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Theorem clwlknon2num 29312
Description: There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 29049, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
Hypothesis
Ref Expression
clwlknon2num.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
clwlknon2num ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋   𝑤,𝐾

Proof of Theorem clwlknon2num
StepHypRef Expression
1 rusgrusgr 28512 . . . . . 6 (𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph)
2 usgruspgr 28129 . . . . . 6 (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph)
31, 2syl 17 . . . . 5 (𝐺 RegUSGraph 𝐾𝐺 ∈ USPGraph)
433ad2ant2 1134 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ USPGraph)
5 clwlknon2num.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
65eleq2i 2829 . . . . . 6 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
76biimpi 215 . . . . 5 (𝑋𝑉𝑋 ∈ (Vtx‘𝐺))
873ad2ant3 1135 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝑋 ∈ (Vtx‘𝐺))
9 2nn 12226 . . . . 5 2 ∈ ℕ
109a1i 11 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ)
11 clwwlknonclwlknonen 29307 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 2 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
124, 8, 10, 11syl3anc 1371 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2))
131anim2i 617 . . . . . . . . 9 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1413ancomd 462 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
155isfusgr 28266 . . . . . . . 8 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1614, 15sylibr 233 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → 𝐺 ∈ FinUSGraph)
17163adant3 1132 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 𝐺 ∈ FinUSGraph)
18 2nn0 12430 . . . . . . 7 2 ∈ ℕ0
1918a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → 2 ∈ ℕ0)
20 wlksnfi 28852 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 2 ∈ ℕ0) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
2117, 19, 20syl2anc 584 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2} ∈ Fin)
22 clwlkswks 28724 . . . . . . 7 (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
2322a1i 11 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (ClWalks‘𝐺) ⊆ (Walks‘𝐺))
24 simp2l 1199 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) ∧ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋) ∧ 𝑤 ∈ (ClWalks‘𝐺)) → (♯‘(1st𝑤)) = 2)
2523, 24rabssrabd 4041 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ⊆ {𝑤 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑤)) = 2})
2621, 25ssfid 9211 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin)
275clwwlknonfin 29038 . . . . 5 (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
28273ad2ant1 1133 . . . 4 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin)
29 hashen 14247 . . . 4 (({𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ∈ Fin ∧ (𝑋(ClWWalksNOn‘𝐺)2) ∈ Fin) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
3026, 28, 29syl2anc 584 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → ((♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) ↔ {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)2)))
3112, 30mpbird 256 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = (♯‘(𝑋(ClWWalksNOn‘𝐺)2)))
327anim2i 617 . . . 4 ((𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
33323adant1 1130 . . 3 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)))
34 clwwlknon2num 29049 . . 3 ((𝐺 RegUSGraph 𝐾𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3533, 34syl 17 . 2 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
3631, 35eqtrd 2776 1 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  wss 3910   class class class wbr 5105  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  cen 8880  Fincfn 8883  0cc0 11051  cn 12153  2c2 12208  0cn0 12413  chash 14230  Vtxcvtx 27947  USPGraphcuspgr 28099  USGraphcusgr 28100  FinUSGraphcfusgr 28264   RegUSGraph crusgr 28504  Walkscwlks 28544  ClWalkscclwlks 28718  ClWWalksNOncclwwlknon 29031
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-xadd 13034  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-edg 27999  df-uhgr 28009  df-ushgr 28010  df-upgr 28033  df-umgr 28034  df-uspgr 28101  df-usgr 28102  df-fusgr 28265  df-nbgr 28281  df-vtxdg 28414  df-rgr 28505  df-rusgr 28506  df-wlks 28547  df-clwlks 28719  df-wwlks 28775  df-wwlksn 28776  df-clwwlk 28926  df-clwwlkn 28969  df-clwwlknon 29032
This theorem is referenced by:  numclwlk1lem1  29313
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