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Theorem fusgreghash2wspv 27489
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 27486. (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 27488 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝑀𝑣) = 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
43fveq2d 6356 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (♯‘(𝑀𝑣)) = (♯‘ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
54adantr 472 . . . 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 26475 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
86, 7sylan2b 493 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
98adantr 472 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
10 eqid 2760 . . . . 5 ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) = ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})
11 snfi 8203 . . . . . 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 26435 . . . . . . . . . . 11 (𝐺 NeighbVtx 𝑣) ⊆ 𝑉
1413a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
1514ssdifd 3889 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}))
16 iunss1 4684 . . . . . . . . 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 3104 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
19 simpr 479 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → 𝑣𝑉)
20 s3iunsndisj 13908 . . . . . . . 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 4775 . . . . . . 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 472 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣) 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2519adantr 472 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣𝑉)
2625anim1i 593 . . . . . . 7 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑣𝑉𝑐 ∈ (𝐺 NeighbVtx 𝑣)))
2726ancomd 466 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉))
28 s3sndisj 13907 . . . . . 6 ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
2927, 28syl 17 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
30 s3cli 13826 . . . . . 6 ⟨“𝑐𝑣𝑑”⟩ ∈ Word V
31 hashsng 13351 . . . . . 6 (⟨“𝑐𝑣𝑑”⟩ ∈ Word V → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
3230, 31mp1i 13 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
339, 10, 12, 24, 29, 32hash2iun1dif1 14755 . . . 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)))
34 fusgrusgr 26413 . . . . . . 7 (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
351hashnbusgrvd 26634 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑣𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
3634, 35sylan 489 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
37 id 22 . . . . . . 7 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
38 oveq1 6820 . . . . . . 7 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) − 1) = (((VtxDeg‘𝐺)‘𝑣) − 1))
3937, 38oveq12d 6831 . . . . . 6 ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
4036, 39syl 17 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
41 id 22 . . . . . 6 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → ((VtxDeg‘𝐺)‘𝑣) = 𝐾)
42 oveq1 6820 . . . . . 6 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) − 1) = (𝐾 − 1))
4341, 42oveq12d 6831 . . . . 5 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)) = (𝐾 · (𝐾 − 1)))
4440, 43sylan9eq 2814 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
455, 33, 443eqtrd 2798 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1)))
4645ex 449 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
4746ralrimiva 3104 1 (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  wss 3715  {csn 4321   ciun 4672  Disj wdisj 4772  cmpt 4881  cfv 6049  (class class class)co 6813  Fincfn 8121  1c1 10129   · cmul 10133  cmin 10458  2c2 11262  chash 13311  Word cword 13477  ⟨“cs3 13787  Vtxcvtx 26073  USGraphcusgr 26243  FinUSGraphcfusgr 26407   NeighbVtx cnbgr 26423  VtxDegcvtxdg 26571   WSPathsN cwwspthsn 26931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-ac2 9477  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-disj 4773  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-oi 8580  df-card 8955  df-ac 9129  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-xnn0 11556  df-z 11570  df-uz 11880  df-rp 12026  df-xadd 12140  df-fz 12520  df-fzo 12660  df-seq 12996  df-exp 13055  df-hash 13312  df-word 13485  df-concat 13487  df-s1 13488  df-s2 13793  df-s3 13794  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-sum 14616  df-vtx 26075  df-iedg 26076  df-edg 26139  df-uhgr 26152  df-ushgr 26153  df-upgr 26176  df-umgr 26177  df-uspgr 26244  df-usgr 26245  df-fusgr 26408  df-nbgr 26424  df-vtxdg 26572  df-wlks 26705  df-wlkson 26706  df-trls 26799  df-trlson 26800  df-pths 26822  df-spths 26823  df-pthson 26824  df-spthson 26825  df-wwlks 26933  df-wwlksn 26934  df-wwlksnon 26935  df-wspthsn 26936  df-wspthsnon 26937
This theorem is referenced by:  fusgreghash2wsp  27492
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