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Theorem numclwlk1lem1 29889
Description: Lemma 1 for numclwlk1 29891 (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 1093 . . . . . . 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 563 . . . . . . . 8 ((((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((2nd β€˜π‘€)β€˜0) = 𝑋)
32anbi2i 621 . . . . . . 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 3436 . . . . 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 6893 . . . 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 481 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝑉 ∈ Fin)
8 simpr 483 . . . . 5 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 RegUSGraph 𝐾)
9 simpl 481 . . . . 5 ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) β†’ 𝑋 ∈ 𝑉)
10 numclwlk1.v . . . . . 6 𝑉 = (Vtxβ€˜πΊ)
1110clwlknon2num 29888 . . . . 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 1420 . . . 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 2782 . . 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 29088 . . . . . . . . 9 (𝐺 RegUSGraph 𝐾 β†’ 𝐺 ∈ USGraph)
1514anim2i 615 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝑉 ∈ Fin ∧ 𝐺 ∈ USGraph))
1615ancomd 460 . . . . . . 7 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1710isfusgr 28842 . . . . . . 7 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
1816, 17sylibr 233 . . . . . 6 ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ 𝐺 ∈ FinUSGraph)
19 ne0i 4333 . . . . . . 7 (𝑋 ∈ 𝑉 β†’ 𝑉 β‰  βˆ…)
2019adantr 479 . . . . . 6 ((𝑋 ∈ 𝑉 ∧ 𝑁 = 2) β†’ 𝑉 β‰  βˆ…)
2110frusgrnn0 29095 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 β‰  βˆ…) β†’ 𝐾 ∈ β„•0)
2218, 8, 20, 21syl2an3an 1420 . . . . 5 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ 𝐾 ∈ β„•0)
2322nn0red 12537 . . . 4 (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ 𝐾 ∈ ℝ)
24 ax-1rid 11182 . . . 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 29887 . . . . . . 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 724 . . . . . 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 6894 . . . . 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 5463 . . . . . 6 βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩ ∈ V
30 hashsng 14333 . . . . . 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 2787 . . . 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 7427 . . 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 2776 . 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 2742 . . . . . . . 8 (𝑁 = 2 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘€)) = 2))
37 oveq1 7418 . . . . . . . . . 10 (𝑁 = 2 β†’ (𝑁 βˆ’ 2) = (2 βˆ’ 2))
38 2cn 12291 . . . . . . . . . . 11 2 ∈ β„‚
3938subidi 11535 . . . . . . . . . 10 (2 βˆ’ 2) = 0
4037, 39eqtrdi 2786 . . . . . . . . 9 (𝑁 = 2 β†’ (𝑁 βˆ’ 2) = 0)
4140fveqeq2d 6898 . . . . . . . 8 (𝑁 = 2 β†’ (((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘€)β€˜0) = 𝑋))
4236, 413anbi13d 1436 . . . . . . 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 3438 . . . . . 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 2782 . . . . 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 6894 . . . 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 2741 . . . . . . . . 9 (𝑁 = 2 β†’ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ↔ (β™―β€˜(1st β€˜π‘€)) = 0))
4847anbi1d 628 . . . . . . . 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 3438 . . . . . . 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 2782 . . . . . 6 (𝑁 = 2 β†’ 𝐹 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)})
5150fveq2d 6894 . . . . 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 7427 . . . 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 2746 . . 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 725 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  {crab 3430  Vcvv 3472  βˆ…c0 4321  {csn 4627  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  Fincfn 8941  β„cr 11111  0cc0 11112  1c1 11113   Β· cmul 11117   βˆ’ cmin 11448  2c2 12271  β„•0cn0 12476  β™―chash 14294  Vtxcvtx 28523  USGraphcusgr 28676  FinUSGraphcfusgr 28840   RegUSGraph crusgr 29080  ClWalkscclwlks 29294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  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 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  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 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-rp 12979  df-xadd 13097  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625  df-vtx 28525  df-iedg 28526  df-edg 28575  df-uhgr 28585  df-ushgr 28586  df-upgr 28609  df-umgr 28610  df-uspgr 28677  df-usgr 28678  df-fusgr 28841  df-nbgr 28857  df-vtxdg 28990  df-rgr 29081  df-rusgr 29082  df-wlks 29123  df-clwlks 29295  df-wwlks 29351  df-wwlksn 29352  df-clwwlk 29502  df-clwwlkn 29545  df-clwwlknon 29608
This theorem is referenced by:  numclwlk1  29891
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