MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwlk1lem1 Structured version   Visualization version   GIF version

Theorem numclwlk1lem1 27768
Description: Lemma 1 for numclwlk1 27770 (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 1120 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
2 anidm 560 . . . . . . . 8 ((((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((2nd𝑤)‘0) = 𝑋)
32anbi2i 616 . . . . . . 7 (((♯‘(1st𝑤)) = 2 ∧ (((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
41, 3bitri 267 . . . . . 6 (((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋))
54rabbii 3398 . . . . 5 {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}
65fveq2i 6440 . . . 4 (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)})
7 simpl 476 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝑉 ∈ Fin)
8 simpr 479 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺RegUSGraph𝐾)
9 simpl 476 . . . . 5 ((𝑋𝑉𝑁 = 2) → 𝑋𝑉)
10 numclwlk1.v . . . . . 6 𝑉 = (Vtx‘𝐺)
1110clwlknon2num 27767 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
127, 8, 9, 11syl2an3an 1549 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
136, 12syl5eq 2873 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
14 rusgrusgr 26869 . . . . . . . . 9 (𝐺RegUSGraph𝐾𝐺 ∈ USGraph)
1514anim2i 610 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1615ancomd 455 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1710isfusgr 26622 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1816, 17sylibr 226 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) → 𝐺 ∈ FinUSGraph)
19 ne0i 4152 . . . . . . 7 (𝑋𝑉𝑉 ≠ ∅)
2019adantr 474 . . . . . 6 ((𝑋𝑉𝑁 = 2) → 𝑉 ≠ ∅)
2110frusgrnn0 26876 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺RegUSGraph𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
2218, 8, 20, 21syl2an3an 1549 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℕ0)
2322nn0red 11686 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 𝐾 ∈ ℝ)
24 ax-1rid 10329 . . . 4 (𝐾 ∈ ℝ → (𝐾 · 1) = 𝐾)
2523, 24syl 17 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = 𝐾)
2610wlkl0 27766 . . . . . . 7 (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2726ad2antrl 719 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})
2827fveq2d 6441 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}))
29 opex 5155 . . . . . 6 ⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V
30 hashsng 13456 . . . . . 6 (⟨∅, {⟨0, 𝑋⟩}⟩ ∈ V → (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1)
3129, 30ax-mp 5 . . . . 5 (♯‘{⟨∅, {⟨0, 𝑋⟩}⟩}) = 1
3228, 31syl6req 2878 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → 1 = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
3332oveq2d 6926 . . 3 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (𝐾 · 1) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
3413, 25, 333eqtr2d 2867 . 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 2836 . . . . . . . 8 (𝑁 = 2 → ((♯‘(1st𝑤)) = 𝑁 ↔ (♯‘(1st𝑤)) = 2))
37 oveq1 6917 . . . . . . . . . 10 (𝑁 = 2 → (𝑁 − 2) = (2 − 2))
38 2cn 11433 . . . . . . . . . . 11 2 ∈ ℂ
3938subidi 10680 . . . . . . . . . 10 (2 − 2) = 0
4037, 39syl6eq 2877 . . . . . . . . 9 (𝑁 = 2 → (𝑁 − 2) = 0)
4140fveqeq2d 6445 . . . . . . . 8 (𝑁 = 2 → (((2nd𝑤)‘(𝑁 − 2)) = 𝑋 ↔ ((2nd𝑤)‘0) = 𝑋))
4236, 413anbi13d 1566 . . . . . . 7 (𝑁 = 2 → (((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋) ↔ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)))
4342rabbidv 3402 . . . . . 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 2873 . . . . 5 (𝑁 = 2 → 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘0) = 𝑋)})
4544fveq2d 6441 . . . 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 2835 . . . . . . . . 9 (𝑁 = 2 → ((♯‘(1st𝑤)) = (𝑁 − 2) ↔ (♯‘(1st𝑤)) = 0))
4847anbi1d 623 . . . . . . . 8 (𝑁 = 2 → (((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋) ↔ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)))
4948rabbidv 3402 . . . . . . 7 (𝑁 = 2 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)} = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5046, 49syl5eq 2873 . . . . . 6 (𝑁 = 2 → 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})
5150fveq2d 6441 . . . . 5 (𝑁 = 2 → (♯‘𝐹) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)}))
5251oveq2d 6926 . . . 4 (𝑁 = 2 → (𝐾 · (♯‘𝐹)) = (𝐾 · (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)})))
5345, 52eqeq12d 2840 . . 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 720 . 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)}))))
5534, 54mpbird 249 1 (((𝑉 ∈ Fin ∧ 𝐺RegUSGraph𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wne 2999  {crab 3121  Vcvv 3414  c0 4146  {csn 4399  cop 4405   class class class wbr 4875  cfv 6127  (class class class)co 6910  1st c1st 7431  2nd c2nd 7432  Fincfn 8228  cr 10258  0cc0 10259  1c1 10260   · cmul 10264  cmin 10592  2c2 11413  0cn0 11625  chash 13417  Vtxcvtx 26301  USGraphcusgr 26455  FinUSGraphcfusgr 26620  RegUSGraphcrusgr 26861  ClWalkscclwlks 27079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-rp 12120  df-xadd 12240  df-fz 12627  df-fzo 12768  df-seq 13103  df-exp 13162  df-hash 13418  df-word 13582  df-lsw 13630  df-concat 13638  df-s1 13663  df-substr 13708  df-pfx 13757  df-vtx 26303  df-iedg 26304  df-edg 26353  df-uhgr 26363  df-ushgr 26364  df-upgr 26387  df-umgr 26388  df-uspgr 26456  df-usgr 26457  df-fusgr 26621  df-nbgr 26637  df-vtxdg 26771  df-rgr 26862  df-rusgr 26863  df-wlks 26904  df-clwlks 27080  df-wwlks 27136  df-wwlksn 27137  df-clwwlk 27318  df-clwwlkn 27370  df-clwwlknon 27459
This theorem is referenced by:  numclwlk1  27770
  Copyright terms: Public domain W3C validator