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Theorem fusgreghash2wspv 29853
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 29850. (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 = (Vtxβ€˜πΊ)
fusgreg2wsp.m 𝑀 = (π‘Ž ∈ 𝑉 ↦ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (π‘€β€˜1) = π‘Ž})
Assertion
Ref Expression
fusgreghash2wspv (𝐺 ∈ FinUSGraph β†’ βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1))))
Distinct variable groups:   𝐺,π‘Ž   𝑉,π‘Ž   𝑀,𝐺,π‘Ž,𝑣
Allowed substitution hints:   𝐾(𝑀,𝑣,π‘Ž)   𝑀(𝑀,𝑣,π‘Ž)   𝑉(𝑀,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 = (Vtxβ€˜πΊ)
2 fusgreg2wsp.m . . . . . . 7 𝑀 = (π‘Ž ∈ 𝑉 ↦ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (π‘€β€˜1) = π‘Ž})
31, 2fusgr2wsp2nb 29852 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (π‘€β€˜π‘£) = βˆͺ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
43fveq2d 6896 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (β™―β€˜(π‘€β€˜π‘£)) = (β™―β€˜βˆͺ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}))
54adantr 479 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (β™―β€˜(π‘€β€˜π‘£)) = (β™―β€˜βˆͺ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}))
61eleq2i 2823 . . . . . . 7 (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtxβ€˜πΊ))
7 nbfiusgrfi 28897 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtxβ€˜πΊ)) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
86, 7sylan2b 592 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
98adantr 479 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
10 eqid 2730 . . . . 5 ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) = ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐})
11 snfi 9048 . . . . . 6 {βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} ∈ Fin
1211a1i 11 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐})) β†’ {βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} ∈ Fin)
131nbgrssvtx 28864 . . . . . . . . . . 11 (𝐺 NeighbVtx 𝑣) βŠ† 𝑉
1413a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ (𝐺 NeighbVtx 𝑣) βŠ† 𝑉)
1514ssdifd 4141 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) βŠ† (𝑉 βˆ– {𝑐}))
16 iunss1 5012 . . . . . . . . 9 (((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) βŠ† (𝑉 βˆ– {𝑐}) β†’ βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} βŠ† βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
1715, 16syl 17 . . . . . . . 8 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} βŠ† βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
1817ralrimiva 3144 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ βˆ€π‘ ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} βŠ† βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
19 simpr 483 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
20 s3iunsndisj 14921 . . . . . . . 8 (𝑣 ∈ 𝑉 β†’ Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
22 disjss2 5117 . . . . . . 7 (βˆ€π‘ ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} βŠ† βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} β†’ (Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ (𝑉 βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©} β†’ Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}))
2318, 21, 22sylc 65 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
2423adantr 479 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
2519adantr 479 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ 𝑣 ∈ 𝑉)
2625anim1ci 614 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉))
27 s3sndisj 14920 . . . . . 6 ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉) β†’ Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
2826, 27syl 17 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©})
29 s3cli 14838 . . . . . 6 βŸ¨β€œπ‘π‘£π‘‘β€βŸ© ∈ Word V
30 hashsng 14335 . . . . . 6 (βŸ¨β€œπ‘π‘£π‘‘β€βŸ© ∈ Word V β†’ (β™―β€˜{βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}) = 1)
3129, 30mp1i 13 . . . . 5 ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐})) β†’ (β™―β€˜{βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}) = 1)
329, 10, 12, 24, 28, 31hash2iun1dif1 15776 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (β™―β€˜βˆͺ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)βˆͺ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}){βŸ¨β€œπ‘π‘£π‘‘β€βŸ©}) = ((β™―β€˜(𝐺 NeighbVtx 𝑣)) Β· ((β™―β€˜(𝐺 NeighbVtx 𝑣)) βˆ’ 1)))
33 fusgrusgr 28844 . . . . . . 7 (𝐺 ∈ FinUSGraph β†’ 𝐺 ∈ USGraph)
341hashnbusgrvd 29050 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£))
3533, 34sylan 578 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£))
36 id 22 . . . . . . 7 ((β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£))
37 oveq1 7420 . . . . . . 7 ((β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£) β†’ ((β™―β€˜(𝐺 NeighbVtx 𝑣)) βˆ’ 1) = (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1))
3836, 37oveq12d 7431 . . . . . 6 ((β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£) β†’ ((β™―β€˜(𝐺 NeighbVtx 𝑣)) Β· ((β™―β€˜(𝐺 NeighbVtx 𝑣)) βˆ’ 1)) = (((VtxDegβ€˜πΊ)β€˜π‘£) Β· (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1)))
3935, 38syl 17 . . . . 5 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ ((β™―β€˜(𝐺 NeighbVtx 𝑣)) Β· ((β™―β€˜(𝐺 NeighbVtx 𝑣)) βˆ’ 1)) = (((VtxDegβ€˜πΊ)β€˜π‘£) Β· (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1)))
40 id 22 . . . . . 6 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾)
41 oveq1 7420 . . . . . 6 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1) = (𝐾 βˆ’ 1))
4240, 41oveq12d 7431 . . . . 5 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) Β· (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1)) = (𝐾 Β· (𝐾 βˆ’ 1)))
4339, 42sylan9eq 2790 . . . 4 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ ((β™―β€˜(𝐺 NeighbVtx 𝑣)) Β· ((β™―β€˜(𝐺 NeighbVtx 𝑣)) βˆ’ 1)) = (𝐾 Β· (𝐾 βˆ’ 1)))
445, 32, 433eqtrd 2774 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1)))
4544ex 411 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1))))
4645ralrimiva 3144 1 (𝐺 ∈ FinUSGraph β†’ βˆ€π‘£ ∈ 𝑉 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  βˆͺ ciun 4998  Disj wdisj 5114   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7413  Fincfn 8943  1c1 11115   Β· cmul 11119   βˆ’ cmin 11450  2c2 12273  β™―chash 14296  Word cword 14470  βŸ¨β€œcs3 14799  Vtxcvtx 28521  USGraphcusgr 28674  FinUSGraphcfusgr 28838   NeighbVtx cnbgr 28854  VtxDegcvtxdg 28987   WSPathsN cwwspthsn 29347
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-inf2 9640  ax-ac2 10462  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-oi 9509  df-dju 9900  df-card 9938  df-ac 10115  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-div 11878  df-nn 12219  df-2 12281  df-3 12282  df-n0 12479  df-xnn0 12551  df-z 12565  df-uz 12829  df-rp 12981  df-xadd 13099  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14034  df-hash 14297  df-word 14471  df-concat 14527  df-s1 14552  df-s2 14805  df-s3 14806  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-vtx 28523  df-iedg 28524  df-edg 28573  df-uhgr 28583  df-ushgr 28584  df-upgr 28607  df-umgr 28608  df-uspgr 28675  df-usgr 28676  df-fusgr 28839  df-nbgr 28855  df-vtxdg 28988  df-wlks 29121  df-wlkson 29122  df-trls 29214  df-trlson 29215  df-pths 29238  df-spths 29239  df-pthson 29240  df-spthson 29241  df-wwlks 29349  df-wwlksn 29350  df-wwlksnon 29351  df-wspthsn 29352  df-wspthsnon 29353
This theorem is referenced by:  fusgreghash2wsp  29856
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