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Theorem clwlknon2num 28156
 Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 27893, 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 27357 . . . . . 6 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
2 usgruspgr 26974 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph)
433ad2ant2 1131 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
65eleq2i 2907 . . . . . 6 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
76biimpi 219 . . . . 5 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
873ad2ant3 1132 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9 2nn 11707 . . . . 5 2 ∈ ℕ
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ)
11 clwwlknonclwlknonen 28151 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
124, 8, 10, 11syl3anc 1368 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
131anim2i 619 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 465 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 27111 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 237 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
17163adant3 1129 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ FinUSGraph)
18 2nn0 11911 . . . . . . 7 2 ∈ ℕ0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ0)
20 wlksnfi 27696 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
2117, 19, 20syl2anc 587 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
22 clwlkswks 27568 . . . . . . 7 (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
2322a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24 simp2l 1196 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) ∧ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st𝑤)) = 2)
2523, 24rabssrabd 4044 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2})
2621, 25ssfid 8738 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin)
275clwwlknonfin 27882 . . . . 5 (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28273ad2ant1 1130 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29 hashen 13712 . . . 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 587 . . 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 260 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
327anim2i 619 . . . 4 ((𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
33323adant1 1127 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
34 clwwlknon2num 27893 . . 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 2859 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  {crab 3137   ⊆ wss 3919   class class class wbr 5052  ‘cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683   ≈ cen 8502  Fincfn 8505  0cc0 10535  ℕcn 11634  2c2 11689  ℕ0cn0 11894  ♯chash 13695  Vtxcvtx 26792  USPGraphcuspgr 26944  USGraphcusgr 26945  FinUSGraphcfusgr 27109   RegUSGraph crusgr 27349  Walkscwlks 27389  ClWalkscclwlks 27562  ClWWalksNOncclwwlknon 27875 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-rp 12387  df-xadd 12505  df-fz 12895  df-fzo 13038  df-seq 13374  df-exp 13435  df-hash 13696  df-word 13867  df-lsw 13915  df-concat 13923  df-s1 13950  df-substr 14003  df-pfx 14033  df-edg 26844  df-uhgr 26854  df-ushgr 26855  df-upgr 26878  df-umgr 26879  df-uspgr 26946  df-usgr 26947  df-fusgr 27110  df-nbgr 27126  df-vtxdg 27259  df-rgr 27350  df-rusgr 27351  df-wlks 27392  df-clwlks 27563  df-wwlks 27619  df-wwlksn 27620  df-clwwlk 27770  df-clwwlkn 27813  df-clwwlknon 27876 This theorem is referenced by:  numclwlk1lem1  28157
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