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Theorem numclwlk1lem1 27555
Description: Lemma 1 for numclwlk1 27557 (Statement 9 in [Huneke] p. 2 for n=2): "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)". (Contributed by AV, 23-May-2022.)
Hypotheses
Ref Expression
numclwlk1.v 𝑉 = (Vtx‘𝐺)
numclwlk1.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
numclwlk1.f 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}
Assertion
Ref Expression
numclwlk1lem1 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))
Distinct variable groups:   𝑤,𝐺   𝑤,𝐾   𝑤,𝑁   𝑤,𝑉   𝑤,𝑋
Allowed substitution hints:   𝐶(𝑤)   𝐹(𝑤)

Proof of Theorem numclwlk1lem1
StepHypRef Expression
1 3anass 1079 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
2 anidm 546 . . . . . . . 8 ((((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((2nd𝑤)‘0) = 𝑋)
32anbi2i 601 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
41, 3bitri 264 . . . . . 6 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
54rabbii 3334 . . . . 5 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}
65fveq2i 6335 . . . 4 (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)})
7 simpl 468 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝑉 ∈ Fin)
8 simpr 471 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺RegUSGraph𝐾)
9 simpl 468 . . . . 5 ((𝑋𝑉𝑁 = 2) → 𝑋𝑉)
10 numclwlk1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1110clwlknon2num 27554 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
127, 8, 9, 11syl2an3an 1531 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
136, 12syl5eq 2816 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
14 rusgrusgr 26694 . . . . . . . . 9 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
1514anim2i 595 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1615ancomd 453 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1710isfusgr 26432 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1816, 17sylibr 224 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
19 ne0i 4067 . . . . . . 7 (𝑋𝑉𝑉 ≠ ∅)
2019adantr 466 . . . . . 6 ((𝑋𝑉𝑁 = 2) → 𝑉 ≠ ∅)
2110frusgrnn0 26701 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
2218, 8, 20, 21syl2an3an 1531 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℕ0)
2322nn0red 11553 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℝ)
24 ax-1rid 10207 . . . 4 (𝐾 ∈ ℝ → (𝐾 · 1) = 𝐾)
2523, 24syl 17 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = 𝐾)
2610wlkl0 27553 . . . . . . 7 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2726ad2antrl 699 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2827fveq2d 6336 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29 opex 5060 . . . . . 6 ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30 hashsng 13360 . . . . . 6 (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
3129, 30ax-mp 5 . . . . 5 (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
3228, 31syl6req 2821 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
3332oveq2d 6808 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
3413, 25, 333eqtr2d 2810 . 2 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
35 numclwlk1.c . . . . . 6 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}
36 eqeq2 2781 . . . . . . . 8 (𝑁 = 2 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑤)) = 2))
37 oveq1 6799 . . . . . . . . . . 11 (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38 2cn 11292 . . . . . . . . . . . 12 2 ∈ ℂ
3938subidi 10553 . . . . . . . . . . 11 (2 − 2) = 0
4037, 39syl6eq 2820 . . . . . . . . . 10 (𝑁 = 2 → (𝑁 − 2) = 0)
4140fveq2d 6336 . . . . . . . . 9 (𝑁 = 2 → ((2nd𝑤)‘(𝑁 − 2)) = ((2nd𝑤)‘0))
4241eqeq1d 2772 . . . . . . . 8 (𝑁 = 2 → (((2nd𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑤)‘0) = 𝑋))
4336, 423anbi13d 1548 . . . . . . 7 (𝑁 = 2 → (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
4443rabbidv 3338 . . . . . 6 (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)})
4535, 44syl5eq 2816 . . . . 5 (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)})
4645fveq2d 6336 . . . 4 (𝑁 = 2 → (♯‘𝐶) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}))
47 numclwlk1.f . . . . . . 7 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}
4840eqeq2d 2780 . . . . . . . . 9 (𝑁 = 2 → ((♯‘(1st𝑤)) = (𝑁 − 2) ↔ (♯‘(1st𝑤)) = 0))
4948anbi1d 607 . . . . . . . 8 (𝑁 = 2 → (((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)))
5049rabbidv 3338 . . . . . . 7 (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5147, 50syl5eq 2816 . . . . . 6 (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5251fveq2d 6336 . . . . 5 (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
5352oveq2d 6808 . . . 4 (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
5446, 53eqeq12d 2785 . . 3 (𝑁 = 2 → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))))
5554ad2antll 700 . 2 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → ((♯‘𝐶) = (𝐾 · (♯‘𝐹)) ↔ (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))))
5634, 55mpbird 247 1 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  {crab 3064  Vcvv 3349  c0 4061  {csn 4314  cop 4320   class class class wbr 4784  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Fincfn 8108  cr 10136  0cc0 10137  1c1 10138   · cmul 10142  cmin 10467  2c2 11271  0cn0 11493  chash 13320  Vtxcvtx 26094  USGraphcusgr 26265  FinUSGraphcfusgr 26430  RegUSGraphcrusgr 26686  ClWalkscclwlks 26900
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-vtx 26096  df-iedg 26097  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:  numclwlk1  27557
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