Theorem fusgreghash2wspv 28600

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 28597. (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 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
fusgreghash2wspv	⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (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.

Step	Hyp	Ref	Expression
1		frgrhash2wsp.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
2		fusgreg2wsp.m	. . . . . . 7 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
3	1, 2	fusgr2wsp2nb 28599	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝑀‘𝑣) = ∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
4	3	fveq2d 6760	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝑀‘𝑣)) = (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
5	4	adantr 480	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀‘𝑣)) = (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}))
6	1	eleq2i 2830	. . . . . . 7 ⊢ (𝑣 ∈ 𝑉 ↔ 𝑣 ∈ (Vtx‘𝐺))
7		nbfiusgrfi 27645	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
8	6, 7	sylan2b 593	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
9	8	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐺 NeighbVtx 𝑣) ∈ Fin)
10		eqid 2738	. . . . 5 ⊢ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) = ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})
11		snfi 8788	. . . . . 6 ⊢ {⟨“𝑐𝑣𝑑”⟩} ∈ Fin
12	11	a1i 11	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → {⟨“𝑐𝑣𝑑”⟩} ∈ Fin)
13	1	nbgrssvtx 27612	. . . . . . . . . . 11 ⊢ (𝐺 NeighbVtx 𝑣) ⊆ 𝑉
14	13	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝐺 NeighbVtx 𝑣) ⊆ 𝑉)
15	14	ssdifd 4071	. . . . . . . . 9 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}))
16		iunss1 4935	. . . . . . . . 9 ⊢ (((𝐺 NeighbVtx 𝑣) ∖ {𝑐}) ⊆ (𝑉 ∖ {𝑐}) → ∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
17	15, 16	syl 17	. . . . . . . 8 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → ∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
18	17	ralrimiva 3107	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ∀𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩} ⊆ ∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
19		simpr 484	. . . . . . . 8 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉)
20		s3iunsndisj 14607	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑉 → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
21	19, 20	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ (𝑉 ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
22		disjss2 5038	. . . . . . 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”⟩}))
23	18, 21, 22	sylc 65	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
24	23	adantr 480	. . . . 5 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → Disj 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
25	19	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → 𝑣 ∈ 𝑉)
26	25	anim1ci 615	. . . . . 6 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → (𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉))
27		s3sndisj 14606	. . . . . 6 ⊢ ((𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑣 ∈ 𝑉) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
28	26, 27	syl 17	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)) → Disj 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩})
29		s3cli 14522	. . . . . 6 ⊢ ⟨“𝑐𝑣𝑑”⟩ ∈ Word V
30		hashsng 14012	. . . . . 6 ⊢ (⟨“𝑐𝑣𝑑”⟩ ∈ Word V → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
31	29, 30	mp1i 13	. . . . 5 ⊢ ((((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) ∧ 𝑐 ∈ (𝐺 NeighbVtx 𝑣) ∧ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐})) → (♯‘{⟨“𝑐𝑣𝑑”⟩}) = 1)
32	9, 10, 12, 24, 28, 31	hash2iun1dif1 15464	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘∪ 𝑐 ∈ (𝐺 NeighbVtx 𝑣)∪ 𝑑 ∈ ((𝐺 NeighbVtx 𝑣) ∖ {𝑐}){⟨“𝑐𝑣𝑑”⟩}) = ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)))
33		fusgrusgr 27592	. . . . . . 7 ⊢ (𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph)
34	1	hashnbusgrvd 27798	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
35	33, 34	sylan 579	. . . . . 6 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
36		id 22	. . . . . . 7 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → (♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣))
37		oveq1 7262	. . . . . . 7 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) − 1) = (((VtxDeg‘𝐺)‘𝑣) − 1))
38	36, 37	oveq12d 7273	. . . . . 6 ⊢ ((♯‘(𝐺 NeighbVtx 𝑣)) = ((VtxDeg‘𝐺)‘𝑣) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)))
39	35, 38	syl 17	. . . . 5 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → ((♯‘(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 7262	. . . . . 6 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) − 1) = (𝐾 − 1))
42	40, 41	oveq12d 7273	. . . . 5 ⊢ (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (((VtxDeg‘𝐺)‘𝑣) · (((VtxDeg‘𝐺)‘𝑣) − 1)) = (𝐾 · (𝐾 − 1)))
43	39, 42	sylan9eq 2799	. . . 4 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → ((♯‘(𝐺 NeighbVtx 𝑣)) · ((♯‘(𝐺 NeighbVtx 𝑣)) − 1)) = (𝐾 · (𝐾 − 1)))
44	5, 32, 43	3eqtrd 2782	. . 3 ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) ∧ ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))
45	44	ex 412	. 2 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑣 ∈ 𝑉) → (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))
46	45	ralrimiva 3107	1 ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  {csn 4558  ∪ ciun 4921  Disj wdisj 5035   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  1c1 10803   · cmul 10807   − cmin 11135  2c2 11958  ♯chash 13972  Word cword 14145  ⟨“cs3 14483  Vtxcvtx 27269  USGraphcusgr 27422  FinUSGraphcfusgr 27586   NeighbVtx cnbgr 27602  VtxDegcvtxdg 27735   WSPathsN cwwspthsn 28094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-oi 9199  df-dju 9590  df-card 9628  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-rp 12660  df-xadd 12778  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-vtx 27271  df-iedg 27272  df-edg 27321  df-uhgr 27331  df-ushgr 27332  df-upgr 27355  df-umgr 27356  df-uspgr 27423  df-usgr 27424  df-fusgr 27587  df-nbgr 27603  df-vtxdg 27736  df-wlks 27869  df-wlkson 27870  df-trls 27962  df-trlson 27963  df-pths 27985  df-spths 27986  df-pthson 27987  df-spthson 27988  df-wwlks 28096  df-wwlksn 28097  df-wwlksnon 28098  df-wspthsn 28099  df-wspthsnon 28100

This theorem is referenced by:  fusgreghash2wsp  28603