MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlk1 Structured version   Visualization version   GIF version

Theorem numclwwlk1 29603
Description: Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝐺 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but not for n=2, since 𝐹 = βˆ…, but (𝑋𝐢2), the set of closed walks with length 2 on 𝑋, see 2clwwlk2 29590, needs not be βˆ… in this case. This is because of the special definition of 𝐹 and the usage of words to represent (closed) walks, and does not contradict Huneke's statement, which would read "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty closed walks on v, see numclwlk1lem1 29611. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 29613. This case, however, is not required to prove the friendship theorem. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.) (Proof shortened by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
extwwlkfab.v 𝑉 = (Vtxβ€˜πΊ)
extwwlkfab.c 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
extwwlkfab.f 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
Assertion
Ref Expression
numclwwlk1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐢𝑁)) = (𝐾 Β· (β™―β€˜πΉ)))
Distinct variable groups:   𝑛,𝐺,𝑣,𝑀   𝑛,𝑁,𝑣,𝑀   𝑛,𝑉,𝑣,𝑀   𝑛,𝑋,𝑣,𝑀   𝑀,𝐹
Allowed substitution hints:   𝐢(𝑀,𝑣,𝑛)   𝐹(𝑣,𝑛)   𝐾(𝑀,𝑣,𝑛)

Proof of Theorem numclwwlk1
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 rusgrusgr 28810 . . . . 5 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
21ad2antlr 725 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐺 ∈ USGraph)
3 simprl 769 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑋 ∈ 𝑉)
4 simprr 771 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑁 ∈ (β„€β‰₯β€˜3))
5 extwwlkfab.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
6 extwwlkfab.c . . . . 5 𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})
7 extwwlkfab.f . . . . 5 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))
85, 6, 7numclwwlk1lem2 29602 . . . 4 ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) β‰ˆ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
92, 3, 4, 8syl3anc 1371 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝑋𝐢𝑁) β‰ˆ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
10 hasheni 14304 . . 3 ((𝑋𝐢𝑁) β‰ˆ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)) β†’ (β™―β€˜(𝑋𝐢𝑁)) = (β™―β€˜(𝐹 Γ— (𝐺 NeighbVtx 𝑋))))
119, 10syl 17 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐢𝑁)) = (β™―β€˜(𝐹 Γ— (𝐺 NeighbVtx 𝑋))))
12 eqid 2732 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1312clwwlknonfin 29336 . . . . . 6 ((Vtxβ€˜πΊ) ∈ Fin β†’ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ∈ Fin)
145eleq1i 2824 . . . . . 6 (𝑉 ∈ Fin ↔ (Vtxβ€˜πΊ) ∈ Fin)
157eleq1i 2824 . . . . . 6 (𝐹 ∈ Fin ↔ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)) ∈ Fin)
1613, 14, 153imtr4i 291 . . . . 5 (𝑉 ∈ Fin β†’ 𝐹 ∈ Fin)
1716adantr 481 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐹 ∈ Fin)
1817adantr 481 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐹 ∈ Fin)
195finrusgrfusgr 28811 . . . . . . 7 ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) β†’ 𝐺 ∈ FinUSGraph)
2019ancoms 459 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 ∈ FinUSGraph)
21 fusgrfis 28576 . . . . . 6 (𝐺 ∈ FinUSGraph β†’ (Edgβ€˜πΊ) ∈ Fin)
2220, 21syl 17 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (Edgβ€˜πΊ) ∈ Fin)
2322adantr 481 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (Edgβ€˜πΊ) ∈ Fin)
24 eqid 2732 . . . . 5 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
255, 24nbusgrfi 28620 . . . 4 ((𝐺 ∈ USGraph ∧ (Edgβ€˜πΊ) ∈ Fin ∧ 𝑋 ∈ 𝑉) β†’ (𝐺 NeighbVtx 𝑋) ∈ Fin)
262, 23, 3, 25syl3anc 1371 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (𝐺 NeighbVtx 𝑋) ∈ Fin)
27 hashxp 14390 . . 3 ((𝐹 ∈ Fin ∧ (𝐺 NeighbVtx 𝑋) ∈ Fin) β†’ (β™―β€˜(𝐹 Γ— (𝐺 NeighbVtx 𝑋))) = ((β™―β€˜πΉ) Β· (β™―β€˜(𝐺 NeighbVtx 𝑋))))
2818, 26, 27syl2anc 584 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝐹 Γ— (𝐺 NeighbVtx 𝑋))) = ((β™―β€˜πΉ) Β· (β™―β€˜(𝐺 NeighbVtx 𝑋))))
295rusgrpropnb 28829 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 β†’ (𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘₯ ∈ 𝑉 (β™―β€˜(𝐺 NeighbVtx π‘₯)) = 𝐾))
30 oveq2 7413 . . . . . . . . . . . 12 (π‘₯ = 𝑋 β†’ (𝐺 NeighbVtx π‘₯) = (𝐺 NeighbVtx 𝑋))
3130fveqeq2d 6896 . . . . . . . . . . 11 (π‘₯ = 𝑋 β†’ ((β™―β€˜(𝐺 NeighbVtx π‘₯)) = 𝐾 ↔ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3231rspccv 3609 . . . . . . . . . 10 (βˆ€π‘₯ ∈ 𝑉 (β™―β€˜(𝐺 NeighbVtx π‘₯)) = 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
33323ad2ant3 1135 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ 𝐾 ∈ β„•0* ∧ βˆ€π‘₯ ∈ 𝑉 (β™―β€˜(𝐺 NeighbVtx π‘₯)) = 𝐾) β†’ (𝑋 ∈ 𝑉 β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3429, 33syl 17 . . . . . . . 8 (𝐺 RegUSGraph 𝐾 β†’ (𝑋 ∈ 𝑉 β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3534adantl 482 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝑋 ∈ 𝑉 β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3635com12 32 . . . . . 6 (𝑋 ∈ 𝑉 β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3736adantr 481 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾))
3837impcom 408 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑋)) = 𝐾)
3938oveq2d 7421 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ ((β™―β€˜πΉ) Β· (β™―β€˜(𝐺 NeighbVtx 𝑋))) = ((β™―β€˜πΉ) Β· 𝐾))
40 hashcl 14312 . . . . 5 (𝐹 ∈ Fin β†’ (β™―β€˜πΉ) ∈ β„•0)
41 nn0cn 12478 . . . . 5 ((β™―β€˜πΉ) ∈ β„•0 β†’ (β™―β€˜πΉ) ∈ β„‚)
4218, 40, 413syl 18 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜πΉ) ∈ β„‚)
4320adantr 481 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐺 ∈ FinUSGraph)
44 simplr 767 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐺 RegUSGraph 𝐾)
45 ne0i 4333 . . . . . . . 8 (𝑋 ∈ 𝑉 β†’ 𝑉 β‰  βˆ…)
4645adantr 481 . . . . . . 7 ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑉 β‰  βˆ…)
4746adantl 482 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝑉 β‰  βˆ…)
485frusgrnn0 28817 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 β‰  βˆ…) β†’ 𝐾 ∈ β„•0)
4943, 44, 47, 48syl3anc 1371 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐾 ∈ β„•0)
5049nn0cnd 12530 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ 𝐾 ∈ β„‚)
5142, 50mulcomd 11231 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ ((β™―β€˜πΉ) Β· 𝐾) = (𝐾 Β· (β™―β€˜πΉ)))
5239, 51eqtrd 2772 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ ((β™―β€˜πΉ) Β· (β™―β€˜(𝐺 NeighbVtx 𝑋))) = (𝐾 Β· (β™―β€˜πΉ)))
5311, 28, 523eqtrd 2776 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐢𝑁)) = (𝐾 Β· (β™―β€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  {crab 3432  βˆ…c0 4321   class class class wbr 5147   Γ— cxp 5673  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   β‰ˆ cen 8932  Fincfn 8935  β„‚cc 11104   Β· cmul 11111   βˆ’ cmin 11440  2c2 12263  3c3 12264  β„•0cn0 12468  β„•0*cxnn0 12540  β„€β‰₯cuz 12818  β™―chash 14286  Vtxcvtx 28245  Edgcedg 28296  USGraphcusgr 28398  FinUSGraphcfusgr 28562   NeighbVtx cnbgr 28578   RegUSGraph crusgr 28802  ClWWalksNOncclwwlknon 29329
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-rp 12971  df-xadd 13089  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-word 14461  df-lsw 14509  df-concat 14517  df-s1 14542  df-substr 14587  df-pfx 14617  df-s2 14795  df-vtx 28247  df-iedg 28248  df-edg 28297  df-uhgr 28307  df-ushgr 28308  df-upgr 28331  df-umgr 28332  df-uspgr 28399  df-usgr 28400  df-fusgr 28563  df-nbgr 28579  df-vtxdg 28712  df-rgr 28803  df-rusgr 28804  df-wwlks 29073  df-wwlksn 29074  df-clwwlk 29224  df-clwwlkn 29267  df-clwwlknon 29330
This theorem is referenced by:  numclwlk1lem2  29612  numclwwlk3  29627
  Copyright terms: Public domain W3C validator