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Theorem fusgreghash2wspv 29588
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 29585. (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 29587 . . . . . 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 482 . . . 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 2826 . . . . . . 7 (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtxβ€˜πΊ))
7 nbfiusgrfi 28632 . . . . . . 7 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtxβ€˜πΊ)) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
86, 7sylan2b 595 . . . . . 6 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
98adantr 482 . . . . 5 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (𝐺 NeighbVtx 𝑣) ∈ Fin)
10 eqid 2733 . . . . 5 ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) = ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐})
11 snfi 9044 . . . . . 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 28599 . . . . . . . . . . 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 3147 . . . . . . 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 486 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
20 s3iunsndisj 14915 . . . . . . . 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 482 . . . . 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 482 . . . . . . 7 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ 𝑣 ∈ 𝑉)
2625anim1ci 617 . . . . . 6 ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉))
27 s3sndisj 14914 . . . . . 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 14832 . . . . . 6 βŸ¨β€œπ‘π‘£π‘‘β€βŸ© ∈ Word V
30 hashsng 14329 . . . . . 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 15770 . . . 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 28579 . . . . . . 7 (𝐺 ∈ FinUSGraph β†’ 𝐺 ∈ USGraph)
341hashnbusgrvd 28785 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) β†’ (β™―β€˜(𝐺 NeighbVtx 𝑣)) = ((VtxDegβ€˜πΊ)β€˜π‘£))
3533, 34sylan 581 . . . . . 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 7416 . . . . . . 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 7427 . . . . . 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 7416 . . . . . 6 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1) = (𝐾 βˆ’ 1))
4240, 41oveq12d 7427 . . . . 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 2793 . . . 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 2777 . . 3 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾) β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1)))
4544ex 414 . 2 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (β™―β€˜(π‘€β€˜π‘£)) = (𝐾 Β· (𝐾 βˆ’ 1))))
4645ralrimiva 3147 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629  βˆͺ ciun 4998  Disj wdisj 5114   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  1c1 11111   Β· cmul 11115   βˆ’ cmin 11444  2c2 12267  β™―chash 14290  Word cword 14464  βŸ¨β€œcs3 14793  Vtxcvtx 28256  USGraphcusgr 28409  FinUSGraphcfusgr 28573   NeighbVtx cnbgr 28589  VtxDegcvtxdg 28722   WSPathsN cwwspthsn 29082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-oi 9505  df-dju 9896  df-card 9934  df-ac 10111  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-rp 12975  df-xadd 13093  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546  df-s2 14799  df-s3 14800  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-sum 15633  df-vtx 28258  df-iedg 28259  df-edg 28308  df-uhgr 28318  df-ushgr 28319  df-upgr 28342  df-umgr 28343  df-uspgr 28410  df-usgr 28411  df-fusgr 28574  df-nbgr 28590  df-vtxdg 28723  df-wlks 28856  df-wlkson 28857  df-trls 28949  df-trlson 28950  df-pths 28973  df-spths 28974  df-pthson 28975  df-spthson 28976  df-wwlks 29084  df-wwlksn 29085  df-wwlksnon 29086  df-wspthsn 29087  df-wspthsnon 29088
This theorem is referenced by:  fusgreghash2wsp  29591
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