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Theorem clwlknon2num 29885
Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 29622, 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 29085 . . . . . 6 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
2 usgruspgr 28702 . . . . . 6 (𝐺 ∈ USGraph β†’ 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USPGraph)
433ad2ant2 1133 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtxβ€˜πΊ)
65eleq2i 2824 . . . . . 6 (𝑋 ∈ 𝑉 ↔ 𝑋 ∈ (Vtxβ€˜πΊ))
76biimpi 215 . . . . 5 (𝑋 ∈ 𝑉 β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
873ad2ant3 1134 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (Vtxβ€˜πΊ))
9 2nn 12290 . . . . 5 2 ∈ β„•
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 2 ∈ β„•)
11 clwwlknonclwlknonen 29880 . . . 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 1370 . . 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 616 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 461 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 28839 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 233 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 ∈ FinUSGraph)
17163adant3 1131 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 𝐺 ∈ FinUSGraph)
18 2nn0 12494 . . . . . . 7 2 ∈ β„•0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ 2 ∈ β„•0)
20 wlksnfi 29425 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ β„•0) β†’ {𝑀 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 2} ∈ Fin)
2117, 19, 20syl2anc 583 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ {𝑀 ∈ (Walksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 2} ∈ Fin)
22 clwlkswks 29297 . . . . . . 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 1198 . . . . . 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 4082 . . . . 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 9270 . . . 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 29611 . . . . 5 (𝑉 ∈ Fin β†’ (𝑋(ClWWalksNOnβ€˜πΊ)2) ∈ Fin)
28273ad2ant1 1132 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)2) ∈ Fin)
29 hashen 14312 . . . 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 583 . . 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 616 . . . 4 ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtxβ€˜πΊ)))
33323adant1 1129 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtxβ€˜πΊ)))
34 clwwlknon2num 29622 . . 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 2771 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {crab 3431   βŠ† wss 3949   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977   β‰ˆ cen 8939  Fincfn 8942  0cc0 11113  β„•cn 12217  2c2 12272  β„•0cn0 12477  β™―chash 14295  Vtxcvtx 28520  USPGraphcuspgr 28672  USGraphcusgr 28673  FinUSGraphcfusgr 28837   RegUSGraph crusgr 29077  Walkscwlks 29117  ClWalkscclwlks 29291  ClWWalksNOncclwwlknon 29604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-rp 12980  df-xadd 13098  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-word 14470  df-lsw 14518  df-concat 14526  df-s1 14551  df-substr 14596  df-pfx 14626  df-edg 28572  df-uhgr 28582  df-ushgr 28583  df-upgr 28606  df-umgr 28607  df-uspgr 28674  df-usgr 28675  df-fusgr 28838  df-nbgr 28854  df-vtxdg 28987  df-rgr 29078  df-rusgr 29079  df-wlks 29120  df-clwlks 29292  df-wwlks 29348  df-wwlksn 29349  df-clwwlk 29499  df-clwwlkn 29542  df-clwwlknon 29605
This theorem is referenced by:  numclwlk1lem1  29886
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