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Theorem fusgreghash2wspv 28120
Description: According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For directed simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths, see also comment of frgrhash2wsp 28117. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 12-Feb-2022.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
fusgreghash2wspv (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺,𝑎,𝑣
Allowed substitution hints:   𝐾(𝑤,𝑣,𝑎)   𝑀(𝑤,𝑣,𝑎)   𝑉(𝑤,𝑣)

Proof of Theorem fusgreghash2wspv
Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrhash2wsp.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
2 fusgreg2wsp.m . . . . . . 7 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
31, 2fusgr2wsp2nb 28119 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
43fveq2d 6649 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (♯‘(𝑀𝑣)) = (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
54adantr 484 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀𝑣)) = (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
61eleq2i 2881 . . . . . . 7 (𝑣𝑉𝑣 ∈ (Vtx‘𝐺))
7 nbfiusgrfi 27165 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
86, 7sylan2b 596 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
98adantr 484 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
10 eqid 2798 . . . . 5 ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) = ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})
11 snfi 8577 . . . . . 6 {⟨“𝑐𝑣𝑑”⟩} ∈ Fin
1211a1i 11 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
131nbgrssvtx 27132 . . . . . . . . . . 11 (𝐺 NeighbVtx 𝑣) ⊆ 𝑉
1413a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
1514ssdifd 4068 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}))
16 iunss1 4895 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}) → 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
1715, 16syl 17 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
1817ralrimiva 3149 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
19 simpr 488 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 𝑣𝑉)
20 s3iunsndisj 14319 . . . . . . . 8 (𝑣𝑉Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
22 disjss2 4998 . . . . . . 7 (∀𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} → (Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
2318, 21, 22sylc 65 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2423adantr 484 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2519adantr 484 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣𝑉)
2625anim1ci 618 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉))
27 s3sndisj 14318 . . . . . 6 ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2826, 27syl 17 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
29 s3cli 14234 . . . . . 6 ⟨“𝑐𝑣𝑑”⟩ ∈ Word V
30 hashsng 13726 . . . . . 6 (⟨“𝑐𝑣𝑑”⟩ ∈ Word V → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
3129, 30mp1i 13 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
329, 10, 12, 24, 28, 31hash2iun1dif1 15171 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)))
33 fusgrusgr 27112 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
341hashnbusgrvd 27318 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑣𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
3533, 34sylan 583 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
36 id 22 . . . . . . 7 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
37 oveq1 7142 . . . . . . 7 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) − 1) = (((VtxDeg‘𝐺)‘𝑣) − 1))
3836, 37oveq12d 7153 . . . . . 6 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
3935, 38syl 17 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
40 id 22 . . . . . 6 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
41 oveq1 7142 . . . . . 6 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) − 1) = (𝐾 − 1))
4240, 41oveq12d 7153 . . . . 5 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)) = (𝐾 · (𝐾 − 1)))
4339, 42sylan9eq 2853 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
445, 32, 433eqtrd 2837 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1)))
4544ex 416 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
4645ralrimiva 3149 1 (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  {crab 3110  Vcvv 3441  cdif 3878  wss 3881  {csn 4525   ciun 4881  Disj wdisj 4995  cmpt 5110  cfv 6324  (class class class)co 7135  Fincfn 8492  1c1 10527   · cmul 10531  cmin 10859  2c2 11680  chash 13686  Word cword 13857  ⟨“cs3 14195  Vtxcvtx 26789  USGraphcusgr 26942  FinUSGraphcfusgr 27106   NeighbVtx cnbgr 27122  VtxDegcvtxdg 27255   WSPathsN cwwspthsn 27614
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-ac2 9874  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-oi 8958  df-dju 9314  df-card 9352  df-ac 9527  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-rp 12378  df-xadd 12496  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-vtx 26791  df-iedg 26792  df-edg 26841  df-uhgr 26851  df-ushgr 26852  df-upgr 26875  df-umgr 26876  df-uspgr 26943  df-usgr 26944  df-fusgr 27107  df-nbgr 27123  df-vtxdg 27256  df-wlks 27389  df-wlkson 27390  df-trls 27482  df-trlson 27483  df-pths 27505  df-spths 27506  df-pthson 27507  df-spthson 27508  df-wwlks 27616  df-wwlksn 27617  df-wwlksnon 27618  df-wspthsn 27619  df-wspthsnon 27620
This theorem is referenced by:  fusgreghash2wsp  28123
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