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Theorem fusgreghash2wspv 30364
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 30361. (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 30363 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
43fveq2d 6911 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (♯‘(𝑀𝑣)) = (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
54adantr 480 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀𝑣)) = (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
61eleq2i 2831 . . . . . . 7 (𝑣𝑉𝑣 ∈ (Vtx‘𝐺))
7 nbfiusgrfi 29407 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
86, 7sylan2b 594 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
98adantr 480 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
10 eqid 2735 . . . . 5 ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) = ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})
11 snfi 9082 . . . . . 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 29374 . . . . . . . . . . 11 (𝐺 NeighbVtx 𝑣) ⊆ 𝑉
1413a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
1514ssdifd 4155 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}))
16 iunss1 5011 . . . . . . . . 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 3144 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
19 simpr 484 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 𝑣𝑉)
20 s3iunsndisj 15004 . . . . . . . 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 5118 . . . . . . 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 480 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2519adantr 480 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣𝑉)
2625anim1ci 616 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉))
27 s3sndisj 15003 . . . . . 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 14917 . . . . . 6 ⟨“𝑐𝑣𝑑”⟩ ∈ Word V
30 hashsng 14405 . . . . . 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 15857 . . . 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 29354 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
341hashnbusgrvd 29561 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑣𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
3533, 34sylan 580 . . . . . 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 7438 . . . . . . 7 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) − 1) = (((VtxDeg‘𝐺)‘𝑣) − 1))
3836, 37oveq12d 7449 . . . . . 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 7438 . . . . . 6 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) − 1) = (𝐾 − 1))
4240, 41oveq12d 7449 . . . . 5 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)) = (𝐾 · (𝐾 − 1)))
4339, 42sylan9eq 2795 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
445, 32, 433eqtrd 2779 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1)))
4544ex 412 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
4645ralrimiva 3144 1 (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  {csn 4631   ciun 4996  Disj wdisj 5115  cmpt 5231  cfv 6563  (class class class)co 7431  Fincfn 8984  1c1 11154   · cmul 11158  cmin 11490  2c2 12319  chash 14366  Word cword 14549  ⟨“cs3 14878  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348   NeighbVtx cnbgr 29364  VtxDegcvtxdg 29498   WSPathsN cwwspthsn 29858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-disj 5116  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-oi 9548  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-xadd 13153  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-vtx 29030  df-iedg 29031  df-edg 29080  df-uhgr 29090  df-ushgr 29091  df-upgr 29114  df-umgr 29115  df-uspgr 29182  df-usgr 29183  df-fusgr 29349  df-nbgr 29365  df-vtxdg 29499  df-wlks 29632  df-wlkson 29633  df-trls 29725  df-trlson 29726  df-pths 29749  df-spths 29750  df-pthson 29751  df-spthson 29752  df-wwlks 29860  df-wwlksn 29861  df-wwlksnon 29862  df-wspthsn 29863  df-wspthsnon 29864
This theorem is referenced by:  fusgreghash2wsp  30367
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