 1128 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9 2nn 11386 . . . . 5 2 ∈ ℕ
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → 2 ∈ ℕ)
11 clwwlknonclwlknonen 27548 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
124, 8, 10, 11syl3anc 1475 . . 3 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
131anim2i 595 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 453 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 26432 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 224 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
17163adant3 1125 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → 𝐺 ∈ FinUSGraph)
18 2nn0 11510 . . . . . . 7 2 ∈ ℕ0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → 2 ∈ ℕ0)
20 wlksnfi 27049 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
2117, 19, 20syl2anc 565 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
22 clwlkswks 26906 . . . . . . 7 (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
2322a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24 simp2l 1240 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) ∧ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st𝑤)) = 2)
2523, 24rabssrabd 3836 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2})
2621, 25ssfid 8338 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin)
275clwwlknonfin 27265 . . . . 5 (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28273ad2ant1 1126 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29 hashen 13338 . . . 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 565 . . 3 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → ((♯‘{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 247 . 2 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
327anim2i 595 . . . 4 ((𝐺RegUSGraph𝐾𝑋𝑉) → (𝐺RegUSGraph𝐾𝑋 ∈ (Vtx‘𝐺)))
33323adant1 1123 . . 3 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (𝐺RegUSGraph𝐾𝑋 ∈ (Vtx‘𝐺)))
34 clwwlknon2num 27277 . . 3 ((𝐺RegUSGraph𝐾𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3533, 34syl 17 . 2 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3631, 35eqtrd 2804 1 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  {crab 3064   ⊆ wss 3721   class class class wbr 4784  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313   ≈ cen 8105  Fincfn 8108  0cc0 10137  ℕcn 11221  2c2 11271  ℕ0cn0 11493  ♯chash 13320  Vtxcvtx 26094  USPGraphcuspgr 26264  USGraphcusgr 26265  FinUSGraphcfusgr 26430  RegUSGraphcrusgr 26686  Walkscwlks 26726  ClWalkscclwlks 26900  ClWWalksNOncclwwlknon 27256 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-ifp 1049  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-n0 11494  df-xnn0 11565  df-z 11579  df-uz 11888  df-rp 12035  df-xadd 12151  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-edg 26160  df-uhgr 26173  df-ushgr 26174  df-upgr 26197  df-umgr 26198  df-uspgr 26266  df-usgr 26267  df-fusgr 26431  df-nbgr 26447  df-vtxdg 26596  df-rgr 26687  df-rusgr 26688  df-wlks 26729  df-clwlks 26901  df-wwlks 26957  df-wwlksn 26958  df-clwwlk 27129  df-clwwlkn 27173  df-clwwlknon 27257 This theorem is referenced by:  numclwlk1lem1  27555
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