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Theorem fusgreghash2wspv 29321
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 29318. (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 29320 . . . . . 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 6851 . . . . 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 2830 . . . . . . 7 (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtxβ€˜πΊ))
7 nbfiusgrfi 28365 . . . . . . 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 2737 . . . . 5 ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) = ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐})
11 snfi 8995 . . . . . 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 28332 . . . . . . . . . . 11 (𝐺 NeighbVtx 𝑣) βŠ† 𝑉
1413a1i 11 . . . . . . . . . 10 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ (𝐺 NeighbVtx 𝑣) βŠ† 𝑉)
1514ssdifd 4105 . . . . . . . . 9 (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) β†’ ((𝐺 NeighbVtx 𝑣) βˆ– {𝑐}) βŠ† (𝑉 βˆ– {𝑐}))
16 iunss1 4973 . . . . . . . . 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 486 . . . . . . . 8 ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) β†’ 𝑣 ∈ 𝑉)
20 s3iunsndisj 14860 . . . . . . . 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 5078 . . . . . . 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 14859 . . . . . 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 14777 . . . . . 6 βŸ¨β€œπ‘π‘£π‘‘β€βŸ© ∈ Word V
30 hashsng 14276 . . . . . 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 15716 . . . 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 28312 . . . . . . 7 (𝐺 ∈ FinUSGraph β†’ 𝐺 ∈ USGraph)
341hashnbusgrvd 28518 . . . . . . 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 7369 . . . . . . 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 7380 . . . . . 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 7369 . . . . . 6 (((VtxDegβ€˜πΊ)β€˜π‘£) = 𝐾 β†’ (((VtxDegβ€˜πΊ)β€˜π‘£) βˆ’ 1) = (𝐾 βˆ’ 1))
4240, 41oveq12d 7380 . . . . 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 2797 . . . 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 2781 . . 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 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3410  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  {csn 4591  βˆͺ ciun 4959  Disj wdisj 5075   ↦ cmpt 5193  β€˜cfv 6501  (class class class)co 7362  Fincfn 8890  1c1 11059   Β· cmul 11063   βˆ’ cmin 11392  2c2 12215  β™―chash 14237  Word cword 14409  βŸ¨β€œcs3 14738  Vtxcvtx 27989  USGraphcusgr 28142  FinUSGraphcfusgr 28306   NeighbVtx cnbgr 28322  VtxDegcvtxdg 28455   WSPathsN cwwspthsn 28815
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-oi 9453  df-dju 9844  df-card 9882  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-rp 12923  df-xadd 13041  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-word 14410  df-concat 14466  df-s1 14491  df-s2 14744  df-s3 14745  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578  df-vtx 27991  df-iedg 27992  df-edg 28041  df-uhgr 28051  df-ushgr 28052  df-upgr 28075  df-umgr 28076  df-uspgr 28143  df-usgr 28144  df-fusgr 28307  df-nbgr 28323  df-vtxdg 28456  df-wlks 28589  df-wlkson 28590  df-trls 28682  df-trlson 28683  df-pths 28706  df-spths 28707  df-pthson 28708  df-spthson 28709  df-wwlks 28817  df-wwlksn 28818  df-wwlksnon 28819  df-wspthsn 28820  df-wspthsnon 28821
This theorem is referenced by:  fusgreghash2wsp  29324
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