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Theorem numclwlk1lem1 29313
Description: Lemma 1 for numclwlk1 29315 (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 = (Vtxβ€˜πΊ)
numclwlk1.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
numclwlk1.f 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
Assertion
Ref Expression
numclwlk1lem1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
Distinct variable groups:   𝑀,𝐺   𝑀,𝐾   𝑀,𝑁   𝑀,𝑉   𝑀,𝑋
Allowed substitution hints:   𝐢(𝑀)   𝐹(𝑀)

Proof of Theorem numclwlk1lem1
StepHypRef Expression
1 3anass 1095 . . . . . . 7 (((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ (((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)))
2 anidm 565 . . . . . . . 8 ((((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((2nd β€˜π‘€)β€˜0) = 𝑋)
32anbi2i 623 . . . . . . 7 (((β™―β€˜(1st β€˜π‘€)) = 2 ∧ (((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)) ↔ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋))
41, 3bitri 274 . . . . . 6 (((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋))
54rabbii 3413 . . . . 5 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
65fveq2i 6845 . . . 4 (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
7 simpl 483 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝑉 ∈ Fin)
8 simpr 485 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 RegUSGraph 𝐾)
9 simpl 483 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) β†’ 𝑋 ∈ 𝑉)
10 numclwlk1.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
1110clwlknon2num 29312 . . . . 5 ((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)
127, 8, 9, 11syl2an3an 1422 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = 𝐾)
136, 12eqtrid 2788 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = 𝐾)
14 rusgrusgr 28512 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
1514anim2i 617 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1615ancomd 462 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1710isfusgr 28266 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1816, 17sylibr 233 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 ∈ FinUSGraph)
19 ne0i 4294 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ 𝑉 β‰  βˆ…)
2019adantr 481 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) β†’ 𝑉 β‰  βˆ…)
2110frusgrnn0 28519 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 β‰  βˆ…) β†’ 𝐾 ∈ β„•0)
2218, 8, 20, 21syl2an3an 1422 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ 𝐾 ∈ β„•0)
2322nn0red 12474 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ 𝐾 ∈ ℝ)
24 ax-1rid 11121 . . . 4 (𝐾 ∈ ℝ β†’ (𝐾 Β· 1) = 𝐾)
2523, 24syl 17 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (𝐾 Β· 1) = 𝐾)
2610wlkl0 29311 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩})
2726ad2antrl 726 . . . . . 6 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩})
2827fveq2d 6846 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (β™―β€˜{βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩}))
29 opex 5421 . . . . . 6 βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ∈ V
30 hashsng 14269 . . . . . 6 (βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ∈ V β†’ (β™―β€˜{βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩}) = 1)
3129, 30ax-mp 5 . . . . 5 (β™―β€˜{βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩}) = 1
3228, 31eqtr2di 2793 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ 1 = (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}))
3332oveq2d 7373 . . 3 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (𝐾 Β· 1) = (𝐾 Β· (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})))
3413, 25, 333eqtr2d 2782 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (𝐾 Β· (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})))
35 numclwlk1.c . . . . . 6 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
36 eqeq2 2748 . . . . . . . 8 (𝑁 = 2 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘€)) = 2))
37 oveq1 7364 . . . . . . . . . 10 (𝑁 = 2 β†’ (𝑁 βˆ’ 2) = (2 βˆ’ 2))
38 2cn 12228 . . . . . . . . . . 11 2 ∈ β„‚
3938subidi 11472 . . . . . . . . . 10 (2 βˆ’ 2) = 0
4037, 39eqtrdi 2792 . . . . . . . . 9 (𝑁 = 2 β†’ (𝑁 βˆ’ 2) = 0)
4140fveqeq2d 6850 . . . . . . . 8 (𝑁 = 2 β†’ (((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘€)β€˜0) = 𝑋))
4236, 413anbi13d 1438 . . . . . . 7 (𝑁 = 2 β†’ (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)))
4342rabbidv 3415 . . . . . 6 (𝑁 = 2 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
4435, 43eqtrid 2788 . . . . 5 (𝑁 = 2 β†’ 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
4544fveq2d 6846 . . . 4 (𝑁 = 2 β†’ (β™―β€˜πΆ) = (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}))
46 numclwlk1.f . . . . . . 7 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
4740eqeq2d 2747 . . . . . . . . 9 (𝑁 = 2 β†’ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ↔ (β™―β€˜(1st β€˜π‘€)) = 0))
4847anbi1d 630 . . . . . . . 8 (𝑁 = 2 β†’ (((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)))
4948rabbidv 3415 . . . . . . 7 (𝑁 = 2 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
5046, 49eqtrid 2788 . . . . . 6 (𝑁 = 2 β†’ 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
5150fveq2d 6846 . . . . 5 (𝑁 = 2 β†’ (β™―β€˜πΉ) = (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}))
5251oveq2d 7373 . . . 4 (𝑁 = 2 β†’ (𝐾 Β· (β™―β€˜πΉ)) = (𝐾 Β· (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})))
5345, 52eqeq12d 2752 . . 3 (𝑁 = 2 β†’ ((β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)) ↔ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (𝐾 Β· (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}))))
5453ad2antll 727 . 2 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ ((β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)) ↔ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = (𝐾 Β· (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}))))
5534, 54mpbird 256 1 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2943  {crab 3407  Vcvv 3445  βˆ…c0 4282  {csn 4586  βŸ¨cop 4592   class class class wbr 5105  β€˜cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Fincfn 8883  β„cr 11050  0cc0 11051  1c1 11052   Β· cmul 11056   βˆ’ cmin 11385  2c2 12208  β„•0cn0 12413  β™―chash 14230  Vtxcvtx 27947  USGraphcusgr 28100  FinUSGraphcfusgr 28264   RegUSGraph crusgr 28504  ClWalkscclwlks 28718
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-vtx 27949  df-iedg 27950  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:  numclwlk1  29315
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