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Theorem numclwlk1lem1 28075
Description: Lemma 1 for numclwlk1 28077 (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 ∧ 𝐺 RegUSGraph 𝐾) ∧ (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 1087 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
2 anidm 565 . . . . . . . 8 ((((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((2nd𝑤)‘0) = 𝑋)
32anbi2i 622 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
41, 3bitri 276 . . . . . 6 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
54rabbii 3471 . . . . 5 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}
65fveq2i 6666 . . . 4 (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)})
7 simpl 483 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝑉 ∈ Fin)
8 simpr 485 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 RegUSGraph 𝐾)
9 simpl 483 . . . . 5 ((𝑋𝑉𝑁 = 2) → 𝑋𝑉)
10 numclwlk1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1110clwlknon2num 28074 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
127, 8, 9, 11syl2an3an 1414 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
136, 12syl5eq 2865 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
14 rusgrusgr 27273 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
1514anim2i 616 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1615ancomd 462 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1710isfusgr 27027 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1816, 17sylibr 235 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
19 ne0i 4297 . . . . . . 7 (𝑋𝑉𝑉 ≠ ∅)
2019adantr 481 . . . . . 6 ((𝑋𝑉𝑁 = 2) → 𝑉 ≠ ∅)
2110frusgrnn0 27280 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
2218, 8, 20, 21syl2an3an 1414 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℕ0)
2322nn0red 11944 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℝ)
24 ax-1rid 10595 . . . 4 (𝐾 ∈ ℝ → (𝐾 · 1) = 𝐾)
2523, 24syl 17 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = 𝐾)
2610wlkl0 28073 . . . . . . 7 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2726ad2antrl 724 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2827fveq2d 6667 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29 opex 5347 . . . . . 6 ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30 hashsng 13718 . . . . . 6 (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
3129, 30ax-mp 5 . . . . 5 (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
3228, 31syl6req 2870 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
3332oveq2d 7161 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
3413, 25, 333eqtr2d 2859 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (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 2830 . . . . . . . 8 (𝑁 = 2 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑤)) = 2))
37 oveq1 7152 . . . . . . . . . 10 (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38 2cn 11700 . . . . . . . . . . 11 2 ∈ ℂ
3938subidi 10945 . . . . . . . . . 10 (2 − 2) = 0
4037, 39syl6eq 2869 . . . . . . . . 9 (𝑁 = 2 → (𝑁 − 2) = 0)
4140fveqeq2d 6671 . . . . . . . 8 (𝑁 = 2 → (((2nd𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑤)‘0) = 𝑋))
4236, 413anbi13d 1429 . . . . . . 7 (𝑁 = 2 → (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
4342rabbidv 3478 . . . . . 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)})
4435, 43syl5eq 2865 . . . . 5 (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)})
4544fveq2d 6667 . . . 4 (𝑁 = 2 → (♯‘𝐶) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}))
46 numclwlk1.f . . . . . . 7 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}
4740eqeq2d 2829 . . . . . . . . 9 (𝑁 = 2 → ((♯‘(1st𝑤)) = (𝑁 − 2) ↔ (♯‘(1st𝑤)) = 0))
4847anbi1d 629 . . . . . . . 8 (𝑁 = 2 → (((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)))
4948rabbidv 3478 . . . . . . 7 (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5046, 49syl5eq 2865 . . . . . 6 (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5150fveq2d 6667 . . . . 5 (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
5251oveq2d 7161 . . . 4 (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
5345, 52eqeq12d 2834 . . 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)}))))
5453ad2antll 725 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (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)}))))
5534, 54mpbird 258 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  {crab 3139  Vcvv 3492  c0 4288  {csn 4557  cop 4563   class class class wbr 5057  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Fincfn 8497  cr 10524  0cc0 10525  1c1 10526   · cmul 10530  cmin 10858  2c2 11680  0cn0 11885  chash 13678  Vtxcvtx 26708  USGraphcusgr 26861  FinUSGraphcfusgr 27025   RegUSGraph crusgr 27265  ClWalkscclwlks 27478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-vtx 26710  df-iedg 26711  df-edg 26760  df-uhgr 26770  df-ushgr 26771  df-upgr 26794  df-umgr 26795  df-uspgr 26862  df-usgr 26863  df-fusgr 27026  df-nbgr 27042  df-vtxdg 27175  df-rgr 27266  df-rusgr 27267  df-wlks 27308  df-clwlks 27479  df-wwlks 27535  df-wwlksn 27536  df-clwwlk 27687  df-clwwlkn 27730  df-clwwlknon 27794
This theorem is referenced by:  numclwlk1  28077
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