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Theorem clwlknon2num 29659
Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 29396, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
Hypothesis
Ref Expression
clwlknon2num.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
clwlknon2num ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = 𝐾)
Distinct variable groups:   𝑀,𝐺   𝑀,𝑉   𝑀,𝑋   𝑀,𝐾

Proof of Theorem clwlknon2num
StepHypRef Expression
1 rusgrusgr 28859 . . . . . 6 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
2 usgruspgr 28476 . . . . . 6 (𝐺 ∈ USGraph β†’ 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USPGraph)
433ad2ant2 1134 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
65eleq2i 2825 . . . . . 6 (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtxβ€˜πΊ))
76biimpi 215 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
873ad2ant3 1135 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
9 2nn 12287 . . . . 5 2 ∈ β„•
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 2 ∈ β„•)
11 clwwlknonclwlknonen 29654 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtxβ€˜πΊ) ∧ 2 ∈ β„•) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β‰ˆ (𝑋(ClWWalksNOnβ€˜πΊ)2))
124, 8, 10, 11syl3anc 1371 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β‰ˆ (𝑋(ClWWalksNOnβ€˜πΊ)2))
131anim2i 617 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 462 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 28613 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 233 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 ∈ FinUSGraph)
17163adant3 1132 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝐺 ∈ FinUSGraph)
18 2nn0 12491 . . . . . . 7 2 ∈ β„•0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 2 ∈ β„•0)
20 wlksnfi 29199 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ β„•0) β†’ {𝑀 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 2} ∈ Fin)
2117, 19, 20syl2anc 584 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ {𝑀 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 2} ∈ Fin)
22 clwlkswks 29071 . . . . . . 7 (ClWalksβ€˜πΊ) βŠ† (Walksβ€˜πΊ)
2322a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (ClWalksβ€˜πΊ) βŠ† (Walksβ€˜πΊ))
24 simp2l 1199 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) ∧ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ∧ 𝑀 ∈ (ClWalksβ€˜πΊ)) β†’ (β™―β€˜(1st β€˜π‘€)) = 2)
2523, 24rabssrabd 4081 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} βŠ† {𝑀 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 2})
2621, 25ssfid 9269 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∈ Fin)
275clwwlknonfin 29385 . . . . 5 (𝑉 ∈ Fin β†’ (𝑋(ClWWalksNOnβ€˜πΊ)2) ∈ Fin)
28273ad2ant1 1133 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)2) ∈ Fin)
29 hashen 14309 . . . 4 (({𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∈ Fin ∧ (𝑋(ClWWalksNOnβ€˜πΊ)2) ∈ Fin) β†’ ((β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)) ↔ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β‰ˆ (𝑋(ClWWalksNOnβ€˜πΊ)2)))
3026, 28, 29syl2anc 584 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ ((β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)) ↔ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β‰ˆ (𝑋(ClWWalksNOnβ€˜πΊ)2)))
3112, 30mpbird 256 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)))
327anim2i 617 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtxβ€˜πΊ)))
33323adant1 1130 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtxβ€˜πΊ)))
34 clwwlknon2num 29396 . . 3 ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)) = 𝐾)
3533, 34syl 17 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)) = 𝐾)
3631, 35eqtrd 2772 1 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3948   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976   β‰ˆ cen 8938  Fincfn 8941  0cc0 11112  β„•cn 12214  2c2 12269  β„•0cn0 12474  β™―chash 14292  Vtxcvtx 28294  USPGraphcuspgr 28446  USGraphcusgr 28447  FinUSGraphcfusgr 28611   RegUSGraph crusgr 28851  Walkscwlks 28891  ClWalkscclwlks 29065  ClWWalksNOncclwwlknon 29378
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-rp 12977  df-xadd 13095  df-fz 13487  df-fzo 13630  df-seq 13969  df-exp 14030  df-hash 14293  df-word 14467  df-lsw 14515  df-concat 14523  df-s1 14548  df-substr 14593  df-pfx 14623  df-edg 28346  df-uhgr 28356  df-ushgr 28357  df-upgr 28380  df-umgr 28381  df-uspgr 28448  df-usgr 28449  df-fusgr 28612  df-nbgr 28628  df-vtxdg 28761  df-rgr 28852  df-rusgr 28853  df-wlks 28894  df-clwlks 29066  df-wwlks 29122  df-wwlksn 29123  df-clwwlk 29273  df-clwwlkn 29316  df-clwwlknon 29379
This theorem is referenced by:  numclwlk1lem1  29660
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